Continuous symmetries and approximate quantum error correction

TL;DR

This paper explores the relationship between continuous symmetries and quantum error correction, deriving bounds on error infidelity, constructing covariant codes, and providing insights into holographic duality and fault-tolerant quantum computation.

Contribution

It introduces bounds on error correction for covariant codes, constructs new codes with symmetry properties, and connects these ideas to holography and the Eastin-Knill theorem.

Findings

Lower bounds on infidelity decrease with more subsystems or larger local dimension.

Codes can achieve near-optimal scaling of error correction performance.

The approach offers new insights into holographic duality and fault-tolerant quantum computing.

Abstract

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is covariant with respect to a symmetry group $G$ if a $G$ transformation on the logical system can be realized by performing transformations on the individual subsystems. For a $G$-covariant code with $G$ a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems $n$ or the dimension $d$ of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with $n$ or $d$ as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large $d$ (using random codes) or $n$ (using codes based on $W$-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

Figures

Tables

Table 1. Table 1: Summary of the codes considered in VI : thethree-rotor secret-sharing code, the five-rotor perfect code, and an n 𝑛 n -qubit“thermodynamic code” with codewords consisting of Dicke states (and a , b 𝑎 𝑏 a,b chosen appropriately).

	Covariance	Dimen.	Error correction
${\left[[3,1,2]\right]}_{\mathbb{Z}}$
sharp cutoff	$U\left(1\right)$	Finite	Approximate
smooth cutoff	$U\left(1\right)$	Infinite	Approximate
${\left[[5,1,3]\right]}_{\mathbb{Z}}$
qudit version	${\mathbb{Z}}_D$	Finite	Exact
smooth cutoff	$U\left(1\right)$	Infinite	Approximate
$\left[[n,a\log n,b\log n]\right]$
finite $n$	$U\left(1\right)$	Finite	Approximate

Equations

$${ϵ}_{\mathrm{worst}}=\sqrt{1-f_{\mathrm{worst}}^2}.$$

$${ϵ}_{\mathrm{worst}}$$

$${ϵ}_{\mathrm{worst}}⩾\frac{1}{2n{\max }_i\ln d_i}+O\left(\frac{1}{n{d}_L}\right),$$

$$F_{\mathrm{e}}^2(\mathcal{K})$$

$$F_{\mathrm{worst}}^2(\mathcal{K})$$

$$F_{|\varphi ⟩}^2(\mathcal{K},{\mathcal{K}}^{\prime })$$

$$F_{\mathrm{e}}(\mathcal{K})=F_{|\hat{\varphi }⟩}(\mathcal{K},\mathrm{id}),F_{\mathrm{worst}}(\mathcal{K})=\underset{|\varphi ⟩}{\min }{F}_{|\varphi ⟩}(\mathcal{K},\mathrm{id}).$$

$$f_{\mathrm{e}}(\mathcal{N}\circ \mathcal{E})$$

$$f_{\mathrm{worst}}(\mathcal{N}\circ \mathcal{E})$$

$${ϵ}_{\mathrm{e}}(\mathcal{N}\circ \mathcal{E})$$

$${ϵ}_{\mathrm{worst}}(\mathcal{N}\circ \mathcal{E})$$

$${\mathcal{N}}_{A\to AC}^{(1)}(\cdot )=\sum q_i|i⟩⟨i{|}_C\otimes |{\varphi }_i⟩⟨{\varphi }_i{|}_{A_i}\otimes {\mathrm{tr}}_{A_i}(\cdot ),$$

$${\mathcal{N}}_{A\to AC}(\cdot )$$

$${\mathcal{N}}_{A\to A}^{\alpha }(\cdot )$$

$$f_{\mathrm{e}}(\mathcal{N}\circ \mathcal{E})$$

$$f_{\mathrm{worst}}(\mathcal{N}\circ \mathcal{E})$$

$$\begin{gathered} W_{A\to AC{C}^{\prime }E}= \\ \sum \sqrt{q_i}|i{⟩}_C\otimes |i{⟩}_{C^{\prime }}\otimes |{\varphi }_i{⟩}_{A_i}\otimes \text{mathds}{1}_{A_i\to E}\otimes \text{mathds}{1}_{A\setminus A_i}, \end{gathered}$$

$${\widehat{\mathcal{N}\circ \mathcal{E}}}_{L\to C^{\prime }EF}(\cdot )$$

$$=\sum q_i|i⟩⟨i{|}_{C^{\prime }}\otimes {\mathrm{tr}}_{A\setminus A_i}(V(\cdot )V^{†}),$$

$${\widehat{\mathcal{N}\circ \mathcal{E}}}_{L\to C^{\prime }EF}(\cdot )$$

$${\mathcal{E}}_{L\to A}\text{bBigg@}1.2(U_L(g)(\cdot )U_L^{†}(g)\text{bBigg@}1.2)=U_A(g){\mathcal{E}}_{L\to A}(\cdot )U_A^{†}(g).$$

$$U_L(g)$$

$$V[T_L,(\cdot )]V^{†}=[T_A,V\left(\cdot \right)V^{†}].$$

$$(t-t^{\prime })|{\psi }_{t,j}⟩⟨{\psi }_{t^{\prime },j^{\prime }}|=[T_A,|{\psi }_{t,j}⟩⟨{\psi }_{t^{\prime },j^{\prime }}|],$$

$$V^{†}{T}_{A}V=T_L-\nu \text{mathds}{1}_L.$$

$$T_{A}V=V(T_L-\nu \text{mathds}1).$$

$$\mathrm{tr}(T_L|x⟩⟨x{|}_L)=\mathrm{tr}(T_A|{\psi }_x⟩⟨{\psi }_x{|}_A)=\sum _i\mathrm{tr}(T_i{\rho }_i^x),$$

$$\mathrm{tr}(T_L|x⟩⟨x{|}_L)=\mathrm{tr}(Z_{C^{\prime }E}\widehat{\mathcal{N}\circ \mathcal{E}}(|x⟩⟨x{|}_L)),$$

$$\delta \text{bBigg@}1.5(\widehat{\mathcal{N}\circ \mathcal{E}}(|{\varphi }_{-}⟩⟨{\varphi }_{-}{|}_L),\widehat{\mathcal{N}\circ \mathcal{E}}(|{\varphi }_{+}⟩⟨{\varphi }_{+}{|}_L)\text{bBigg@}1.5)⩾\frac{\Delta T_L}{2\Vert Z_{C^{\prime }E}{\Vert }_{\infty }},$$

$$\displaystyle\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})\geqslant\frac{1}{2}\delta\Bigl{(}\widehat{\mathcal{N}\circ\mathcal{E}}(\lvert{x}\rangle\langle{x}\rvert_{L}),\widehat{\mathcal{N}\circ\mathcal{E}}(\lvert{x^{\prime}}\rangle\langle{x^{\prime}}\rvert_{L})\Bigr{)}\ .$$

$${ϵ}_{\mathrm{worst}}(\mathcal{N}\circ \mathcal{E})⩾\frac{1}{2n}\frac{\Delta T_L}{{\max }_i\Delta T_i}.$$

$${ϵ}_{\mathrm{e}}(\mathcal{N}\circ \mathcal{E})⩾\frac{1}{n}\frac{\Vert T_L-\mathrm{tr}(T_L)\text{mathds}{1}_L/d_L{\Vert }_1/(2d_L)}{{\max }_i\Delta T_i},$$

$$\left|\mathrm{tr}\left(\sum (T_{\alpha }-t_{\alpha }){\Pi }_{\alpha }^{\perp }\mathcal{E}({\sigma }_L)\right)\right|⩽\eta ,$$

Full text

I Introduction

Quantum error-correcting codes protect fragile quantum states againstnoise Nielsen and Chuang (2000).If quantum information is cleverly encoded in a highly entangled state of many physical subsystems, then damage inflicted by local interactions with the environment can be reversed by a suitable recovery operation.Aside from their applications to resilient quantum computing, quantumerror-correcting codes appear in a wide variety of physical settings where quantum states are delocalized over many subsystems, such astopological phases ofmatter Kitaev (2003); Dennis et al. (2002); Nayak et al. (2008); Zeng et al. (2015) and the AdS/CFTcorrespondence in holographic quantumgravity Almheiri et al. (2015); Pastawski et al. (2015).On the other hand, naturally occurring physical systems often respect symmetries, and phases of matter can be classified according to how these symmetries are realized in equilibrium states.Likewise, quantum error-correcting codes often have approximate or exact symmetries with important implications. In the case of a time-translation-invariant many-body system, for example,certain energy subspaces are known to formapproximate quantum error-correcting codes Brandão et al. (2017); Gschwendtner et al. (2019), which are preserved under time evolution. Limits to sensitivity in quantum metrology are related to the degree of asymmetry of probe states, a notion formalized in the resource theory of asymmetry andreference frames Bartlett et al. (2007); Marvian and Spekkens (2014). Thus, reference frame information can be protected against noise using quantum codes withsuitable symmetry properties Hayden et al. (2017).Furthermore, recent developments in quantum gravity have shown that theAdS/CFT correspondence can be viewed as a quantum error-correcting code which isexpected to be compatible with the natural physical symmetries of the system,such as time-translation invariance Harlow (2017); Susskind (2018); Pastawski et al. (2015); Harlow and Ooguri (2018a, b).Finally, the Eastin-Knill theorem Eastin and Knill (2009); Zeng et al. (2011); Chen et al. (2008), which complicates theconstruction of fault-tolerant schemes for quantum computation by forbiddingquantum error-correcting codes from admitting a universal set of transversalgates, can be viewed as the statement that finite-dimensional quantum codeswhich correct erasure have no continuoussymmetries Hayden et al. (2017). Thus, there are loopholes to theEastin-Knill theorem that are naturally exploited by holographic theories ofquantum gravity. This article provides a detailed quantitative investigation ofthose loopholes, critically evaluating their potential for application toquantum fault-tolerance.A continuous symmetry, as opposed to a discrete symmetry, allows forinfinitesimally small transformations that are arbitrarily close to the identityoperation. Such symmetry transformations are generated by conserved operators called charges.For instance, consider a particle in three-dimensional space that we rotateabout the $Z$-axis by an angle $\theta$. Acting on the Hilbert space, thissymmetry transformation is represented by a unitary $U_{\theta}$ that is generatedby the $Z$-component of the Hermitian angular momentum operator $J_{z}$, i.e.,$U_{\theta}=e^{-iJ_{z}\theta}$. Crucially, a unitary operation $U$ that iscovariant with respect to rotations about the $Z$ axis must conserve thephysical quantity $J_{z}$. In particular, if the initial state $\lvert{\psi}\rangle$ is aneigenstate of $J_{z}$ with eigenvalue $m$, then the transformed state $U\lvert{\psi}\rangle$must also be an eigenstate of $J_{z}$ with the same eigenvalue (up to a constantshift in all the eigenvalues).

Here, we study the accuracy of quantum error-correcting codes that are covariant withrespect to continuous symmetries (Fig. 1). Our resultsbuild on earlier work showing that infinite-dimensional covariant quantum codes exist, while finite-dimensional covariant codes cannot correct erasure errors perfectly Preskill (2000); Hayden et al. (2017).A finite-dimensional error-correcting code that is covariant with respect to a continuoussymmetry cannot correct erasure of a subsystem exactly, because an adversary who steals the erased subsystem could acquire some information about the encoded state, hence driving irreversible decoherence of the logical quantum information Preskill (2000); Hayden et al. (2017).More concretely, if $\Pi$ is the projector onto the code space, then theerror-correctionconditions Knill and Laflamme (1997); Bennett et al. (1996) statethat any operator $O$ supported on the erased subsystem must act trivially within the codespace, i.e.,$\Pi O\Pi\propto\Pi$. If the symmetry acts transversally, the correspondinggenerator $T_{A}$ is a sum of strictly local terms, $T_{A}=\sum T_{i}$, where each $T_{i}$ is supported on a single subsystem. However, thisimplies that $\Pi T_{A}\Pi=\sum\Pi T_{i}\Pi\propto\Pi$, and hence it follows from the error-correction condition that any such$T_{A}$ must act trivially on thecodewords.Crucially for the considerations in this paper, the above argument makes twoimplicit assumptions: that the sum over $i$ is finite (bounded number ofsubsystems), and that the codewords are normalizable (finite-dimensionalsubsystems). If both assumptions are relaxed, then quantum codes covariant withrespect to a continuous symmetry are possible, as shownin Hayden et al. (2017).Our main task in this paper is to explore quantitatively the case where the number of subsystems and the dimension of each subsystem are finite, using the tools of approximate quantumerror correction Leung et al. (1997); Crépeau et al. (2005); Bény and Oreshkov (2010). That is, we will quantify the deviation from perfect correctability in this case, for a code covariant with respect to a continuous symmetry. Assuming that the symmetry acts transversally and thatthe noise acts by erasing one or more subsystems, we provide upper bounds on the accuracyof the code, characterized using either the average entanglement fidelity or theworst-case entanglement fidelity of the error-corrected state. Our proof strategy is to show that in the presence of a continuous symmetry, the environment necessarily learns someinformation about the logical charge, which implies that the code necessarilyperforms imperfectly as an error-correcting code Hayden et al. (2008); Bény and Oreshkov (2010); Bény et al. (2018).In fact, some of these assumptions may be relaxed in our main technical theorem; forinstance, the generating charge may be a sum of $k$-local terms, instead of a sum of strictly local terms as for a transversal symmetry action, and the code only needs to beapproximately rather than exactly covariant.Our lower bound on infidelity vanishes in two interesting regimes: as the dimension $d$ of the physicalsubsystems gets large, or as the number $n$ of physical subsystems gets large. In these limits we can find error-correcting codes whose infidelity approximately matches thescaling of our bound with $d$ or $n$. We construct explicit examples based on normalizedversions of the rotor code presented in Ref. Hayden et al. (2017), andnote that codes considered in Ref. Brandão et al. (2017) providefurther examples. We also discuss a 5-rotor code that can be stacked to construct acovariant holographic code Pastawski et al. (2015).Furthermore, our results provide an approximate version of theEastin-Knill theorem Eastin and Knill (2009); Zeng et al. (2011); Chen2008; Hayden et al. (2017), which statesthat a universal set of transversal logical gates cannot exist for afinite-dimensional encoding that protects perfectly against erasure.By applying our bounds and exploiting the nonabelian nature of the full unitarygroup on the logical space, we derive a lower bound on infidelity which scalesas $1/\log d$, where $d$ is the subsystem dimension, for a code that admitsuniversal transversal logical gates. We also find that if a code admits auniversal set of transversal logical gates, then there are strong lower boundson the subsystem dimension $d$ that depend on the code’s infidelity, and whichin some regimes are even exponential in the logical system dimension $d_{L}$.Using randomized code constructions, we prove the existence of codes which approximately achieve this relationship between $d$ and $d_{L}$. In addition, we exhibit codes with universal transversal logical gates which achieve arbitrarily small infidelity when the number $n$ of subsystems becomes large with the logical dimension $d_{L}$ fixed.

We also provide a general framework for constructing codes that are covariant withrespect to general symmetry groups, by encoding logical information into theso-called regular representation of the groups. Using this frameworkwe can generalize several widely-known codes (bit-flip, phase-flip, $[[4,2,2]]$code, etc.) to infinite-dimensional covariant codes based on oscillators or rotors.Finally, we discuss the interpretation of our results in the context of quantum gravity and, in particular, the AdS/CFT correspondence. Time evolution itself provides an example of a symmetry that must be reconciled with the error-correcting properties of the system.The remainder of the manuscript is organized as follows(Fig. 2). In II, we summarize our mainresults. We set up notation in III and prove a bound on theperformance of codes covariant with respect to a $\mathit{U}(1)$ symmetry inIV. A criterion certifying code performance is derived inV. In VI, we apply our bounds andcriterion to the following examples of $\mathit{U}(1)$-covariant encodings: aninfinite-dimensional rotor extension of the qutrit $[[3,1,2]]$ and qubit$[[5,1,3]]$ codes as well as a many-body Dicke-state code. We apply our bound tocodes admitting universal transversal gates in VII, discussing a$\mathit{U}(d)$-invariant encoding based on $W$-states inVII.2. Erasure-correcting codes whose transversal gates form ageneral group $G$ are introduced in VIII. InIX we study applications to quantum gravity. We concludewith a discussion in X.

II Summary of main results

II.1 Bound on the accuracy of codes covariant with respect to acontinuous symmetry

Our first main result is a bound on the accuracy of any approximate quantumerror-correcting code that is covariant with respect to a continuous symmetry. We consider an encoding map from a logical system $L$ to a physical system $A$ consisting of $n$ subsystems denoted $A_{1},A_{2},\dots A_{n}$. A one-parameter family of continuous unitary symmetries acting on $L$ is generated by the logical charge observable $T_{L}$, which corresponds to the physical charge observable $T_{A}$ acting on $A$. We assume that the symmetry acts transversally, so that $T_{A}=\sum_{i=1}^{n}T_{i}$, where $T_{i}$ acts on subsystem $T_{i}$.How well does this code protect the logical system against erasure of one of thesubsystems? To quantify the code’s performance we may use the worst-caseentanglement fidelity, where “worst-case” means the minimal fidelity forany entangled state shared by the logical system and a reference system. (SeeIII for a precise definition.) Then we consider the value$f_{\mathrm{worst}}$ of this worst-case entanglement fidelity which is achievedby the best possible recovery map applied after an erasure error. A measure ofthe residual error after recovery is

[TABLE]

Our result is a lower bound on $\epsilon_{\mathrm{worst}}$ which limits theperformance of any covariant quantum code:

[TABLE]

where $\Delta T$ denotes the difference between the maximal and minimaleigenvalue of $T$. That is, the code’s accuracy is constrained by the range ofcharges one wishes to be able to encode, by the size of the charge fluctuationswithin each subsystem, and by the number of physical subsystems.We also find that \tagform@2 can be generalized in a number of ways. We can express the limit on code performance in terms of other measures besides worst-case entanglement fidelity, such as average entanglement fidelity, or the entanglement fidelity of a fixed input state. We can derive bounds that apply in the case where more than one subsystem is erased, or where the erasure occurs for an unknown subsystem rather than a known subsystem. We can consider cases where the charge distribution for a subsystem has infinite range, but with a normalizable tail. We can also treat the case where the covariance of the code is approximate, or where the physical charge operator is not strictly transversal.

II.2 Regimes where our bound is circumvented and criterion for code performance

The idea underlying \tagform@2 is that for erasurecorrection to work well one should not be able to learn much about the globalvalue of the charge by performing a local measurement on a subsystem. Hence, tobe able to correct the errors to good accuracy, we need either large localcharge fluctuations ($\Delta T_{i}\to\infty$), or many subsystems ($n\to\infty$)so that the global charge is a sum of many local contributions. In fact, codescan be constructed in either limit for which $\epsilon_{\mathrm{worst}}$approximately matches the scaling in $\Delta T_{i}$ and $n$ of the lowerbound \tagform@2.To study the case of large $\Delta T_{i}$, we consider a normalized variant of theinfinite-dimensional covariant code constructedin Hayden et al. (2017). The infinite-dimensional version encodes onelogical rotor (with unbounded $U(1)$ charge) in a code block of three rotors. Inthe modified version of this code, we either truncate the charge of the logicalsystem to $\{-h,-h+1,\ldots,+h\}$ or use a Gaussian envelope of width $w$ tonormalize the physical codewords. The value of $\epsilon_{\mathrm{worst}}$achieved by this code, and our lower bound, both scale like $h/w$ up to alogarithmic factor.Regarding the limit of a large number of subsystems, we observe that a codediscussed in Ref. Brandão et al. (2017) matches the $1/n$ scalingof our lower bound on $\epsilon_{\mathrm{worst}}$. Here the subsystems arequbits, regarded as spin-$1/2$ particles, and the code space is two-dimensional,spanned by two Dicke states with different values of the total angular momentum$J_{z}$ along the $z$-axis. (A Dicke state is a symmetrized superposition of allbasis states with a specified $J_{z}$). This code is covariant with respect to$z$-axis rotations by construction, and can be shown to achieve$\epsilon_{\mathrm{worst}}$ scaling like $1/n$, where $n$ is the number ofphysical qubits.A further result of independent interest is a general criterion used in ouranalysis for certifying the performance of an error-correcting code againstarbitrary noise. Stated informally, this criterion asserts that if the reduceddensity operator on each subsystem is approximately the same for all codewords,and if the environment does not get any information from the off-diagonal termsin the logical density operator, then the code performs well. While thiscriterion is sufficient to certify the performance of an approximateerror-correcting code, it is not necessary—there may be codes achieving small$\epsilon_{\mathrm{worst}}$ that do not satisfy it.

II.3 Approximate Eastin-Knill theorem and random $\mathit{U}(d)$-covariant codes

Quantum error-correcting codes are essential for realizing scalable quantum computing using realistic noisy physical gates. In a fault-tolerant quantum computation, logical quantum gates are applied to encoded quantum data, and error recovery is performed repeatedly to prevent errors due to faulty gates from accumulating and producing uncorrectable errors at the logical level. For this purpose, transversal logical gates are especially convenient. For example, if a logical gate on an $n$-qubit code block can be achieved by applying $n$ single-qubit gates in parallel, then each faulty physical gate produces only a single error in the code block. Nontransversal logical gates, on other hand, either require substantially more computational overhead, or propagate errors more egregiously, allowing a single faulty gate to produce multiple errors in a code block.A nontrivial transversal logical gate can be regarded as a covariant symmetry operation acting on the code. If all the logical gates in a complete universal gate set could be chosen to be transversal, then the Lie group of transversal logical gates would coincide with the group $\mathit{U}(d_{L})$ of unitary gates acting on the $d_{L}$-dimensional logical system (up to an irrelevant overall phase). It then follows that any generator $T_{A}$ of $\mathit{U}(d_{L})$ acting on the physical system $A$ could be expressed as a sum of terms, where each term in the sum has support on a single subsystem.Unfortunately, the Eastin-Knill theorem rules out this appealing scenario, if erasure of each subsystem is correctable and the code is finite-dimensional. But now that we have seen that there are parameter regimes in which covariance can be compatible with good performance of approximate quantum error-correcting codes, one wonders whether a universal transversal logical gate set is possible after all, at the cost of a small but nonzero value of $\epsilon_{\mathrm{worst}}$.We have found, however, that a fully $\mathit{U}(d_{L})$-covariant code requires a value of $\epsilon_{\mathrm{worst}}$ which scales quite unfavorably with the local subsystem dimension.Leveraging tools from representation theory, we show that the lower bound on$\epsilon_{\mathrm{worst}}$ becomes

[TABLE]

where $d_{i}$ is the dimension of the $i$th physical subsystem.We also find lower bounds for the local subsystem dimension that depend on thenumber of logical qubits and the code’s infidelity. This result also applies tothe case when each gate can be approximated with a discrete sequence oftransversal operations to arbitrary accuracy, as in the context of theSolovay-Kitaev theorem.Furthermore, using randomized constructions, we prove the existence of$\mathit{U}(d_{L})$-covariant code families which achieve arbitrarily small infidelity inthe limit of large subsystem dimension. In addition, we exhibit a simple$\mathit{U}(d_{L})$-covariant code family, whose codewords are generalized $W$-states,such that $\epsilon_{\mathrm{worst}}$ approaches zero as the number ofsubsystems $n$ approaches infinity.

II.4 Framework for constructing covariant codes

We also develop a general framework for constructing codes that are covariantwith respect to any group $G$ admitting a Haar measure, where both the logicalsystem and the physical subsystems transform as the regular representation of$G$. In this construction, the dimension of each subsystem is the order $\lvert{G}\rvert$ ofthe group when $G$ is finite, and infinite when $G$ is a Lie group.Using this formalism we construct natural generalizations of well-known familiesof qubit codes, such as the bit-flip and phase-flip codes, with the qubitsreplaced by $\lvert{G}\rvert$-dimensional systems. These codes admit transversal logicalgates representing each element of $G$.When $G$ is a Lie group, the qubits are replaced by infinite-dimensional systemssuch as rotors or oscillators. These infinite-dimensional codes circumvent theEastin-Knill theorem—they are covariant with respect to a continuous symmetrygroup, yet erasure of a subsystem is perfectly correctable.

III Setup & notation

III.1 Approximate error correction

Consider a code, which to each logical state $\lvert{x}\rangle_{L}$ on some abstractlogical system $L$ associates a state $\lvert{\psi_{x}}\rangle_{A_{1}A_{2}\ldots A_{n}}$ on aphysical system $A$ consisting of $n$ subsystems$A=A_{1}\otimes A_{2}\otimes\cdots\otimes A_{n}$ (Fig. 3). The spanof all codewords $\{\lvert{\psi_{x}}\rangle_{A}\}$ forms the code subspace. Moregenerally, we denote by $\mathcal{E}_{L\to A}(\cdot)$ the encoding channel whichassociates to any logical state the corresponding encoded physical state. Inthis work, the encoding is usually an isometry, i.e., the encoding itself doesnot introduce noise into the system.

The noise channel is the process to which the physical system is exposed, whichmight cause the information encoded in it to get degraded. It is a quantumchannel $\mathcal{N}_{A\to B}$ mapping the physical system to physical system$B$. (The system $B$ might be the same as $A$, but it might be different; forinstance, $B$ might include a register which remembers which type of erroroccurred or which subsystem was lost.)To study the approximate error correction properties of a code, we need toquantify the approximation quality using distance measures between states andchannels.Proximity between quantum states can be quantified using the trace distance$\delta(\rho,\sigma)=\lVert{\rho-\sigma}\rVert_{1}/2$, or using the fidelity111Throughout this paper, we stick to the convention that the fidelity and itsderived quantities refer to an amplitude rather than a probability, i.e., weuse the convention of ref. Nielsen and Chuang (2000). In the literature,the quantity that we denote by $F^{2}$ is also referred to as “fidelity,”while the quantity we represent by $F$ is sometimes called “root fidelity.”$F(\rho,\sigma)=\lVert{\sqrt{\rho}\sqrt{\sigma}}\rVert_{1}$ Nielsen and Chuang (2000). We need toquantify how close a quantum channel $\mathcal{K}_{L\to L}$ is to the identitychannel. Two standard measures to achieve this are the average entanglementfidelity $F_{\mathrm{e}}$ and the worst-case entanglement fidelity$F_{\mathrm{worst}}$ Schumacher (1996); Gilchrist et al. (2005),defined as

[TABLE]

Here the input state appearing in the definition of $F_{\mathrm{e}}$ is$\lvert{\hat{\phi}}\rangle_{LR}=\sum_{k=0}^{d_{L}-1}\lvert{k}\rangle\otimes\lvert{k}\rangle/\sqrt{d_{L}}$,the maximally entangled state of $L$ and a reference system $R$; the system $R$has the same dimension as $L$, which we denote by $d_{L}$. The optimization in thedefinition of $F_{\mathrm{worst}}$ ranges over all bipartite states of $L$ and$R$. We may also use the state fidelity $F(\rho,\sigma)$ to compare twochannels $\mathcal{K}$ and $\mathcal{K^{\prime}}$; the entanglement fidelity between$\mathcal{K}$ and $\mathcal{K^{\prime}}$ for a fixed bipartite input state$\lvert{\phi}\rangle_{LR}$ is defined as

[TABLE]

thus

[TABLE]

By optimizing over the input state, we may define$F_{\mathrm{worst}}(\mathcal{K},\mathcal{K^{\prime}})$, which is closely related to thediamond distance between thechannels Schumacher (1996); Gilchrist et al. (2005).We now ask how well one can recover the logical state after theencoding and the application of the noise channel. That is, we seek acompletely positive map $\mathcal{R}_{B\to L}$ (the recovery map), suchthat $\mathcal{R}_{B\to L}\circ\mathcal{N}_{A\to B}\circ\mathcal{E}_{L\to A}$ isas close as possible to the identity channel $\operatorname{id}_{L\to L}$. Theresilience of a code $\mathcal{E}_{L\to A}$ to errors caused by a noise map$\mathcal{N}_{A\to B}$ is thus quantified by the proximity to the identitychannel of the combined process$\mathcal{R}_{B\to L}\circ\mathcal{N}_{A\to B}\circ\mathcal{E}_{L\to A}$ for thebest possible recovery map $\mathcal{R}_{B\to L}$.Using either the entanglement fidelity with fixed input $\lvert{\phi}\rangle_{LR}$ or theworst-case entanglement fidelity measures, the quality of the code$\mathcal{E}_{L\to A}$ under the noise $\mathcal{N}_{A\to B}$ is quantified as

[TABLE]

We will also find it convenient to work with the alternative quantities

[TABLE]

which are closely related to the infidelity and Bures distance measures. A codewhich performs well has $f\approx 1$ and correspondingly $\epsilon\approx 0$.

III.2 Erasures at known locations

In this work, we consider the noise model consisting of erasures which occur atknown locations. (Our bound then naturally applies also to erasures at unknownlocations, since the latter are necessarily harder to correct against.) Forinstance, if the $i$th physical subsystem is lost to the environment withprobability $q_{i}$, then the corresponding noise map is

[TABLE]

where we have introduced a classical register $C$ which records which one of the$n$ systems was lost, and where ${\lvert{\phi}\rangle_{i}}$ are some fixed states.One can also consider more general erasure scenarios, where any givencombination of subsystems can be lost with a given probability. For instance,one might assume that systems $A_{1}$ and $A_{2}$ are simultaneously lost withprobability $q_{\{1,2\}}$, systems $A_{2}$ and $A_{3}$ are simultaneously lost withprobability $q_{\{2,3\}}$, and systems $A_{1}$ and $A_{3}$ are lost with probability$q_{\{1,3\}}$. More generally, a combination of subsystems, which we labelgenerically by $\alpha$, can be lost with probability $q_{\alpha}$; we assume weknow exactly which systems were lost. The corresponding general noise map isthen

[TABLE]

where the register $C$ encodes the exact locations at which simultaneouserasures have occurred, where $A_{\alpha}$ denotes the physical systems labeled by$\alpha$ (for instance, if $\alpha=\{2,3\}$ then $A_{\alpha}=A_{2}\otimes A_{3}$),and where $\{\lvert{\phi}\rangle_{\alpha}\}$ are fixed states. The sum ranges overa set $K$ of possible $\alpha$’s corresponding to erasures which may occur.Technically, $K$ is any set of subsets of $\{1,2,\ldots,n\}$. Situationswhich can be described using this setting include for instance any $k$consecutive erasures, or the erasure of any $k$ subsystems.

III.3 Characterization via the environment

A very useful characterization of the quantities \tagform@7 isprovided by Bény and Oreshkov Bény and Oreshkov (2010), building upon thedecoupling approach to error correction Hayden et al. (2008). Therecoverability of the logical information can be characterized by studying howmuch information is leaked to the environment, as represented by a complementarychannel $\widehat{\mathcal{N}\circ\mathcal{E}}$ of$\mathcal{N}\circ\mathcal{E}$. Recall that a complementary channel$\hat{\mathcal{F}}_{A\to C}$ of a quantum channel $\mathcal{F}_{A\to B}$ is achannel of the form$\hat{\mathcal{F}}_{A\to C}(\cdot)=\operatorname{tr}_{B}(W_{A\to BC}(\cdot)W^{\dagger})$, where$W_{A\to BC}$ is a Stinespring dilation isometry for the map $\mathcal{F}$,i.e., $\mathcal{F}_{A\to B}(\cdot)=\operatorname{tr}_{C}(W_{A\to BC}(\cdot)W^{\dagger})$.Bény and Oreshkov show that the fidelity with which one can reverse theaction of the encoding and the noise is exactly the fidelity of the totalcomplementary channel to a constant channel:

[TABLE]

where $\mathcal{T}_{\zeta}(\cdot)=\operatorname{tr}(\cdot)\,\zeta$ is the constant channeloutputting the state $\zeta$ and where the maximizations range over all quantumstates $\zeta$ on the output system of $\widehat{\mathcal{N}\circ\mathcal{E}}$.Now we determine a complementary channel $\widehat{\mathcal{N}\circ\mathcal{E}}$to the encoding and noise channels. Consider first the single-erasure noisechannel \tagform@9. A Stinespring dilation of$\mathcal{N}^{(1)}_{A\to AC}$ on two additional systems $C^{\prime}\otimes E$ is givenas $\mathcal{N}^{(1)}_{A\to AC}=\operatorname{tr}_{C^{\prime}E}(W(\cdot)W^{\dagger})$, with

[TABLE]

where $\mathds{1}_{A_{i}\to E}$ is an isometric embedding of $A_{i}$ into $E$ and$\mathds{1}_{A\setminus A_{i}}$ is the identity operator on all systems $A$ except$A_{i}$. Now consider a Stinespring dilation of $\mathcal{E}_{L\to A}$ as$\mathcal{E}_{L\to A}=\operatorname{tr}_{F}(V_{L\to AF}\,(\cdot)\,V^{\dagger})$. Then, wemay take

[TABLE]

where $\operatorname{tr}_{A\setminus A_{i}}$ denotes the partial trace over all systems except$A_{i}$ (the latter is then embedded in the $E$ system).Hence, the complementary channel to the single erasure channel simply gives theerased information to the environment with the corresponding erasureprobability. It is straightforward to see that for the more general noisechannel \tagform@10 a complementary channel is given by

[TABLE]

where the register $C^{\prime}$ now remembers which combination of systems were lost.This channel provides the environment with the systems that were erased, whereeach erasure combination $\alpha$ appears with probability $q_{\alpha}$.

III.4 Covariant codes

The final ingredient we introduce is covariance with respect to a symmetry group(Fig. 1). Let $G$ be any Lie group acting unitarily onthe logical and physical systems, with representing unitaries $U_{L}(g)$ and$U_{A}(g)$, respectively, for any $g\in G$. A code $\mathcal{E}_{L\to A}$ iscovariant if it commutes with the group action:

[TABLE]

On either logical and physical systems, we can expand the unitary action of $G$in terms of generators of the corresponding Lie algebra, i.e., for a given $g$there is a generator $T_{L}$ on $L$ and a generator $T_{A}$ on $A$ such that

[TABLE]

for some $\theta\in\mathbb{R}$ that we can choose to normalize our generators.The generators are Hermitian matrices, and they can be interpreted as physicalobservables. (For instance, the generators of the rotations in 3-D space arethe angular momenta.)If the encoding map is isometric,$\mathcal{E}_{L\to A}(\cdot)=V_{L\to A}\,(\cdot)\,V^{\dagger}$, then anyeigenstate $\lvert{t}\rangle$ of $T_{L}$ with eigenvalue $t$ must necessarily be encodedinto an eigenstate of $T_{A}$ with the same eigenvalue $t$ (up to a constantoffset). This can be seen as follows. Expanding thecondition \tagform@17 for small $\theta$ yields

[TABLE]

Let $\{\lvert{t,j}\rangle_{L}\}$ be a basis of eigenstates of $T_{L}$ where $t$ is the eigenvalue andwhere $j$ is a degeneracy index. Inserting in the place of $(\cdot)$ the operator$\lvert{t,j}\rangle\langle{t^{\prime},j^{\prime}}\rvert$, we obtain

[TABLE]

where $\lvert{\psi_{t,j}}\rangle=V\lvert{t,j}\rangle$. Setting $t=t^{\prime},j=j^{\prime}$, we see that$\lvert{\psi_{t,j}}\rangle$ is necessarily an eigenstate of $T_{A}$; let $u_{t,j}$ be itscorresponding eigenvalue. Setting $t=t^{\prime},j\neq j^{\prime}$ in\tagform@20 implies$0=(u_{t,j}-u_{t,j^{\prime}})\lvert{\psi_{t,j}}\rangle\langle{\psi_{t,j^{\prime}}}\rvert$ and hence$u_{t,j}=u_{t,j^{\prime}}=:u_{t}$. Now\tagform@20 tells us for any $t,t^{\prime},j,j^{\prime}$that $t-t^{\prime}=u_{t}-u_{t^{\prime}}$. It follows that $u_{t}=t-\nu$ for all $t$, for someconstant offset $\nu$; in other words, the codewords must have the same charge as thelogical state, except for a possible constant offset $\nu$. We may condense thiscondition into the constraint $[T_{A},VV^{\dagger}]=0$ along with the identity

[TABLE]

Equivalently, acting with $V$ on\tagform@21 we have

[TABLE]

This is a crucial property of covariant codes, and is a central ingredient ofthe proof of our main result.Our main result, in its simplified form, further assumes that the action of thegroup is transversal on the physical systems, meaning that$U_{A}(g)=U_{1}(g)\otimes U_{2}(g)\otimes\cdots U_{n}(g)$. In this case, thecorresponding generator is strictly local, $T_{A}=T_{1}+T_{2}+\cdots+T_{n}$,where each of the $T_{i}$’s act only on$A_{i}$.As opposed to covariant isometries, covariant channels in general do notconserve charge since they may exchange charge with the environment. Forinstance, the fully depolarizing channel is covariant with respect to anysymmetry but it changes the charge of its input. Our main result in its fullygeneral form is formulated for approximately charge-conserving channelencodings, which is a superset of covariant isometries.

IV Inaccuracy of covariant codes for a continuous symmetry

Our first main result is a general characterization of how poorly a codenecessarily performs against erasures at known locations, given that the codemust be covariant with respect to a continuous symmetry.For the sake of clarity, we first present a simplified version of our generalbound. Consider an encoding map $\lvert{x}\rangle_{L}\to\lvert{\psi_{x}}\rangle_{A}$ with respect tosome basis $\{\lvert{x}\rangle_{L}\}$, which we may represent by an isometry$V_{L\to A}=\sum_{x}\lvert{\psi_{x}}\rangle_{A}\langle{x}\rvert_{L}$. Denote the correspondingencoding channel by $\mathcal{E}_{L\to A}(\cdot)=V(\cdot)V^{\dagger}$.Pick any generator $T_{L}$ from the Lie algebra of the symmetry acting on $L$.Let $T_{i}$ be the corresponding generator acting on the $i$th physicalsubsystem $A_{i}$, with the total generator on $A$ being $T_{A}=\sum_{i}T_{i}$. AsHermitian matrices, these are quantum mechanical observables whose eigenvalueswe may think of as abstract “charges.” (These charges mightcorrespond to the component of angular momentum in a given direction, the number of particles, orsome other physical quantity.) Crucially, since the code $\mathcal{E}$ iscovariant, a logical charge eigenstate $\lvert{t}\rangle_{L}$ must be encoded into acodeword $\lvert{\psi_{t}}\rangle_{A}$ which is an eigenvector of $T_{A}$ with the sameeigenvalue $t$, up to a constant offset $\nu$. Let us assume for simplicitythat $\nu=0$.Assume the environment erases a subsystem $i$ chosen at random with probability$q_{i}=1/n$. Then the environment gets the information represented by thecomplementary channel \tagform@15. That is, if theoriginal state was $\lvert{x}\rangle_{L}$, then the environment gets the state$\rho^{x}_{i}=\operatorname{tr}_{A\setminus A_{i}}(\lvert{\psi_{x}}\rangle\langle{\psi_{x}}\rvert_{L})$ on subsystem $i$ withprobability $1/n$.Yet, because the charge observable is local, the environment can learn theexpectation value of the charge. Indeed, for any $\lvert{x}\rangle_{L}$,

[TABLE]

where the first equality holds because the code is covariant, and the secondbecause the charge is local. Hence, if we define the observable$Z_{C^{\prime}E}=n\sum_{i}\lvert{i}\rangle\langle{i}\rvert_{C^{\prime}}\otimes T_{i}$ on the environment systems, wehave

[TABLE]

making it clear that the environment can measure the average charge using theinformation it has available.Surely, if the charge expectation value leaks to the environment, then the codemust be bad. However, the accuracy of the code is measured in terms of anentanglement fidelity (worst-case or fixed input) to the identity channel.Hence, it still remains to relate the accuracy of the code to the environment’sability to access the codeword’s total charge.On one hand, we observe that the difference in expectation value of $Z_{C^{\prime}E}$on the environment can be translated into a distinguishability of codewords interms of the trace distance. More precisely and in general, for any two states$\rho,\sigma$, if there is an observable $Q$ for which $\rho,\sigma$ havedifferent expectation values, then$\delta(\rho,\sigma)\geqslant\lvert{\operatorname{tr}(Q\rho)-\operatorname{tr}(Q\sigma)}\rvert/(2\lVert{Q}\rVert_{\infty})$. In our case, consider two logical chargeeigenstates $\lvert{\phi_{\pm}}\rangle_{L}$ corresponding to the maximal and minimaleigenvalues of $T_{L}$; then it holds that

[TABLE]

where $\Delta T_{L}$ is the spectral range of $T_{L}$, i.e., the difference betweenthe maximal and minimal eigenvalue of $T_{L}$. We assume here for simplicity thatthe maximal and minimal eigenvalues of $T_{i}$ are equal in magnitude, such that$\Delta T_{i}=2\lVert{T_{i}}\rVert_{\infty}$; hence$2\,\lVert{Z_{C^{\prime}E}}\rVert_{\infty}=2n\max_{i}\lVert{T_{i}}\rVert_{\infty}=n\max_{i}\Delta T_{i}$.On the other hand, if the environment’s states are distinguishable fordifferent codewords, then the accuracy of the code is bad; specifically, we showin the Appendix (Appendix G) that for any two logicalstates $\lvert{x}\rangle_{L},\lvert{x^{\prime}}\rangle_{L}$, we have

[TABLE]

Finally, we have proven our simplified main result.

Theorem 1**.**

The performance of the covariant code $\mathcal{E}(\cdot)=V(\cdot)V^{\dagger}$ under the aboveassumptions, quantified by the worst-case entanglement fidelity, is bounded asfollows:

[TABLE]

A similar analysis leads to a bound for the figure of merit$\epsilon_{\mathrm{e}}$ based on the average entanglement fidelity,

[TABLE]

The right hand side of \tagform@28 is simply a different measurefor the spread of eigenvalues; unlike $\Delta T_{L}$, it takes contributions fromall eigenvalues of $T_{L}$. The argument of the norm is simply the chargeoperator $T_{L}$ with a global shift that makes the operator traceless.Equation 28 is proven as a special case ofIV.In Appendix B, we provide an alternative proof for thebound \tagform@28 using a different approach: We quantify theinformation leaked to the environment by studying the connected correlationfunctions between the subsystems. In fact, we lower bound the sum of thecorrelation functions between the logical qubit and individual physicalsubsystems, and since this total correlation is non-zero, we deduce that theenvironment is correlated with the logical information, which translates to anupper bound on the fidelity of recovery.In short, a covariant code with respect to a local charge may not performwell for correcting a single erasure at a known location, unless it eitherencodes the information into large physical systems, with a large range ofpossible charge values ($\max_{i}\Delta T_{i}\to\infty$), or it encodes theinformation into many physical systems ($n\to\infty$).The following theorem generalizes IV in anumber of ways. It allows for the code to only approximately conserve charge,considers erasures affecting multiple systems with arbitrary erasureprobabilities, and does not require the charge to be strictly local; finally, itcan be applied in situations in which the codewords have most of their weight ona finite charge range (but may have distribution tails extending to arbitrarilylarge charge values). The setting of IV is depicted inFig. 4.

Theorem 2**.**

Let $L$ and $A=A_{1}\otimes\cdots\otimes A_{n}$ be the logical and physicalsystems, respectively, and let $\mathcal{E}_{L\to A}$ be any completelypositive, trace-preserving map. Consider logical and physical observables$T_{L}$ and $T_{A}$. We assume that:

1. [label=()]

There isa $\nu\in\mathbb{R}$ and a $\delta\geqslant 0$ such that$\lVert{(T_{L}-\nu\mathds{1}_{L})-\mathcal{E}^{\dagger}(T_{A})}\rVert_{\infty}\leqslant\delta$, i.e., the code is approximately charge-conserving up to a constantshift; 3. 2.

We can write $T_{A}=\sum_{\alpha}T_{\alpha}$, where each term$T_{\alpha}$* acts on a subset of physical systems labeled by $\alpha$;* 4. 3.

Fixing cut-offs $t_{\alpha}^{\pm}$ for each $\alpha$, there is$\eta\geqslant 0$* such that for any state $\sigma_{L}$, we have*

[TABLE]

where $\Pi_{\alpha}^{\perp}$ projects onto the eigenspaces of $T_{\alpha}$ whoseeigenvalues are outside $[{t_{\alpha}^{-}\mathclose{},\mathopen{}t_{\alpha}^{+}}]$, and where$t_{\alpha}=(t_{\alpha}^{-}+t_{\alpha}^{+})/2$. That is, when chopping off partsof the codewords exceeding charge $t_{\alpha}^{\pm}$ on term $\alpha$ andshifting the charge term to center it around zero, the total averagecharge chopped off does not exceed $\eta$; 5. 4.

The noise acts as per \tagform@10 by erasingsubsystems labeled by $\alpha$ with probability $q_{\alpha}>0$, for each$\alpha$* for which there is a corresponding term $T_{\alpha}$ in the globalgenerator $T_{A}$.*

Then the accuracy of the code $\mathcal{E}_{L\to A}$ against the noise$\mathcal{N}$ is bounded as

[TABLE]

where $\Delta T_{\alpha}=t_{\alpha}^{+}-t_{\alpha}^{-}$, where $d_{L}$ is thedimension of $L$, where $\langle{\cdot}\rangle_{\alpha}=\sum_{\alpha}q_{\alpha}(\cdot)$,and where $\mu(T_{L})$ is a median eigenvalue of $T_{L}$. We define a medianeigenvalue of $T_{L}$ to be a number $\mu$ such that the length-$d_{L}$ vector ofeigenvalues of $T_{L}$ counted with multiplicity has at least$\lceil d_{L}/2\rceil$ components that are less than or equal to $\mu$, and atleast $\lceil d_{L}/2\rceil$ components that are greater than or equal to $\mu$.Additionally, the first term in the numeratorof \tagform@30c may be replaced by$\lVert{T_{L}-\operatorname{tr}(T_{L})\mathds{1}/d_{L}}\rVert_{1}/(2d_{L})$.

The bound \tagform@30c is intuitively sensitive tothe “average amount of logical charge” in absolute value, up to an arbitrarycharge offset; this makes sense since the average entanglement fidelity “onlysamples the average case.” On the other hand, the worst-case entanglementfidelity picks up the worst possible situation, noticing that there are twostates with maximally different charges; thebound \tagform@30d reflects that the code willperform the worst for those input states.The median eigenvalue in \tagform@30c appears as anoptimal solution to the optimization $\min_{\mu}\lVert{T_{L}-\mu\mathds{1}_{L}}\rVert_{1}$.For an operator $T$ that has the same number of positive eigenvalues as negativeones (with multiplicity), such as a component of spin, we can set $\mu(T)=0$.For isometric encodings,condition 1 reallymeans that the encoding is approximately covariant. However our theorem holdsmore generally for encodings that are not an isometry, as long as theyapproximately conserve charge. The latter condition is stricter than beingcovariant. However, an approximately covariant channel encoding that does notapproximately preserve charge can still fit in the context ofIV, by explicitly considering instead its covariantStinespring dilation Scutaru (1979); Keyl and Werner (1999); Marvian Mashhad (2012); Faist et al. (2018) into anancilla system which is then erased by the environment with certainty as part ofthe noise channel.The proof of IV is provided inAppendix A. The proof is split into two parts. A first partshows that there exists an observable accessible to the environment which isable to infer the global logical charge to a good approximation. The secondpart deduces from the existence of such an observable that the code mustnecessarily have limited performance, as quantified by various entanglementfidelity measures.

V Criterion for certifying code performance

Here we introduce a criterion that allows us to certify a given encoding asperforming accurately as an approximate error-correcting code against any givennoise channel, as measured by the worst-case entanglement fidelity.Proving that a code has a good average-case entanglement fidelity (i.e., showingthat $\epsilon_{\mathrm{e}}(\mathcal{N}\circ\mathcal{E})$ is small) is perhapscomparatively easier, as one can attempt to guess a suitable recovery map for amaximally entangled input state and directly compute the fidelity of recovery.The method we present provides an upper bound to the stricter measure$\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})$ and does not require us tocome up with explicit recovery procedures.Intuitively, if we consider erasures at known locations, we can expect that ifall local reduced states of codewords look alike independently of the logicalinformation, then the code performs well. That is, if for each individualsubsystem each codeword has the same reduced state, then because the environmentgets access only to those individual reduced states, it obtains no informationabout the codeword and the erasure is thus correctable. This intuition iscorrect in the exact case, but in the approximate case the fact that theentanglement fidelity is defined with a “stabilization” over a referencesystem poses an additional challenge Bény (2017). Our solution is toconsider how logical operators of the form $\lvert{x}\rangle\langle{x^{\prime}}\rvert$ are encoded, where$\{\lvert{x}\rangle\}$ is any fixed basis of the logical system. In the case of asingle erasure at a known location, we define

[TABLE]

noting that $\rho_{i}^{x,x^{\prime}}$ is a quantum state if $x=x^{\prime}$ but is not evennecessarily Hermitian for $x\neq x^{\prime}$. Our criterion then states the following:If the states $\rho_{i}^{x,x}$ are approximately independent of $x$, and if each$\rho_{i}^{x,x^{\prime}}$ for $x\neq x^{\prime}$ has a very small norm, then the code is a good approximateerror-correcting code against erasure of subsystem $i$.

Theorem 3**.**

For any encoding channel $\mathcal{E}$ and for any noise channel$\mathcal{N}$, let $\widehat{\mathcal{N}\circ\mathcal{E}}$ be a complementarychannel of $\mathcal{N}\circ\mathcal{E}$.Fixing a basis of logical states $\{\lvert{x}\rangle\}$, we define

[TABLE]

Assume that there exists a state $\zeta$, as well as constants$\epsilon,\nu\geqslant 0$ such that

[TABLE]

Then, the code $\mathcal{E}$ is an approximate error-correcting code with anapproximation parameter satisfying

[TABLE]

where $d_{L}$ is the logical system dimension.If one of several noise channels is applied at random but it is known whichone occurred, then \tagform@34 holds forthe overall noise channel if the assumptions above are satisfied for eachindividual noise channel.

Note that the criterion holds for any arbitrary noise channel, not only forerasures at known locations. The proof of Vis given in Appendix C.Our criterion can be seen as an expression of the approximate Knill-Laflammeconditions Bény and Oreshkov (2010) in a particular basis, but where we providesimple and practical conditions on how to bound the error parameter$\epsilon_{\mathrm{worst}}$ of the code.Our criterion is a sufficient condition for a code to be approximatelyerror-correcting, but the condition is not necessary. When thecriterion does not apply we cannot draw any conclusion about the code’sperformance.We note that our criterion does not make reference to individual Kraus operatorsof the noise channel, as the Knill-Laflamme conditions or their approximateverisons do Knill and Laflamme (1997); Bény and Oreshkov (2010). This propertyeases its application to large-dimensional physical quantum systems.

VI Examples of covariant codes

Here we study three classes of covariant codes that illustrate the behavior ofour bound in either regimes of large subsystem dimensions, or large number ofphysical subsystems (Table 1).

VI.1 Three-rotor secret-sharing code

In this subsection, we apply our criterion to a truncated version of a codeintroduced by Hayden et al. Hayden et al. (2017), linking thatcode to the well-known three-qutrit secret-sharing quantum polynomialcode Aharonov and Ben-Or (1997); Cleve et al. (1999). While illustrating how to use ourcriterion, it also provides a covariant code which performs well in the limit ofcodewords covering a large range of physical charge on the subsystems.

VI.1.1 Rotor version of the qutrit secret-sharing code

For our purposes, a quantum rotor (also, an $O(2)$ or planar quantum rotor) is simply asystem with a basis $\{{\lvert{x}\rangle}\}$ that is labeled by an integer $x\in\mathbb{Z}$indexing representations of $U(1)$ Albert et al. (2017). Consider the three-rotor code given inRef. Hayden et al. (2017) defined by the isometry from $L$ to$A=A_{1}\otimes A_{2}\otimes A_{3}$ given as

[TABLE]

where the states $\{{\lvert{x}\rangle}\}$ are eigenstates of the angular momentum operators$T_{L}$ and $T_{A}=T_{1}+T_{2}+T_{3}$. This code can correct against the loss ofany of the three subsystems Hayden et al. (2017). Moreover, the code iscovariant with respect to the charge $T$: A logical state $\lvert{x}\rangle_{L}$ is mappedonto a codeword with the same total charge $x$.Interestingly, this code is a natural rotor generalization of the three-qutrit secret-sharing code Aharonov and Ben-Or (1997); Cleve et al. (1999).The three-qutrit code maps the basis vectors $\lvert{j}\rangle_{L}$ ($j=0,1,2$) of alogical qutrit into the codewords $\sum_{k}\lvert{k,k-j,k+j}\rangle$ where the additionis modulo $3$. Now, substitute each qutrit subsystem with a rotor.We obtain a code defined by the following encoding map:

[TABLE]

This code is not yet covariant with respect to the charge states $\lvert{x}\rangle$, asthe charge of the codeword corresponding to $\lvert{x}\rangle_{L}$ is not $x$. However, wemay apply the isometry mapping $\lvert{(.)}\rangle\to\lvert{-3(.)}\rangle$ on the first rotor and$\lvert{(.)}\rangle\to\lvert{2(.)}\rangle$ on the second, yielding the encodingmap \tagform@44. (In fact, thecode \tagform@45 is covariantwith respect to a different physical charge generator,$T^{\prime}_{A}=-3T_{1}+T_{2}+2T_{3}$, whereas the code \tagform@44 is covariantwith respect to the natural physical charge carried by three rotors,$T_{A}=T_{1}+T_{2}+T_{3}$.)In this sense, the code \tagform@44 is a natural $\mathit{U}(1)$-covariantgeneralization of the qutrit secret-sharing code.In the following sections, we address the problem that the codewordsin \tagform@44 are not normalizable, by building suitable wave packetstates.We normalize the codewords in two different ways: the sharp cutoff selects arange of charges to use for each rotor and discards the rest, while the smoothcutoff imposes a Gaussian envelope on each rotor, thereby keeping all the statesbut making them less prominent as the chargeincreases Gottesman et al. (2001) (see also related recentwork Woods and Alhambra (2019)). Our noise model is one single erasure at aknown location with probabilities $q_{1},q_{2},q_{3}=1/3$, as givenby \tagform@9.

VI.1.2 Sharp cutoff

Let us now truncate the logical system $L$ to a dimension of $2h+1$ for somefixed $h$, so the charge with respect to which the system is $\mathit{U}(1)$-covariantbecomes $T_{L}=\sum_{x=-h}^{h}x\lvert{x}\rangle\langle{x}\rvert_{L}$. The physical subsystems aretruncated in turn to $2m+1$ dimensions, so there are in total two parameters$\{h,m\}$ that determine the ranges of the logical and physicalcharges. Normalizing the codewords, the isometrybecomes

[TABLE]

for $x\in{-h,\cdots,h}$.Since the code is covariant and finite-dimensional, it does not allow forperfect error-correction. In Appendix D, we show that thecode has an accuracy parameter which satisfies

[TABLE]

By comparison, our bound \tagform@27 in this case reads

[TABLE]

There is a difference of a square root between the scaling of our actual codeperformance and of our bound. This is due to switching between the tracedistance and a fidelity-based distance in both of our bounds, and in the way wehave applied our criterion to derive \tagform@47.

VI.1.3 Smooth cutoff

We now consider a different approach to normalizing the codewords: by using aGaussian envelope we can achieve a “smoother” cut-off in contrast to the sharpcut-off considered above (such an envelope is known to be optimal forfinite-sized quantum clocks Woods et al. (2019)).We impose an envelope controlled by a parameter $w>0$ on the code states to makethem normalizable. The encoding isometry $V_{L\to A}^{(w)}$ now acts as

[TABLE]

with a normalization factor $c_{w}=\sum_{y=-\infty}^{\infty}e^{-y^{2}/(2w^{2})}$.Note that the envelope does not disturb the symmetry—the code remainscovariant since all of the basis states used to write each logical state stillhave the same charge. We still consider a $(2h+1)$-dimensional logical system$L$ in order to see how the bound scales in terms of $h/w$. Deferring calculationsto Appendix D, the present code has an accuracy parametersatisfying

[TABLE]

Our bound \tagform@27 in this case reads

[TABLE]

where we have kept only the first order in $h/w$, and where the logarithmic termresults from cutting off the infinite tails of our codewords. Hence, we seethat the present code achieves approximately the scaling of our bound, as bothexpressions scale as $h/w$ up to a logarithmic factor.We may ask for the reason of the discrepancy in the accuracy between thesharp and the smooth cut-off versions of our code. For the sharp cut-off, theerror parameter scales as $\epsilon_{\mathrm{worst}}\sim\sqrt{h/w}$, while forthe smooth cut-off it scales approximately as$\epsilon_{\mathrm{worst}}\sim h/w$. This can be explained from the followingproperty of the infidelity. Loosely speaking, the error parameter$\epsilon_{\mathrm{worst}}$ is related to how much the local reduced state on asingle system varies as a function of the logical state, as measured in terms ofthe infidelity [this can be seen from \tagform@11]. While inboth normalized versions of the above code, using either the sharp or the smoothcut-off, we are careful to ensure that all codewords are close to each other, itturns out that codewords with a sharp cut-off are in a regime where theinfidelity is more sensitive to differences than the smooth cut-off. This isbecause those codewords have incompatible supports. More precisely, for anystate $\rho$, the infidelity $\sqrt{1-F^{2}(\rho,\rho+\varepsilon X)}$, fora small perturbation $\rho\to\rho+\varepsilon X$, can grow like the square rootof $\varepsilon$ if $\rho+\varepsilon X$ has overlap outside of the support of$\rho$, while it grows linearly in $\varepsilon$ in well-behaved cases. Thesharp cut-off belongs to the former regime, while in the case of the smoothcut-off the infidelity is better behaved.

VI.2 Five-rotor perfect code

Here, we provide a rotor extension of the five-qubit perfectcode Bennett et al. (1996); Laflamme et al. (1996) that can be tiled toconstruct holographic codes Pastawski et al. (2015). Whilequdit Chau (1997) and oscillator Braunstein (1998)extensions have been considered, a rotor extension is not as straightforwardbecause one has to take care of preserving the phases in the code states asneeded to error-correct erasures. Our rotor code is the limit of a sequence ofqudit codewords whose constituent phases approach multiples of anirrational number. This same trick has been used to obtain anirrational magnetic flux via a sequence of rational fluxes in thetwo-dimensional electron gas problem Hofstadter (1976) as well as rotorlimits of other Hamiltonians Albert et al. (2017). This limit is meant to be anidealization since there is not enough storage space to measure an irrationalnumber to infinite precision.Let the dimension $D$ of each of the five physical subsystems befinite for the qudit $[[5,1,3]]_{\mathbb{Z}_{D}}$ code and infinitefor the rotor $[[5,1,3]]_{\mathbb{Z}}$ code. The general form ofthe unnormalized encoding for both codes is

[TABLE]

We introduce the rotor code as a limiting case of the qudit code,obtaining a concise expression for the qudit perfect tensor $T^{(D)}$in the process.

VI.2.1 Qudit version

Consider first the known finite-$D$ case, for which222This formula was obtained by constructing the codespace projectionout of powers of products of the code stabilizers, applying it tocanonical basis states $\lvert{x,0,0,0,0}\rangle$, and calculatingthe overlap of the resulting codeword with basis states $\lvert{j,k,l,m,n}\rangle$.

[TABLE]

where $\delta_{a,b}^{(D)}=1$ if $a=b$ modulo $D$ and $\omega$is a primitive $D$-th root of unity. Notice how the above expressionmakes the cyclic permutation symmetry naturally manifest. The deltafunction encodes the state label $x$ into the sum of the physicalqudit variables, with the key difference from the sharply-cutoff $[[3,1,2]]_{\mathbb{Z}}$code being that the sum is modulo $D$. This property makes this codeexactly error-correcting and not covariant with respect toa $U(1)$ symmetry. Instead, this code is covariant with respect toa $\mathbb{Z}_{D}$ symmetry generated by $Z^{\otimes 5}$, where $Z=\sum_{k\in\mathbb{Z}_{D}}\omega^{k}\lvert{k}\rangle\langle{k}\rvert$is the qudit Pauli matrix.

VI.2.2 Smooth cutoff

To take the qudit-to-rotor limit, pick $\omega=\exp(2\pi iL/D)$ withincommensurate integers $L,D\rightarrow\infty$ such that $L/D$ approachesa positive irrational number $\Phi$. The indices in Eq. 52now range over $\mathbb{Z}$,

[TABLE]

and $\delta$ is the usual Kronecker delta function. The finalingredient is to normalize the states, which can be done via a sharpor a smooth cutoff as in the $[[3,1,2]]_{\mathbb{Z}}$ code. We performthe latter using a cyclically-symmetric Gaussian envelope with spread$w$, prepending $\exp[-\frac{1}{4w^{2}}(j^{2}+k^{2}+l^{2}+m^{2}+n^{2})]$to the tensor $T_{jklmn}^{(\infty)}$ in Eq. 52 and thennormalizing the codewords. The resulting code is covariant with respectto a $U(1)$ symmetry generated by the total physical charge $T_{A}=\sum_{i=1}^{5}T_{A_{i}}$,analogous to the three-rotor code \tagform@49. With the addition of the envelope, the resulting tensor becomes approximately perfect. This rotor version can be stacked to form a approximately error-correcting $U(1)$-covariant holographic code in the same way as the qubit perfect tensors were connected in Ref. Pastawski et al. (2015).One can apply the certification criteria to this code to yield the same scalingas for the three-rotor code \tagform@50 for the model of a singleerasure (see Appendix D for details),

[TABLE]

However, this code is capable of correcting any single-subsystem error, so itcan correct for known erasure of any two subsystems. Calculating the bound for the noise channel$\cal{N}$ consisting of erasure of any two sites with equal probability yieldsthe same scaling,

[TABLE]

The larger coefficient is sensible since a code approximately correcting at mosttwo erasures should be better at correcting only one. In both cases, there areadditional corrections of order $O(he^{-cw^{2}})$ for $c>0$ arising from adetailed application of our criterion.

VI.3 Thermodynamic codes for $n\to\infty$

We now investigate a class of covariant codes in the limit where the number ofsubsystems $n$ grows large. We exploit the codes developed inRef. Brandão et al. (2017), relevant for quantum computing withatomic ensembles Saffman et al. (2010).For these codes the basis vectors for the code space can be chosen to be energy eigenstates of a many-body system, with the property that the reduced state on a subsystem appears to be thermal with a nonzero temperature; we therefore call them thermodynamic codes. This thermal behavior of local subsystems is expected for closed quantum systems that satisfy the eigenstate thermalization hypothesis Srednicki (1994) or dynamical typicality Popescu et al. (2006); Riera et al. (2012). Energy eigenstates with slightly different values of the total energy also have slightly different values of the locally measurable temperature; thus the identity of a codeword is imperfectly hidden from a local observer, and therefore erasure of a subsystem is imperfectly correctable.Consider a many-body system, such as a one-dimensional spin chain, and pick outtwo global energy levels $\lvert{E}\rangle_{A},\lvert{E^{\prime}}\rangle_{A}$ in the middle of the spectrum,with a given energy difference $\Delta E=E^{\prime}-E$. Assume, in the spirit of theeigenstate thermalization hypothesis, that the reduced states of both$\lvert{E}\rangle_{A}$ and $\lvert{E^{\prime}}\rangle_{A}$ on each individual system $A_{i}$ are approximatelythermal. The corresponding temperature scales as $T\propto E/n$ since thetemperature is an intensive thermodynamic variable. Then, the temperaturedifference vanishes for $n\to\infty$, and the resulting reduced thermal statesfor these two states are very close. Intuitively, this means that if a system$A_{i}$ is provided to the environment, the latter cannot tell whether the globalstate is $\lvert{E}\rangle$ or $\lvert{E^{\prime}}\rangle$, and hence the two energy levels form atwo-dimensional code space that is approximately error-correcting againsterasures at known locations.For example, consider the code developedin (Brandão et al., 2017, Appendix D), in the context of a 1Dtranslation-invariant Heisenberg spin chain. Here we consider as relevantcharge the total magnetization $M=\sum\sigma_{Z}^{i}$ of the spin chain. Thecodewords $\lvert{h_{m}^{n}}\rangle$ in (Brandão et al., 2017, Appendix D) areDicke states with respect to total magnetization—i.e., they are asuperposition of canonical $n$-spin basis states that all have some fixedmagnetization $m$:

[TABLE]

The code is covariant with respect to total magnetization by construction, bydefining the magnetization charge operator in the abstract logical system tocorrespond to the magnetization of the corresponding codeword. The values $m$are spaced out by steps of $2d+1$, thus ensuring that any errors which changethe magnetization by at most $2d$ cannot cause logical bit flips. Thistrick—using a sufficiently large spacing between codewords so that they arenot mapped into each other by errors—has analogues in CSS codes, relatedmulti-qubit codes Ouyang (2014), and bosonicerror-correction Albert et al. (2018). However, to show that sucherrors are indeed correctable, one still has to make sure that expectationvalues of errors with each codeword do not depend on the codeword in thelarge-$n$ limit.In Appendix D, we show that this code’s approximationparameter as an approximate error-correcting code against the erasure of aconstant number of sites scales as

[TABLE]

On the other hand, our bound \tagform@27 also displays the samescaling,

[TABLE]

In consequence, this code has an approximation parameter that displaysthe same scaling as our bound, meaning that our bound is approximately tight inthe regime $n\to\infty$.

VII Approximate Eastin-Knill theorem

Our second main technical contribution is an approximate version of theEastin-Knill theorem.The Eastin-Knill theorem states that it is not possible for an error-correctingcode to admit a universal set of transversal logical gates, imposing severerestrictions on fault-tolerant quantumcomputation Eastin and Knill (2009). In fact, an approximate versionof the Eastin-Knill theorem naturally follows from our bounds inIV. This is intuitively seen in thesetup of our main theorem depicted in Fig. 1, by choosing thetransformation group to be the full unitary group $\mathit{U}(d_{L})$ on the logicalsystem: To any logical unitary we require that there correspond an transversalunitary on the physical system that achieves the same logical transformation.Hence, our bound provides a limitation to the accuracy of codes that admit auniversal set of transversal logical gates. The goal of this section is tospecialize our main bound \tagform@27 to this situation, in order toobtain a limit expressed in terms of the dimensions of the local physicalsubsystems.There is a subtlety worth noting in the argument above. In the setting of theEastin-Knill theorem, it is not necessarily required that the mapping of logicalto physical unitaries forms an actual representation, i.e., that it iscompatible with the group structure. However, it turns out that we may assumethis without loss of generality. Intuitively, as long as one can generatelogical unitaries that are close to the identity with a transversal physicalunitary, one can show that there are corresponding physical generators whichspan a bona fide representation (Appendix E). That is, ifa code admits a universal transversal gate set, then it is necessarily covariantwith respect to the full logical unitary group for some transversalrepresentation on the physical systems.The bounds derived inIV and IV cannot in general be directlyrelated to the dimension of the local physical subsystems.Indeed, there is no restriction on how large $\Delta T_{i}$ can be. The onlyrestriction that enters the statement of our main theorem is that a logicalcharge eigenstate must be mapped onto a global physical eigenstate of the samecharge (up to a constant offset); the logical charge operator and the localphysical charge operators may otherwise be chosen arbitrarily. For example, therepetition code spanned by $\{\lvert{000}\rangle,\lvert{111}\rangle\}$ with logical charge$\delta\sigma_{z}$, physical charge$M\sigma_{z}^{(1)}-M\sigma_{z}^{(2)}+\delta\sigma_{z}^{(3)}$, and $M\gg\delta$can have a very large range $M$ of charges on each local physical subsystemdespite the systems having only two levels. In the other extreme, a completelydegenerate local physical system will have zero charge range despite a possiblyhuge dimension.The above observation is an expression of the fact that the covariance is withrespect to an abelian symmetry group ($\mathit{U}(1)$). In contrast, for non-abelianLie groups, one may no longer choose the generators arbitrarily because theyhave to obey nontrivial commutation relations with each other. Consider forinstance a code that is covariant with respect to spin, where the group is$\mathit{SU}(2)$. The three generators of the corresponding Lie algebra, $J_{x}$, $J_{y}$,and $J_{z}$, satisfy the commutation relations $[J_{x},J_{y}]=iJ_{z}$ along with thecorresponding cyclic permutations of $x,y,z$. We know in the case of $\mathit{SU}(2)$that the irreducible representations are labeled by a spin quantum number $j$that is a positive integer or half-integer, that the generator $J_{z}$ in thisrepresentation has nondegenerate eigenvalues $m=-j,-j+1,\ldots,+j$, andhence that the dimension of the irreducible representation labeled by $j$ is$2j+1$. By rotational symmetry, the same holds for any other standard generatorin that irreducible representation by choosing an appropriate basis. In otherwords, if the dimension of a physical subsystem is small, we cannot “fit” agenerator on that system with a large range of angular momentum values.More precisely, if $T_{i}^{z}$ is the spin generator corresponding to $J_{z}$ on the$i$th physical subsystem, the largest irreducible representation that canappear in the action of $T_{i}^{z}$ must fit in the physical subsystem, that is, wemay not have any $j$ larger than $(d_{i}-1)/2$, where $d_{i}$ is the dimension ofthe $i$th physical subsystem, or else the representation is too big. In turn,this bounds the range of $J_{z}$ charge values as $\Delta T_{i}^{z}\leqslant d_{i}-1$.Hence, if we encode a qubit using a code that admits a universal set oftransversal logical unitaries, we may apply our bound \tagform@27,choosing $T_{L}=J_{z}=\operatorname{diag}(1/2,-1/2)$ on the logical level with $\Delta T_{L}=1$,with the corresponding $\Delta T_{i}=\Delta T_{i}^{z}\leqslant d_{i}-1$; we thenobtain

[TABLE]

Thus, remarkably, the non-abelian nature of the group $\mathit{SU}(2)$ allows us to bound theexpression in \tagform@27 directly in terms of the dimensions of thelocal physical subsystems.This is because physically, the generators $J_{x,y,z}$ of $\mathit{SU}(2)$ correspond torotations around different axes, and the Lie algebra commutation relationsrequire all of them to be of a similar scale. No such requirement was presentfor $\mathit{U}(1)$ since we were free to rotate around a chosen axis arbitrarily quickly.In the case of a code that is covariant with respect to $\mathit{SU}(d)$ with $d>2$, thedependence on the physical subsystem dimensionsbecomes considerably more restrictive. We provide an overview of our argument,leaving technical details to Appendix E. Irreduciblerepresentations, or irreps, of $\mathit{SU}(d)$ are indexed by $d-1$ nonnegativeintegers $(\lambda_{1},\lambda_{2},\cdots,\lambda_{d-1})\equiv\lambda$ arrangedin decreasing order. These integers determine the largest eigenvalues of the now$d-1$ commuting generators of $\mathit{SU}(d)$. For $\mathit{SU}(2)$, only one generator $J_{z}$is diagonal in the canonical basis and the integer $\lambda=\lambda_{1}=2j$determines the highest spin attainable in that irrep. For the fundamentalrepresentation $\lambda=(1,0)$ of $\mathit{SU}(3)$, the two simultaneouslydiagonalizable generators are the two Gell-Mann matrices that are diagonal inthe canonical basis. Since the entries in $\lambda$ are decreasing, the largesteigenvalue that any generator could have is $\lambda_{1}$, i.e.,$\bBigg@{1.2}\lVert{T_{\lambda}^{(i)}}\bBigg@{1.2}\rVert_{\infty}\leqslant\lambda_{1}$. It turns out that theirrep that minimizes the dimension out of all irreps with fixed $\lambda_{1}$ isthe completely symmetric irrep $(\lambda_{1},0,0,\cdots,0)$. The dimension ofthis irrep is the dimension of the symmetric subspace on $\lambda_{1}$ number of$d$-dimensional systems, which is a polynomial of degree $d-1$ in$\lambda_{1}$. Therefore, in order to fit in a system of dimension $d_{i}$, thelargest possible $\lambda_{1}$ is of order $O\bBigg@{1.2}({d_{i}^{1/(d-1)}}\bBigg@{1.2})$. Now, anygeneral representation can be decomposed into irreps, and a generator $T$ issimply $T=\bigoplus T_{\lambda}$, where $T_{\lambda}$ is the correspondinggenerator for each irrep. We then have$\lVert{T}\rVert_{\infty}=\max_{\lambda}\lVert{T_{\lambda}}\rVert_{\infty}$. So, if arepresentation fits in the system dimension $d_{i}$, then it cannot contain anyirrep $\lambda$ with $\lambda_{1}$ larger than $O\bBigg@{1.2}({d_{i}^{1/(d-1)}}\bBigg@{1.2})$.For a code that is covariant with respect to the full unitary group on thelogical space, we have $d=d_{L}$, and picking a simple standard generator for ourearlier bound \tagform@27, we obtain the following theorem.

Theorem 4** (Approximate Eastin-Knill theorem).**

The performance of an $SU(d_{L})$-covariant code, quantified by the worst-caseentanglement fidelity, is bounded as follows:

[TABLE]

The following bounds also hold:

[TABLE]

Similar bounds can be obtained for the figure of merit$\epsilon_{\mathrm{e}}$, based on the average entanglement fidelity, by makingin \tagform@63 and \tagform@64b the replacement$\epsilon_{\mathrm{worst}}\rightarrow d_{L}\epsilon_{\mathrm{e}}/2$.

In other words, any code that (a) stores a large amount of quantum information,and (b) admits universal transversal gates, has severe restrictions on itsability to recover from erasure errors.The bound \tagform@63 is useful to determine the precision limit of a codethat has a universal set of transversal gates. If we imagine that each physicalsubsystem is composed of $m_{i}$ qubits lumped together, then the error parameterof the code scales at least inversely in the largest number of qubits $m_{i}$ thatwere lumped together. If we set for instance $d_{L}\sim 10^{3}$ (10 logical qubits)that are encoded into $n$ systems consisting of 10 qubits each, i.e.,$d_{i}\sim 10^{3}$, we obtain the rather prohibitive error parameter$\epsilon_{\mathrm{worst}}\gtrsim 0.14/n$. (This estimate can be improved to$\epsilon_{\mathrm{worst}}\geqslant 0.5/n$ using a tighter bound given inAppendix E.)The bound \tagform@64a shows that if$\epsilon_{\mathrm{worst}}$ is kept constant and for $d_{L}\to\infty$, we musthave that $d_{i}$ grows polynomially in $d_{L}$, where the exponent is$1/(2n\epsilon_{\mathrm{worst}})$. If, for instance, we wish to achieve aprecision of $\epsilon_{\mathrm{worst}}\sim 10^{-3}$, then we must have thescaling $d_{i}\sim(d_{L})^{500/n}$. Concretely, for $d_{L}\sim 10^{3}$ (10 logicalqubits) encoded into $n=10$ subsystems, the physical subsystems need to be of arespectable dimension $d_{i}\sim 10^{65}\sim 2^{216}$, i.e., the physicalsubsystem must comprise $216$ qubits lumped together.Our third bound is interesting in the regime of extremely high accuracy.Suppose we wish to accurately resolve individual logical basis states of ahighly mixed logical state. The logical information might, for instance, beentangled with a large reference system. In such a situation, we require$\epsilon_{\mathrm{worst}}\lesssim d_{L}^{-1}$.Bound \tagform@64b then asserts that the physicalsubsystem dimension must grow exponentially in the logical systemdimension.Finally, we note thatEqs. 63, 64a and 64bare obtained from a more general, tighter bound on $\max d_{i}$ which is expressedas a binomial coefficient (see Appendix E for details). In somecases, this bound allows to obtain tighter estimates on the physical dimensionof the subsystems.

VII.1 Random Constructions

The bounds of VII severely limit the error correctioncapability of the unitary $\mathit{SU}(d_{L})$-covariant codes. We now show that it ispossible to find good $\mathit{SU}(d_{L})$-covariant codes in regimes of large physicalsystems that are not excluded by VII.The constructions we present are randomized as well as asymptotic in thedimension of the physical subsystems. More precisely, we consider the encodingof one $d_{L}$-dimensional Hilbert space $\mathscr{H}_{L}$ in a physical space which is atensor product of three Hilbert spaces$\mathscr{H}_{A}=\mathscr{H}_{A_{1}}\otimes\mathscr{H}_{A_{2}}\otimes\mathscr{H}_{A_{3}}$. The encoding is donevia an isometry $V_{L\to A}$, which is $\mathit{U}(d_{L})$ covariant: For all$U\in U(d_{L})$,

[TABLE]

Here $r_{1},r_{2},\text{and }r_{3}$ are three irreps of $\mathit{U}(d_{L})$. Ourconstructions are randomized in the following way:

• •

$V$ is chosen randomly from all possible isometries satisfying the covariance condition \tagform@65.

• •

The irreps $r_{1},r_{2},\text{and }r_{3}$ are chosen randomly, or at leastgenerically. In fact, we only need that the irreducible representationdoes not belong to a small subset of all possible irreducible representations.

In Appendix F, we use randomized constructions to prove existenceof $U(d_{L})$-covariant codes with small error (measured by $\epsilon_{e}$ based onthe average entanglement fidelity), as summarized in the following theorem:

Theorem 5**.**

For $d_{L}\geqslant 4$ and every $\epsilon>0$, there exists a$\mathit{U}(d_{L})$-covariant code with error$\epsilon_{\mathrm{e}}\leqslant\epsilon$ and physical dimensions $d_{i}$,$i\in\{1,2,3\}$, such that

[TABLE]

for some $C_{2}$ which is only a function of $d_{L}$.

It is not clear how to compare the performance of our code givenby \tagform@66 to ourbounds \tagform@64 because our nonconstructiveproof does not specify the behavior of $C_{2}$ as a function of $d_{L}$, which isgiven by details of the representation theory of $\mathit{U}(d_{L})$. It remains openwhether the lower bound can be strengthened or the constructions can beimproved.Our proof technique does not immediately work for $\mathit{U}(2)$-covariant codes, asit is harder to bound the fluctuations of the fidelity of recovery when thelogical Hilbert space is too small. For $\mathit{U}(3)$-covariant codes, our methodslead to codes with a slightly different scaling from Eq. 66. Infact, for the $\mathit{U}(3)$-case, one can provide randomized andnon-asymptotic constructions (which work for known finite physicaldimensions) using the explicit formulas for the Littlewood-Richardsoncoefficients Rassart (2004). These constructions are notincluded in the present paper, as there is little specific interest in the$d_{L}=3$ case.The proof of VII.1 is technical and relies on therepresentation theory of the unitary group (see Appendix F fordetails). The proof starts by connecting the average fidelityrecovery of erasure of a fixed subsystem to the smoothness of theLittlewood-Richardson coefficients.Littlewood-Richardson coefficients are representation theory quantities thatcount the degeneracy of a particular irrep of $\mathit{U}(d_{L})$ in the tensor productof two other irreps, and their smoothness follows from modern results inrepresentation theory of the unitary group Rassart (2004)(Fig. 5).

VII.2 Generalized $W$-state encoding

Here we consider another example of an approximate quantum error-correctingcode, covariant with respect to the full unitary group on the logical system.It is based on the $W$ state, and achieves an arbitrarily small$\epsilon_{\mathrm{worst}}$. in the limit of a large number of subsystems,$n\to\infty$. The logical system $L$ of dimension $d_{L}$ is encoded into aphysical system composed of $n$ copies of a $(d_{L}+1)$-dimensional space, whereeach subsystem is a copy of the logical system with an additional basis vector$\lvert{\perp}\rangle$. The encoding is

[TABLE]

Any logical unitary $U$ can be carried out on the encoded state transversally byapplying the unitary $U\otimes U\otimes\cdots\otimes U$, where we let $U$ acttrivially on the extra state $\lvert{\perp}\rangle$.Remarkably, aside from being $\mathit{U}(d_{L})$-covariant, this trivial code is also effective against random erasures.Intuitively, this is because the environment will only receive access to the logical state $\lvert{\psi}\rangle$ withprobability $1/n$ if it gets access to a singlesubsystem; that is, the environment is unlikely to learn anything aboutthe logical information. This can be formalized with a direct application ofour criterion (V). Given a basis $\lvert{x}\rangle_{L}$of $L$, the reduced state $\rho_{1}^{x}$ on any single physical subsystem of thecodeword \tagform@69 corresponding to $\lvert{x}\rangle_{L}$ is

[TABLE]

and thus $F^{2}(\rho_{1}^{x},\lvert{\perp}\rangle\langle{\perp}\rvert)=1-(1/n)$ for all $x$, and it follows that$\sqrt{1-F^{2}(\rho_{1}^{x},\lvert{\perp}\rangle\langle{\perp}\rvert)}\leqslant\sqrt{2/n}=:\epsilon$.For $x\neq x^{\prime}$ we have accordingto \tagform@31,

[TABLE]

and thus $\lVert{\rho_{1}^{x,x^{\prime}}}\rVert_{1}=1/n=:\nu$. The corresponding reducedstates on the other physical subsystems are the same by symmetry of thecodeword \tagform@69. ThenV asserts that this code has an errorparameter that is at most

[TABLE]

That is, for fixed $d_{L}$, the code becomes a good error-correcting code in thelimit $n\to\infty$.In contrast to the thermodynamic codes presented above, this $W$-state code doesnot saturate our bound on $\epsilon_{\mathrm{worst}}$, which is inverselyproportional to $n$ rather than the square root of $n$. The reason for thisdiscrepancy is the same as for the difference between a sharp and a smoothcut-off for the three-rotor code, discussed inVI.1.3. Again, here, as $n$ becomes large, thelocal reduced state grows close to the rank-deficient state $\lvert{\perp}\rangle\langle{\perp}\rvert$, whichis a regime where the infidelity is particularly sensitive to smallperturbations. In contrast, for instance, our thermodynamic codes ofVI.3 have reduced states that are full-rank, allowingthe code to achieve the same scaling as our accuracy bound as $n\to\infty$.While this code does not achieve the same $1/n$ scaling as the thermodynamiccodes, it does exhibit covariance with repect to the full logical unitary group$\mathit{U}(d_{L})$.

VIII Error-correcting codes for general groups

In this section, we develop a framework for constructing codes that arecovariant with respect to any group $G$ admitting a left- and right-invariantHaar measure, encompassing in particular codes that are based on rotors,oscillators, and qudits.Our construction is based on quantum systems that transform as the regularrepresentation of $G$. Orthonormal basis states $\{{\lvert{g}\rangle}\}_{g\in G}$ for this representation are labeled by group elements; if the group has an infinite number of elements, then thequantum system is infinite-dimensional.A qubit can transform as the regular representation ofthe group $\mathbb{Z}_{2}$, and a qudit as the regular representation of$\mathbb{Z}_{d}$. An oscillator provides a regular representation of the (noncompact) group$\mathbb{R}$, with the group acting by translation in either its position basis$\{{\lvert{x}\rangle}\}$ or its momentum basis $\{{\lvert{p}\rangle}\}$. Similarly, a rotor provides aregular representation of the group $\mathit{U}(1)$, with orthonormal basis states $\{{\lvert{e^{i\phi}}\rangle}\}$; when Fourier transformed, it can transform as a regular representation of$\mathbb{Z}$, where the basis states are the eigenstates of angular momentum$\{{\lvert{\ell}\rangle}\}_{\ell\in\mathbb{Z}}$.For ease of presentation, we will consider codes whose logical systemand whose physical subsystems transform as the regular representation of any compact group $G$, commenting on noncompact groups in VIII.3.Well-known qubit codes such as the bit-flip, phase-flip, and$[[4,2,2]]$ codes, naturally extend to this setting. More generally,we will also discuss extensions of the $[[m^{2},1,m]]$ and $[[2m,2m-2,2]]$ qubit codes.

VIII.1 Bit- and phase-flip codes

For simplicity, let us review bit-flip and phase-flip codes first. An $M$-qubitbit-flip encoding copies the logical basis state index $x\in\mathbb{Z}_{2}$ ineach of the $M$ subsystems. An $M$-qubit phase-flip encoding hides the logicalindex in the sum of the physical qubit states. Taking $M=3$ for concreteness,the two encodings are

[TABLE]

where $\delta_{x,y}=1$ if $x=y$ modulo 2. Bit-flip codes protectagainst single-qubit shifts $x\rightarrow x+1$ while phase-flip codesprotect against single-qubit operators which are diagonal in the canonical basis.By viewing a qubit as a regular representation of the group $G=\mathbb{Z}_{2}$, we can see how to generalize this construction to other groups. For a finite group $G$ with order $|G|$, consider the $|G|$-dimensional Hilbert space $V$ spanned by $\{\lvert{g}\rangle\,|\,g\in G\}$ with inner product $\langle{g}\hskip 0.86108pt|\hskip 0.86108pt{h}\rangle=\delta_{g,h}$, where $\delta_{g,h}=1$ if $g$ and $h$ are the same group element and zero otherwise. For compact continuous groups, the Hilbert space is infinite-dimensional, and $\delta_{g,h}$ becomes the Dirac delta function—infinite when $g=h$ and zero otherwise—and sums $\frac{1}{\left|G\right|}\sum_{h\in G}$ are replaced by integrals $\int dg$, where $dg$ is the group’s normalized Haar measure Sternberg (1995); Arovas . We’ll write sums below for simplicity, with the understanding that the sum is to be replaced by an integral when $G$ is a compact Lie group.The respective $M=3$-subsystem bit- and phase-flip generalizationsof Eq. 73 for finite groups are

[TABLE]

The bit-flip encoding records a group element redundantly, while the phase-flipencoding hides $g$ in a product of three group elements. The error-correctionproperties of these codes are analogous to those for $G=\mathbb{Z}_{2}$:the bit-flip codes correct against errors which take individual subsystemsinto states orthogonal to $\lvert{g}\rangle$ while phase-flip codes correctagainst single-subsystem errors diagonal in the $\lvert{g}\rangle$-basis.To perform an $X$-type gate on these codes, introduce left and right-multipliers,$\overrightarrow{X}_{g}$ and $\overleftarrow{X}_{g}$, which act as

[TABLE]

The sets $\{\overrightarrow{X}_{g}\}_{g\in G}$ and $\{\overleftarrow{X}_{g}\}_{g\in G}$ are permutation matrices formingthe left and right regular representations of $G$. Note that thearrow points towards $h$ from the side that $g$ acts. Since multiplyingfrom the left commutes with multiplying from the right, the two setscommute with each other.For the bit-flip code \tagform@74a, the logical left multiplication gate

[TABLE]

can be implemented transversally:

[TABLE]

For the phase-flip code, which provides no protection against bit-flips at all, logical left multiplication is implemented by acting on a single subsystem:

[TABLE]

where $I$is the subsystem identity. Similar constructions hold for logical right multipliers.For continuous $G$, the code states become nonnormalizable, but thegates work the same way. Therefore, the logical operators $\overrightarrow{X}_{L,k}$ define exact continuous symmetries of theses codes. However, these codes do not correct erasure of a subsystem; rather, each code corrects only a limited set of single-subsystem errors. The same is true for the qubit codes that inspired this construction.We can concatenate the bit-flip code and the phase-flip code for qubits toobtain Bacon-Shor codes Ralph et al. (2005); Bacon (2006), which have theparameters $[[m^{2},1,m]]_{\mathbb{Z}_{2}}$. This notation means that onelogical qubit is encoded in a code block of $m^{2}$ physical qubits, and that thecode distance is $m$; hence erasure of any $m-1$ of the qubits can becorrected. Of the codes in this family, the best known are the$[[4,1,2]]_{\mathbb{Z}_{2}}$ error-detecting code Steane (1996a, b)and Shor’s nine-qubit $[[9,1,3]]_{\mathbb{Z}_{2}}$ error-correctingcode Shor (1995).Likewise, by concatenating the $G$-covariant bit-flip and phase-flip codes, we obtain the $G$-covariant $[[m^{2},1,m]]_{G}$ code. For finite $G$, this is a $G$-covariant encoding of a $|G|$-dimensional logical system in $m^{2}$ $|G|$-dimensional subsystems, protected against erasure of any $m-1$ of the subsystems. If $G$ is a compact Lie group, this code has continuous $G$ symmetry. In that case, as the Eastin-Knill theorem requires, the encoding is infinite-dimensional.Rather than discussing this generalized Bacon-Shor code construction more explicitly here, in VIII.2 we’ll provide a more detailed discussion of a related code, with two rather than just one $|G|$-dimensional logical subsystems.

VIII.2 The $[[4,2,2]]_{G}$ code and its generalizations

There is also a $[[4,2,2]]_{\mathbb{Z}_{2}}$ qubit code Grassl et al. (1997), whichcan be extended to a covariant $[[4,2,2]]_{G}$ code, with encoding map

[TABLE]

In fact, the $[[4,2,2]]_{\mathbb{Z}_{2}}$ code can be viewed as a mimimalversion of Kitaev’s toric code Kitaev (2003), defined by just oneplaquette operator and one star operator, and \tagform@79 defines thecorresponding quantum double code with group $G$.Given $l\in G$, the physical operator $I\otimes I\otimes\overleftarrow{X}_{l}\otimes\overleftarrow{X}_{l}$ has the effect of replacing $g_{2}$ by $g_{2}l$ in Eq. 79, hence mapping the logical state to $|g_{1},g_{2}l\rangle_{L}$. The physical operator $\overrightarrow{X}_{l}\otimes I\otimes\overrightarrow{X}_{l}\otimes I$, after a redefinition of the summation variable ($g\rightarrow l^{-1}g^{\prime}$), has the effect of replacing $g_{1}$ by $lg_{1}$, hence mapping to the logical state to$|lg_{1},g_{2}\rangle_{L}$. Since the left and right multipliers commute, and both logical operations are transversal, the code is covariant with respect to the group $G\times G$.Using the quantum error-correctionconditions Bennett et al. (1996); Knill and Laflamme (1997), we cancheck that this code corrects one erasure. Let $O_{1}$ be an operator acting onthe first subsystem, and consider its matrix element between code states.Plugging into Eq. 79 and contracting indices we find

[TABLE]

This means that the code satisfies the condition for correctability of erasure of the first subsystem.A similar calculation can be performed for operators acting on any of the other subsystems; therefore erasure is correctable for each of the four subsystems.The $[[4,2,2]]_{\mathbb{Z}_{2}}$ qubit code can be generalized to a $[[2m,2m-2,2]]_{\mathbb{Z}_{2}}$ code, which can also be extended to a covariant $[[2m,2m-2,2]]_{G}$ code for any group $G$. To understand this construction, first consider a different $[[4,2,2]]_{G}$ code, which has a smaller covariance group than the code described above. Now we use the encoding map

[TABLE]

Unlike the previously considered code, this code has the property of being invariant under the action of a “stabilizer” operator $S_{l}=\overrightarrow{X}_{l}\otimes\overrightarrow{X}_{l}\otimes\overrightarrow{X}_{l}\otimes\overrightarrow{X}_{l}$for each $l\in G$.The price we pay for this invariance property is a reduction in the number of independent transversal operations which act nontrivially on the code space. There is no nontrivial symmetry of the code acting from the left, but the operator $I\otimes\overleftarrow{X}_{l}\otimes\overleftarrow{X}_{l}\otimes I$ maps $|g_{1},g_{2}\rangle_{L}$ to $|g_{1}l,g_{2}\rangle_{L}$. Therefore, this code is $G$-covariant. We can also check that it satisfies the condition for correctability of erasure for each one of the four subsystems.To illustrate how this code generalizes to a higher-length code with more physical subsystems, we will, to be concrete, describe the corresponding $[[2m,2m-2,2]]_{G}$ code with $m=4$. This code has the stabilizer $S_{l}=\overrightarrow{X}{}_{l}^{\otimes 8}$ for each $l\in G$, and the encoding map

[TABLE]

Aside from being invariant under the action of $S_{l}$, the code has another important property: each codeword is a superposition of states $\lvert{h_{1},h_{2},h_{3},h_{4},h_{5},h_{6},h_{7},h_{8}}\rangle$ of the eight physical subsystems having the property $h_{1}^{-1}h_{2}h_{3}^{-1}h_{4}h_{5}^{-1}h_{6}h_{7}^{-1}h_{8}=1$ (for this to work the code has to have even length). These two properties together suffice to ensure that erasure of each subsystem is correctable.This code is covariant under the group $G^{3}$. The operator

[TABLE]

acts on the code’s basis states according to

[TABLE]

In general, the $[[2m,2m-2,2]]_{G}$ code has a transversal $G^{m-1}$ symmetry, acting similarly.

VIII.3 Further extensions and some limitations

One can extend these constructions to noncompact groups. For example, theoscillator $[[9,1,3]]_{\mathbb{R}}$ code was noticed earlyon Lloyd and Slotine (1998); Braunstein (1998) (seealso Barnes (2004); Bermejo-Vega et al. (2016)). Another example is the rotor$[[4,2,2]]_{\mathbb{Z}}$ encoding

[TABLE]

As done in VI.1, one can impose an envelopeso that the codewords are normalizable. In general, a bi-invariant Haar measureis sufficient to perform the left- and right-multiplier transversal gates aswell as the error-correction, but one would have to approximate the codewords toavoid infinities due to non-normalizable Haar measures. For the oscillator code$[[4,2,2]]_{\mathbb{R}}$, for which the above is an integral over oscillatorposition states, we additionally need to approximate the position states with adisplaced and finitely squeezed vacuum Gottesman et al. (2001). Inother words, noncompactness and the continuous nature of the group mayeach require approximations to achieve normalizability of the codewords.One may also ask if it is possible to extend the secret-sharing code$[[3,1,2]]_{\mathbb{Z}_{3}}$ from VI.1 to amore general group $G$. An extension does indeed work for$G\in\{\mathbb{R},\mathbb{Z},\mathbb{Z}_{2D+1},U(1)\}$, but the code breaks downat, e.g., $\mathbb{Z}_{2D}$ due to there being a non-measure-zero set oforder-two elements in the group. Writing a natural guess for the encoding,

[TABLE]

we see that the third subsystem stores the logical index “in plain sight”whenever $h^{2}=1$. Roughly speaking, for groups with too many such elements,the environment can extract logical information from the code.

IX Symmetries and error correction in quantum gravity

The interplay between continuous symmetries and quantum error correction has implications for holography and quantum gravity.The AdS/CFT correspondence Maldacena (1999); Witten (1998)is a duality between quantum gravity in Anti-de Sitter (AdS) space, and aconformal field theory (CFT) in one fewer spatial dimensions, where the CFTresides on the boundary of the AdS space. It was recently discovered that theduality map from bulk operators to boundary operators may be regarded as theencoding map of a quantum error-correcting code, where the code space is spannedby low-energy states of the CFT. Specifically, local operators deep inside thebulk AdS are encoded as highly nonlocal operators in the boundary CFT which arerobust against erasure errors in the boundarytheory Almheiri et al. (2015); Harlow (2016, 2017). Here, we discuss symmetries of this AdS/CFTcode. First, we reprise a recent analysisfrom Harlow and Ooguri (2018a, b), which rulesout exact global symmetries for quantum gravity in the bulk AdS space. Then weexplain how our results in this paper clarify the correspondence between timeevolution in the bulk and boundary theories.

IX.1 No bulk global symmetries

A longstanding conjecture holds that quantum gravity is incompatible with global symmetry. One argument supporting this claim goes as follows Preskill (1993); Kallosh et al. (1995). According to semiclassical theory, which should be reliable for large black holes, the Hawking radiation emitted by a black hole is not affected by the amount of global charge the black hole might have previously consumed. Therefore, a process in which a black hole arises from the gravitational collapse of an object with large charge, and then evaporates completely, will not obey charge conservation.This argument may not be trustworthy if the symmetry group is a small finite group, in which case the total charge cannot be “large,” and any missing charge might reappear in the late stages of black hole evaporation when semiclassical theory does not apply.But recently, Harlow and Ooguri used AdS/CFT technology to show that even discrete global symmetries are disallowed in the bulk Harlow and Ooguri (2018b). Here we will reprise their argument, expressing it in language that emphasizes the conceptual core of the proof, and that may be more accessible for those familiar with the formalism of quantum error correction.To quantum coding theorists, it sounds strange to hear that the AdS/CFT codecannot have discrete symmetries, because typical quantum codes do. To illustratethis point we’ll revisit a simple quantum-error correcting code that is oftenused to exemplify the structure of the AdS/CFT code: the three-qutritcode Almheiri et al. (2015), which we already discussed inVI.1.1. This encodes a single logical qutrit in a blockof three physical qutrits, and protects against the erasure of any one of thethree qutrits.The three-qutrit code is an example of a stabilizer code—the code space may be defined as the simultaneous eigenspace of a set of generalized Pauli operators. For a qutrit with basis states $\{|j\rangle,j=0,1,2\}$, the generalized Pauli group is generated by operators $X$ and $Z$ defined by

[TABLE]

where $\omega=e^{2\pi i/3}$, which obey the commutation relations

[TABLE]

The code space of the three-qutrit code is the simultaneous eigenspace with eigenvalue 1 of the operators

[TABLE]

acting on the three qutrits in the code block. Note that, although $X$ and $Z$ do not commute, $S_{X}$ and $S_{Z}$ do commute, and can therefore be simultaneously diagonalized. Any nontrivial weight-one Pauli operator (supported on a single qutrit and distinct from the identity) must fail to commute with at least one of $S_{X}$ or $S_{Z}$. Therefore no nontrivial weight-one operator preserves the code space, which is why erasure of a single qutrit is correctable.However, there are weight-two Pauli operators that commute with both $S_{X}$ and $S_{Z}$, and therefore preserve the code space; for example,

[TABLE]

Because they preserve the code space, and act nontrivially on the code space, we say that $X_{L}$ and $Z_{L}$ are nontrivial logical operators for this code. Furthermore, $X_{L}$ and $Z_{L}$ obey the same commutation relations as $X$ and $Z$; they generate the logical Pauli group acting on an encoded qutrit. Note that because $S_{X}$ acts trivially on the code space, the operator $X_{L}$, which is supported on the first two qutrits, acts on the code space in the same way as $X_{L}S_{X}$, which is supported on the first and third qutrit, and also in the same way as $X_{L}S_{X}^{-1}$, which is supported on the second and third qutrit. A similar observation also applies to $Z_{L}$ and $S_{Z}$. This feature illustrates a general property: if $O_{L}$ is a logical operator, and $A$ is a subset of the qutrits in the code block such that erasure of $A$ is correctable, then we may represent $O_{L}$ as a physical operator supported on the complementary set $A^{c}$.Our purpose in describing this code is just to point out that the transversal logical operators $X_{L}$ and $Z_{L}$ may be viewed as global symmetries of the code. The action of each of these operators on the logical system can be realized as a tensor product of single-qutrit operators. Such a symmetry is what Harlow and Ooguri rule out. We need to understand why their argument applies to the AdS/CFT code, but not to the qutrit code or to other stabilizer codes.Harlow and Ooguri use special properties of AdS/CFT in two different ways, and their argument proceeds in two steps. The first step (explained in more detail below) appeals to entanglement wedge reconstruction,together with the structure of global symmetries in quantum field theory, to show that any global symmetry acting on the bulk acts transversally on the boundary. That is, the boundary can be expressed as a union of disjoint subregions $\{A_{k}\}$ such that erasure of each $A_{k}$ is correctable, and any bulk global symmetry operator $U_{L}$, when reconstructed on the boundary, can be expressed as a tensor product $\bigotimes_{k}W_{k}$, where $W_{k}$ is supported on $A_{k}$. (Here we ignore a correction factor supported only where the regions touch, which is inessential to the argument.) This is just the property that we have assumed throughout this paper, and which is exemplified by the three-qutrit code discussed above.The second step of the argument (also explained further below) is the crucial one, which invokes a property of the AdS/CFT code which is not shared by the typical quantum codes which arise in work on fault-tolerant quantum computation. Harlow and Ooguri argue that each $W_{k}$ is itself a logical operator; that is, each $W_{k}$ maps the code space to the code space. The essence of this part of the argument is that the code space is the span of low-energy states in the CFT, and the $W_{k}$’s, perhaps after suitable smoothing, can be chosen so that they do not increase the energy of the CFT by very much. As we have already emphasized, this property does not apply to the three-qutrit code, where $X_{L}$ is a logical operator, yet its weight-one factors $X\otimes I\otimes I$ and $I\otimes X^{-1}\otimes I$ are not logical. Indeed, because $X\otimes I\otimes I$ changes the eigenvalue of the unitary operator $S_{Z}$ by the multiplicative factor $\omega$, it maps the code space (the simultaneous eigenspace of $S_{Z}$ and $S_{X}$ with eigenvalue 1) to a subspace orthogonal to the code space (the eigenspace of $S_{Z}$ with eigenvalue $\omega$).A logical operator supported on a region $A$, where erasure of $A$ is correctable, must be the logical identity. We can easily see that’s true, because otherwise an adversary could steal region $A$ and apply a nontrivial logical operator, altering the encoded state and therefore introducing an uncorrectable error. Now the conclusion of Harlow and Ooguri follows easily. The bulk global symmetry operator $U_{L}$ is a product of logical operators, each of which is trivial; therefore $U_{L}$ must be the identity.As Harlow and Ooguri note (Footnote 68 in Harlow and Ooguri (2018b)),their argument, which excludes discrete symmetries of the AdS/CFT code as wellas continuous symmetries, is quite different than the Eastin-Knill argument,which excludes only continuous symmetries of a code. Both arguments apply in aframework where the symmetry of the code can be applied transversally, as aproduct of local operators. But for the Eastin-Knill argument, there is no needto assume that these local operators preserve the codespace, and therefore theargument applies to general codes. In contrast, Harlow and Ooguri assert thatfor the AdS/CFT code in particular, the local operators do preserve thecode space. Therefore, their argument excluding discrete symmetries applies tothe AdS/CFT code, but not to the typical codes studied by quantum informationtheorists.For completeness, we’ll now sketch the two key steps of the Harlow-Ooguri argument in slightly greater detail, starting with the step which shows that a bulk global symmetry acts transversally on the boundary. We begin by noting that a global symmetry in the bulk AdS space implies a corresponding symmetry acting on the boundary; to see this we need only consider the action of the bulk global symmetry on bulk local operators in the limit where the support of the bulk local operators approaches the boundary. Furthermore, a global symmetry operator of the boundary CFT is splittable; that is, it can be expressed as a tensor product of many operators, each supported on a small region.In coding theory language, the encoding isometry $V$ which maps bulk to boundary has the property

[TABLE]

where $U_{L}$ is the bulk symmetry operator and $U_{\mathrm{CFT}}$ is the corresponding CFT symmetry operator. Because the CFT symmetry is splittable, we may consider decomposing the CFT into small spatial subregions $\{A_{k}\}$, and infer that

[TABLE]

where $W_{k}$ is a CFT operator supported on $A_{k}$.Next we would like to see that the boundary subregions can be chosen so that erasure of any $A_{k}$ is correctable. This point is most naturally discussed using the language of operator algebra quantum error correction Almheiri et al. (2015). We consider the subalgebra $\mathcal{A}$ of logical operators which are supported on a subregion of the bulk. Each logical operator $O_{L}\in\mathcal{A}$ can be “reconstructed” as a physical operator $O_{\mathrm{CFT}}$ acting on the boundary using the encoding isometry $V$:

[TABLE]

What we wish to show is that, for any $O_{L}$ in $\mathcal{A}$, and for each boundary subregion $A_{k}$, the reconstructed boundary operator $O_{\mathrm{CFT}}$ can be chosen to have support on the complementary boundary subregion $A_{k}^{c}$. This property ensures that, for the bulk subalgebra $\mathcal{A}$, erasure of boundary region $A_{k}$ is correctable.The argument showing that erasure of boundary subregion $A_{k}$ is correctable is illustrated in Fig. 6.Associated with each boundary subregion $A_{k}$ is a bulk subregion $a_{k}$ which is called the entanglement wedge of $A_{k}$. The AdS/CFT code has these important properties Almheiri et al. (2015): (1) A bulk operator supported in bulk subregion $a_{k}$ can be reconstructed as a boundary operator supported in boundary subregion $A_{k}$. This property is called subregion duality. (2) Furthermore, a bulk operator supported in the bulk complement $a_{k}^{c}$ of bulk subregion $a_{k}$ can be reconstructed as a boundary operator supported in the boundary complement $A_{k}^{c}$ of boundary subregion $A_{k}$. This property is called complementary recovery.It follows from complementary recovery that if the bulk subalgebra $\mathcal{A}$is supported in $a_{k}^{c}$, then erasure of boundary subregion $A_{k}$ is correctablefor the subalgebra $\mathcal{A}$. This is the key fact that we need. As inFig. 6, for any fixed subregion $a_{0}$ of the bulk, we canchoose the decomposition of the boundary into subregions $\{A_{k}\}$ such that$a_{0}$ lies outside the entanglement wedge of each $A_{k}$. Therefore, the algebra$\mathcal{A}$ of bulk operators supported on $a_{0}$ has the feature that erasureof each $A_{k}$ is correctable for the algebra $\mathcal{A}$. This completes thefirst step of the Harlow-Ooguri argument, showing that a bulk global symmetryoperator $U_{L}$ must be transversal in the sense we have assumed in thispaper—it factorizes as a tensor product of boundary operators, each of whichis supported on a correctable boundary subregion.Actually, so far we have ignored a subtlety in this argument associated with general covariance in the bulk Harlow and Ooguri (2018b). Operators acting in the bulk are not really strictly local; rather a bulk “local” operator is accompanied by gravitational dressing which connects it to the boundary. This dressing is needed in order to enforce invariance under bulk diffeomorphisms. Because the dressing extends to the boundary, it has support on at least one of the $a_{k}$, and its reconstructed counterpart has support on at least one boundary subregion. However, this complication does not invalidate the argument, because the dressing is purely gravitational, and is therefore oblivous to the global charge defined within the bulk subalgebra $\mathcal{A}$.

Now we come to the second part of the Harlow-Ooguri argument, which establishes that the operator $W_{k}$ supported on boundary subregion $A_{k}$ is actually a logical operator. In the holographic correspondence, the choice of code space is actually rather flexible. One possible procedure Almheiri et al. (2015) is to pick a set of local operators deep in the bulk, corresponding to highly nonlocal operators when reconstructed in the CFT. Then the code space is spanned by polynomials of bounded degree in these operators acting on the CFT vacuum state. The motivation for this choice is that each of the highly nonlocal CFT operators raises the energy of the CFT by only a small amount, hence producing only very weak back reaction on the bulk geometry. Logical operators are those that preserve this low-energy sector of the CFT, and Harlow and Ooguri assert that each operator $W_{k}$ can be chosen to have this property.Since $W_{k}$ preserves the code space, and is supported on the correctable boundary subregion $A_{k}$, it must act trivially on the code space. This assertion is affirmed if the code’s logical operators may be regarded as bulk operators which are supported in a bulk region which is outside the entanglement wedge of the $A_{k}$ (such as the region $a_{0}$ in Fig. 6), since in that case each logical operator can be reconstructed on the complementary boundary region $A_{k}^{c}$, where $W_{k}$ acts trivially. Therefore, since each $W_{k}$ is a trivial logical operator, we conclude that the global symmetry operator $U_{L}$ is the identity acting on the code space.In this argument, we assumed that subregion duality and complementary recovery are exact properties of the AdS/CFT code, and thus inferred that erasure of boundary region $A_{k}$ is exactly correctable.In fact, though, these properties of the code hold precisely only in the leading order of a systematic expansion in Newton’s gravitational constant $G_{N}$, and can be modified when corrections higher order in $G_{N}$ are included. Nevertheless, the conclusion that bulk global symmetries are disallowed continues to hold even when these higher-order corrections are taken into account, assuming the corrections are small. A nontrivial global symmetry operation (if one were allowed), acting on a bulk local operator $\phi$, should modify $\phi$ by an amount $\delta\phi$ which is $O(1)$, independent of $G_{N}$. But we have argued that $\delta\phi=0$ to leading order in $G_{N}$ (since exact correctability of $A_{k}$ holds to this order). Higher-order corrections might make an $O(G_{N})$ contribution to $\delta\phi$, but these small corrections do not suffice to restore the proper nontrivial action on $\phi$ of the putative global symmetry.Now we have found that exact bulk local symmetries cannot occur in AdS/CFT. Butwhat can we say about whether approximate discrete global symmetriesare allowed? As we’ve discussed, finite-dimensional quantum error-correctingcodes can have exact discrete symmetries, even though the AdS/CFT codedoes not. In this respect, discrete symmetries are essentially different thancontinuous symmetries, which are disallowed by the Eastin-Knill theorem for anyfinite-dimensional quantum code that can correct erasure of subsystems exactly.Therefore, we can’t expect to make general statements which are directlyanalogous to IV about limitations on approximatediscrete symmetries that apply to general codes.Nevertheless, it may be instructive to study further the properties of approximate quantum error-correcting codes which are approximately covariant with respect to a discrete symmetry. In the setting of AdS/CFT, it is of particular interest to consider the case where the local transformations $\{W_{k}\}$ in Eq. 92 are either precisely or approximately logical.

IX.2 Bulk time evolution

A natural symmetry arising in AdS/CFT is the time-translation invariance of theboundary CFT, which is governed by a local Hamiltonian. Time evolution in thebulk AdS space is a bit subtle because of the general covariance of the bulktheory, but if we fix the gauge by choosing a preferred sequence of bulk timeslices, then time evolution in the bulk corresponds to time evolution on theboundary. From the perspective of quantum error correction, this correspondenceis puzzling, because covariance of the AdS/CFT code with respect to timeevolution seems to be incompatible with perfect correctability of erasure on theboundary Preskill (2000); Hayden et al. (2017).Indeed, the analysis of bulk global symmetries in IX.1, whichis applicable to both discrete and continuous symmetries, builds on theobservation that a boundary global symmetry operator, when restricted to acorrectable boundary subregion, preserves the code space and therefore must be atrivial logical operator. Why can’t we apply similar reasoning to the action ofthe boundary Hamiltonian, concluding (incorrectly) that bulk time evolution istrivial?The answer hinges on a crucial distinction, emphasizedin Harlow and Ooguri (2018a, b), between globalsymmetry and long-range gauge symmetry in the bulk.As we’ve noted, a “local” operator in the bulk is not truly local; it requiresgravitational dressing connecting it to the boundary. For the analysis of bulkglobal symmetries, this dressing could be ignored, because the dressingtransforms trivially under the global symmetry. For the analysis of bulk timeevolution, the dressing cannot be ignored, because the dressing depends on theenergy-momentum of a bulk quantum state. It is the nontrivial action of theboundary Hamiltonian on the asymptotic gravitational dressing of bulk “local”operators which is responsible for the bulk time evolution. Furthermore, becausethe dressing can be detected by localized boundary observers, erasure ofboundary subregions can really be corrected only approximately rather thanexactly.Our IV clarifies the situation by quantifying theincompatibility between continuous symmetries and error correction. In theregime of sufficiently large physical subsystems, or for a large enough numberof subsystems, covariant codes can provide arbitrarily good protection againsterasure errors. The AdS/CFT setting fulfills both of these criteria. Theboundary theory is a field theory, which formally has an unbounded number oflocal physical subsystems. Furthermore, in the “large $N$” limit of the CFT,which corresponds to semiclassical gravity in the bulk, the Hilbert spacedimension of each local subsystem is very large Maldacena (1999).We note that holographic quantum codes, toy models of the bulk which capturesome of the properties of full blown AdS/CFT, have been constructed in whichlocal Hamiltonian evolution in the bulk is realized approximately by a localHamiltonian in the boundary theory Kohler and Cubitt (2018). However, althoughthese codes are approximately covariant, the boundary Hamiltonian is far fromuniform.

X Discussion

In this paper we have studied quantum error-correcting codes that are exactly orapproximately covariant with respect to a continuous symmetry group. A specialcase of our main result applies if the logical charge operator $T_{L}$ whichgenerates a continuous symmetry is a transversal logical operator ofthe code. This means that the logical system $L$ is encoded in a physical system$A$ which can be decomposed as a tensor product of physical subsystems $\{A_{i}\}$such that erasure of each $A_{i}$ is correctable, and that the physical symmetrygenerator $T_{A}$ is a sum $T_{A}=\sum_{i}T_{i}$, such that $T_{i}$ is a localcharge operator supported only on subsystem $A_{i}$.The Eastin-Knill theorem Eastin and Knill (2009); Zeng et al. (2011); Chen et al. (2008) asserts that no quantumerror-correcting code can be covariant with respect to a continuous symmetry ifthe number of physical subsystems is finite, each subsystem isfinite-dimensional, and erasure of each subsystem is exactlycorrectable. However, it was shown in Hayden et al. (2017) that thisconclusion can be evaded by infinite-dimensional codes. Our main objective herehas been to clarify the properties of covariant quantum codes in which thedimension of each physical subsystem is large but finite, and in which thenumber of subsystems is large but finite.In IV, we consider codes that can correcterasure of a subsystem only approximately, and we derive a lower boundon the worst case entanglement infidelity $\epsilon_{\mathrm{worst}}$ that canbe achieved by the best recovery map after an erasure. In keeping with thefindings of Hayden et al. (2017), this lower bound approaches zero whenthe number $n$ of subsystems approaches infinity, or when the fluctuations ofthe local charge of individual subsystems grow without bound. The idea behindthe lower bound is that, if the number of subsystems and the local chargefluctuations are both finite, then some information about the value of theglobal logical charge is available to an adversary who takes possession of asingle physical subsystem, resulting in irreversible decoherence of the logicalstate. In Theorem IV, we extend the result by relaxingthe assumptions. This more general theorem applies when the code is not exactlycovariant, when the logical charge operator is not exactly transversal, and whenmore than one subsystem is erased.While originally derived in the context of fault-tolerant quantum computing, theEastin/Knill theorem has a variety of other applications, for example to quantumreference frames and quantumclocks Preskill (2000); Hayden et al. (2017) (cf.also recent related work Woods and Alhambra (2019)), and to the holographicdictionary relating bulk and boundary physics in the AdS/CFTcorrespondence Hayden et al. (2017). When applied to these settings,our results provide limitations on transmission of reference frames over noisychannels, and help to clarify the relationship between bulk and boundary timeevolution for the AdS/CFT quantum code. Our lower bounds on infidelity alsoapply to the recently discovered quantum codes arising in one-dimensionaltranslation-invariant spin chains Brandão et al. (2017).Our main result hinges on an interplay between the noise model and the structureof the local charge observables. Specifically, Section IVapplies under the following condition: For any term $T_{\alpha}$ that appears inthe physical charge $T_{A}=\sum_{\alpha}T_{\alpha}$, there is a nonzero probabilitythat all physical subsystems supporting $T_{\alpha}$ aresimultaneously lost to the environment. One may wonder whether thiscondition is really necessary—e.g., would a code with a $2$-localcharge operator be allowed if it could correct only a single erasure? It turnsout that such codes do exist, showing that our condition is necessary.As a simple example, the erasure of a single qubit is correctable for the$[[4,2,2]]$ quantum code, but there is also a nontrivial logical operator$Q=X\otimes X\otimes I\otimes I$ supported on the first twoqubits Gottesman (2016). We can exponentiate this 2-localoperator to generate a logical rotation of the first logical qubit. Thisprovides an example of a code that is exactly error-correcting against a singlelocated erasure and that is nevertheless exactly covariant with respect to atwo-local charge.In the lower bound \tagform@2, the range$\Delta T_{L}$ of the logical charge operator and corresponding range $\Delta T_{i}$of the physical charge are not directly related to the corresponding systemdimensions if the symmetry is abelian. The situation is different when we applyour bound \tagform@27 to codes that are covariant with respect tothe full unitary group $\mathit{U}(d_{L})$. In that case, there is a minimal subsystemdimension for each value of $\Delta T_{i}$, and VIItherefore follows from IV.While IV and IV pertain to correction of erasure errors, similar conclusions should apply for more general errors. For dephasing errors in particular, the information leaked to the environment can be explicitly characterized in accord with recent results Bény et al. (2018); Noh et al. .Our work builds on Ref. Hayden et al. (2017), where covariant quantum codes arose in the study of reference frames; i.e., asymmetric states which convey “physical”information Bartlett et al. (2007); Kitaev et al. (2004). As shown in Hayden et al. (2017), exact error correction of reference frames is impossible for finite-dimensional systems, yet in the real world reference frames are always finite-dimensional and communication channels are always imperfect. Nevertheless, in practice we routinely share reference frames over noisy channels, easily reaching agreement about which direction is “up” or what time it is; furthermore quantum technologists can distribute entanglement between nodes of a quantum network, which is possible only if the nodes share a common phase reference. Our results clarify, quantitatively, why accurate communication of reference information is achievable in practice. A quantum reference frame of sufficiently high dimensionality becomes effectively classical, quite robust against the ravages of environment noise. Examples of such systems include highly excited oscillators and rotors, Bose-Einstein condensates, superconductors, and other macroscopic phases of quantum matter.In metrology, quantum error correction provides a promising tool forimproving sensitivity by protecting a probe system against a noisyenvironment Preskill (2000); Kessler et al. (2014); Arrad et al. (2014); Dür et al. (2014); Ozeri (2013). However, there is a delicatebalance to achieve between error-correcting against the noise while stillbeing sensitive to the physical observable $H$ one wishes to measure. In orderto correct against errors, one needs to encode in an appropriatecodespace. Furthermore, in order to measure $H$, it needs to act nontriviallywithin that codespace. The ability to measure $H$ directly by localobservations corresponds, in the language of this paper, to covariance of thecode with respect to the physical charge $H$. In other words, adapting oursetup to one from quantum metrology is straightforward: the goal now is toestimate the continuous parameter $\omega$ in $H=\omega T_{A}$ as accurately aspossible while at the same time being able to correct against relevantnoise. Recent efforts have determined that it is possible to measure $H$ atthe Heisenberg limit using an error-correcting code if $H$ is not a sum of theoperators characterizing the correctablenoise Zhou et al. (2018); Demkowicz-Dobrzański et al. (2017); Layden and Cappellaro (2018); Gorecki et al. (2019). But if the physical charge $T_{A}$ is a sum of local charges,the Eastin-Knilltheorem poses a challengeto the application of error-correcting techniques;namely, we cannot measure what we can correct. The infinite dimensionalcounterexamples of Hayden et al. (2017) show that it is nonethelesspossible to correct against local noise and admit a charge that is asum of such noise operators, granted one has non-normalizable codewords. Thebounds and example codes of this paper provide a quantitative version of thisinfinite-dimensional limit.Our results suggest the possibility that one could sacrifice some errorcorrection precision to achieve better sensitivity with physical, i.e.normalizable, states (cf. also Woods and Alhambra (2019)). However, toproperly apply our results to quantum metrology, there are some additionalsteps that need to be taken, which is the subject of ongoing follow-up work.First, since we are trying to measure an unknown parameter (and not necessarily to protect quantum information per se), we should account for the fact that a code is only required toreconstruct a logical state that would yield a precise reading of said parameter.Second, our results are stated in terms of the worst-case entanglement fidelity,but for applications to metrology one would prefer different figures of merit, such as the precisionat which the probe can sense magnetic fields, or the ability of a quantum clock to tell time accurately.Finally, it would be desirable to consider noise models that are more relevant toquantum metrology, such as fluctuating background magnetic fields that inducedephasing errors.Bény’s characterization of approximate quantum error correction ofalgebras Bény (2009) provides a promising tool for addressing these challenges because one can specify preciselywhich observables need to be faithfully reproduced after action by the noise and a possible recovery operation.Approximate quantum error-correcting codes also arise naturally in many-bodyquantumsystems Brandão et al. (2017); Gschwendtner et al. (2019). Weanticipate that constraints on correlation functions of many-body quantum statescan be derived from the covariance properties of the corresponding codes.Finally, the interplay of symmetry and quantum error correction has a prominentrole in the AdS/CFT holographic correspondence. Although covariance withrespect to a continuous symmetry is incompatibile with perfect correctability oferasure of physical subsystems for any finite-dimensional quantum code,nevertheless we expect that in the AdS/CFT code continuous time evolution of theboundary system corresponds to continuous time evolution of the encoded logicalbulk system. Our results relieve the tension between these two observations,because near perfect correctability can be achieved if either the number ofphysical subsystems, or the dimension of each physical subsystem, becomes verylarge. Both these provisos apply to the continuum limit of a regulatedholographic boundary conformal field theory, as the number of lattice sites perunit volume is very large in this limit, and the number of degrees of freedomper site is also very large if semiclassical gravity accurately describes thebulk geometry (the “large-$N$ limit”).Recent results indicate that not just exact continuous symmetries, but alsoexact discrete symmetries, are incompatible with the quantum error correctionproperties of the AdS/CFTcode Harlow and Ooguri (2018a, b). Anintriguing topic for further research will be investigation of approximatesymmetries, both continuous and discrete, in the context of quantum gravity.

Acknowledgements.

During the preparation of this work, the authors became aware of anindependent effort by Álvaro Alhambra and Mischa Woods to analyze how wellthe Eastin-Knill theorem can be evaded by allowing for a small recoveryerror Woods and Alhambra (2019). We thank them for collegially agreeing tosynchronize our arXiv posts.The authors also thankCédric Bény,Fernando Brandão,Elizabeth Crosson,Steve Flammia,Daniel Harlow,Liang Jiang,Tomas Jochym-O’Connor,Iman Marvian,Hirosi Ooguri,Burak Şahinoğlu, andMichael Walterfor discussions.PhF acknowledges support from the Swiss National Science Foundation (SNSF)through the Early PostDoc.Mobility fellowship No. P2EZP2_165239 hosted by theInstitute for Quantum Information and Matter (IQIM) at Caltech, from the IQIMwhich is a National Science Foundation (NSF) Physics Frontiers Center (NSF GrantPHY-1733907), and from the Department of Energy (DOE) Award DE-SC0018407.VVA acknowledges support from the Walter Burke Institute for Theoretical Physics at Caltech.GS acknowledges support from the IQIM at Caltech, and the Stanford Institute for Theoretical Physics.PH acknowledges support from CIFAR, DOD and the Simons Foundation.JP acknowledges support from ARO, DOE, IARPA, NSF, and the Simons Foundation.Some of this work was done during the 2017 program on “Quantum Physics ofInformation” at the Kavli Institute for Theoretical Physics (NSF GrantPHY-1748958).

SUPPLEMENTAL MATERIAL

List of Appendices

@starttoc

appendixtoc

Appendix A Proof of our bounds for a covariant code

The proof of IV is split into two lemmas. A firstlemma deduces that the environment has access to the logical charge, to a goodapproximation.

Lemma 6**.**

Under the assumptions of IV, and following thelatter’s notation, there exists an observable $Z_{C^{\prime}E}$ satisfying

[TABLE]

where $\nu^{\prime}=\nu+\sum(t_{\alpha}^{-}+t_{\alpha}^{+})/2$ and where thecomplementary channel $\widehat{\mathcal{N}\circ\mathcal{E}}{}_{L\to C^{\prime}EF}$ tothe combined encoding and noise is givenby \tagform@16.

• Proof of Appendix A.  Let $\Pi_{\alpha}=\mathds{1}-\Pi_{\alpha}^{\perp}$ be the projector which projectsonto the eigenspaces of $T_{\alpha}$ whose corresponding eigenvalues are in therange $[{t_{\alpha}^{-}\mathclose{},\mathopen{}t_{\alpha}^{+}}]$. Recall that$t_{\alpha}=(t_{\alpha}^{-}+t_{\alpha}^{+})/2$ is the midpoint of the interval$[{t_{\alpha}^{-}\mathclose{},\mathopen{}t_{\alpha}^{+}}]$.Define the observables$\tilde{T}_{\alpha}=\Pi_{\alpha}(T_{\alpha}-t_{\alpha}\mathds{1})$, and observe that$\tilde{T}_{\alpha}$ has eigenvalues between $-\Delta T_{\alpha}/2$ and$+\Delta T_{\alpha}/2$, and hence$\lVert{\tilde{T}_{\alpha}}\rVert_{\infty}\leqslant\Delta T_{\alpha}/2$. Define theobservable

[TABLE]

Then, for any logical state $\sigma_{L}$, and writing$\rho_{A}=\mathcal{E}(\sigma_{L})$,

[TABLE]

thus

[TABLE]

Noting that$\operatorname{tr}\bBigg@{1.2}({T_{A}\mathcal{E}(\sigma_{L})}\bBigg@{1.2})-\sum t_{\alpha}=\operatorname{tr}\bBigg@{1.2}({\bBigg@{1.2}[{\mathcal{E}^{\dagger}(T_{A})-\sum t_{\alpha}\mathds{1}}\bBigg@{1.2}]\,\sigma_{L}}\bBigg@{1.2})$, wehave

[TABLE]

where we have usedcondition \tagform@29. Recall thatfor any Hermitian operator $X$, we have$\lVert{X}\rVert_{\infty}=\max_{\sigma}\lvert{\operatorname{tr}(X\sigma)}\rvert$ with an optimization overall density matrices $\sigma$. Since \tagform@.6 holds forall $\sigma_{L}$, we have

[TABLE]

Using the approximate charge conservation condition$\lVert{(T_{L}-\nu\mathds{1})-\mathcal{E}^{\dagger}(T_{A})}\rVert_{\infty}\leqslant\delta$and the triangle inequality for the infinity norm, we finally obtain

[TABLE]

setting $\nu^{\prime}=\nu+\sum t_{\alpha}$.Since the infinity norm picks out the largest eigenvalue in absolute value, wesee from \tagform@.3 that$\lVert{Z_{C^{\prime}E}}\rVert_{\infty}=\max_{\alpha}q_{\alpha}^{-1}\lVert{\tilde{T}_{\alpha}}\rVert_{\infty}\leqslant\max_{\alpha}q_{\alpha}^{-1}\Delta{T}_{\alpha}/2$.∎

The second part of the proof of IV is to deduce fromthe environment’s access to the global charge that the code performs poorlywith respect to the various entanglement fidelity measures.We phrase this statement as a more general lemma that applies in fact to anynoise model, and can be used to bound the fixed-input entanglement fidelity forany given fixed input state $\lvert{\phi}\rangle_{LR}$, as long as the environment hasaccess to an observable which yields some information about the logical state.In analogy with $\epsilon_{\mathrm{e}}$ and $\epsilon_{\mathrm{worst}}$, wedefine for any $\lvert{\phi}\rangle_{LR}$ and for any channel $\mathcal{N}^{\prime}$,

[TABLE]

This lemma can be seen as a refinement of Bény’s characterization ofapproximate error correction using operatoralgebras Bény (2009). To formulate the lemma, we define twoauxiliary quantities that depend on a state $\sigma$ and an observable $T$:

[TABLE]

where in the second line the optimization ranges over all $\mu\in\mathbb{R}$.Intuitively, both these quantities $C_{\sigma,T}$ pick up the average chargeabsolute value (where $T$ is the charge and according to the state $\sigma$), upto a constant charge offset $\mu$ or $\operatorname{tr}(T\sigma)$. Special cases of thesequantities will be discussed in the proof of IV.

Lemma 7**.**

Let $(\mathcal{N}\circ\mathcal{E})_{L\to A^{\prime}}$ be the combined encoding andnoise channel with total output system(s) $A^{\prime}$, where both encoding and noisechannels may be any completely positive, trace-preserving maps. Let$\widehat{\mathcal{N}\circ\mathcal{E}}_{L\to E^{\prime}}$ be a complementary channelwith combined output system(s) $E^{\prime}$. (In the context ofIV, we set $A^{\prime}=A\otimes C$ and$E^{\prime}=E\otimes C^{\prime}\otimes F$, but this lemma holds more generally.) Suppose thatthere exists observables $T_{L}$ and $Z_{E^{\prime}}$ on the input and environmentsystems respectively, as well as $\nu^{\prime}\in\mathbb{R}$, $\delta^{\prime}\geqslant 0$,such that:

[TABLE]

Then, for any $\lvert{\phi}\rangle_{LR}$, both$\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})$ and$\epsilon_{\lvert{\phi}\rangle}(\mathcal{N}\circ\mathcal{E})$ are lower bounded by twodifferent independent bounds:

[TABLE]

Finally, if$\widehat{\mathcal{N}\circ\mathcal{E}}(\cdot)=\sum q_{\alpha}\,\widehat{\mathcal{N}_{\alpha}\circ\mathcal{E}}(\cdot)$ for a probabilitydistribution $\{{q_{\alpha}}\}$ and a set of noise channels$\{{\mathcal{N}_{\alpha}}\}$, then for any $\lvert{\phi}\rangle_{LR}$, the same bounds applyto the average of the individual error parameters corresponding to eacherasure event:

[TABLE]

In summary: There are two figures of merit we are interested in,$\epsilon_{\lvert{\phi}\rangle}(\mathcal{N}\circ\mathcal{E})$ and$\bBigg@{1.2}\langle{\epsilon_{\lvert{\phi}\rangle}(\mathcal{N}_{\alpha}\circ\mathcal{E})}\bBigg@{1.2}\rangle_{\alpha}$,and both are bounded from below by the same bound expressed in terms of theauxiliary quantities \tagform@.10aand \tagform@.10b.

• Proof of Appendix A.  We start by showing the following two statements: For any $\lvert{\phi}\rangle_{LR}$, andfor any state $\zeta_{E^{\prime}}$, it holds that

[TABLE]

where $\rho_{E^{\prime}}=\widehat{\mathcal{N}\circ\mathcal{E}}(\phi_{L})$.We recall the following expressions for the one-norm of any Hermitian operator $A$:

[TABLE]

where the first optimization ranges over operators Hermitian $X$, and thesecond over positive semidefinite operators $\Delta_{\pm}$.We start form the left-hand sideof \tagform@.16. By choosing a candidate$X$ in \tagform@.18a of the form$(Z/\lVert{Z}\rVert_{\infty})\otimes X^{\prime}$ with $\lVert{X^{\prime}}\rVert_{\infty}\leqslant 1$, then forany $\lvert{\phi}\rangle_{LR}$ and for any $\zeta_{E^{\prime}}$ we have that

[TABLE]

where the optimization ranges over Hermitian operators $X_{R}^{\prime}$ on the $R$system. Making use of the main assumption of this lemma, and restricting theoptimization to $X_{R}^{\prime}$ such that $\operatorname{tr}(X_{R}^{\prime}\phi_{R})=0$ yields

[TABLE]

using the fact that if $\lVert{A-B}\rVert_{\infty}\leqslant\delta^{\prime}$, then$\operatorname{tr}(AY)\geqslant\operatorname{tr}(BY)-\delta^{\prime}\operatorname{tr}(Y)$ for any Hermitian $A,B$ andpositive semidefinite $Y$, and that furthermore here$\operatorname{tr}(Y)=\operatorname{tr}({X^{\prime}_{R}\phi_{LR}})\leqslant\operatorname{tr}({\phi_{LR}})=1$. Without loss ofgenerality, we may assume that $R\simeq L$ (if $R$ is smaller, then embed ittrivially in a larger system of same dimension as $L$; if $R$ is larger, thenremove unused dimensions on which $\phi_{R}$ has no support, noting that thesupport of $\phi_{R}$ may not exceed the dimension of $L$). Let$\left\{{\lvert{k}\rangle_{L}}\right\}$, $\left\{{\lvert{k}\rangle_{R}}\right\}$ be Schmidt bases of $L$ and $R$corresponding to $\lvert{\phi}\rangle_{LR}$, and recall that we have the relations$\lvert{\phi}\rangle_{LR}=\phi_{L}^{1/2}\,\lvert{\Phi}\rangle_{L:R}=\phi_{R}^{1/2}\,\lvert{\Phi}\rangle_{L:R}$, where$\lvert{\Phi}\rangle_{L:R}=\sum\lvert{k}\rangle_{L}\otimes\lvert{k}\rangle_{R}$ and where as before$\phi_{L}=\operatorname{tr}_{R}(\phi_{LR})$ and $\phi_{R}=\operatorname{tr}_{L}(\phi_{LR})$. Note that forany operator $X^{\prime}_{R}$, we have $X^{\prime}_{R}\lvert{\Phi}\rangle_{L:R}=X_{L}\lvert{\Phi}\rangle_{L:R}$ where$X_{L}$ is related to $X^{\prime}_{R}$ by a transpose with respect to the bases used todefine $\lvert{\Phi}\rangle_{L:R}$, which implies also$\lVert{X_{L}}\rVert_{\infty}=\lVert{X_{R}}\rVert_{\infty}$. Consequently,$\operatorname{tr}_{R}(X^{\prime}_{R}\phi_{LR})=\operatorname{tr}_{R}(X^{\prime}_{R}\,\phi_{L}^{1/2}\,\Phi_{L:R}\,\phi_{L}^{1/2})=\phi_{L}^{1/2}\,X_{L}\,\phi_{L}^{1/2}$. Finally, note that$\operatorname{tr}(X_{R}^{\prime}\phi_{R})=\operatorname{tr}(X_{R}^{\prime}\phi_{LR})=\operatorname{tr}(\phi_{L}^{1/2}\,X_{L}\,\phi_{L}^{1/2})=\operatorname{tr}(X_{L}\phi_{L})$. So we obtain

[TABLE]

The optimization \tagform@.21 is a semidefinite program,and we proceed to compute its dual program Watrous (2009). In termsof the variables $X_{L}=X_{L}^{\dagger}$, $A,B\geqslant 0$, and $\mu\in\mathbb{R}$,and writing for short $T_{L}^{\prime}=T_{L}-\nu^{\prime}\mathds{1}_{L}$, we have

[TABLE]

Strong duality holds because of Slater’s conditions. Indeed $X_{L}=0$ isstrictly feasible in the primal problem; the dual is actually also strictlyfeasible by choosing (say) $\mu=0$ and $A$ and $B$ to be the positive andnegative parts respectively of the Hermitian operator$\phi_{L}^{1/2}T_{L}^{\prime}\phi_{L}^{1/2}$ plus a constant times the identity.For fixed $\mu$ in \tagform@.22h, we recognize the dualsemidefinite program for the one-norm of a Hermitianmatrix \tagform@.18b, and hence weactually obtain the same expression asin \tagform@.10b,

[TABLE]

Then

[TABLE]

noting that the constant shift $\nu^{\prime}\mathds{1}_{L}$ can be absorbed into theoptimization over $\mu$. Thisproves \tagform@.16.Now we show \tagform@.17.Similarly to how we started above, we write

[TABLE]

Define $Z_{L}=\widehat{\mathcal{N}\circ\mathcal{E}}{}^{\dagger}(Z_{E^{\prime}})$, andusing the same procedure to define $\lvert{\Phi}\rangle_{L:R}$ as above with $X_{L}$ inone-to-one correspondence with $X_{R}^{\prime}$ via the transpose operation and with$\operatorname{tr}(X_{L}\phi_{L})=\operatorname{tr}(X_{R}\phi_{R})$, we obtain

[TABLE]

By assumption, we have$Z_{L}=\widehat{\mathcal{N}\circ\mathcal{E}}{}^{\dagger}(Z_{E^{\prime}})=T^{\prime}_{L}+\Delta_{L}$ with $T_{L}^{\prime}=T_{L}-\nu^{\prime}\mathds{1}_{L}$ and$\lVert{\Delta_{L}}\rVert_{\infty}\leqslant\delta^{\prime}$, so this implies that

[TABLE]

where in the last line we use the fact that$\operatorname{tr}(T^{\prime}_{L}\phi_{L})=\operatorname{tr}(T_{L}\phi_{L})-\nu^{\prime}$. Thisproves \tagform@.17.Now, following Bény and Oreshkov Bény and Oreshkov (2010), we have theduality also for a fixed input state, and there exists a state $\zeta_{E^{\prime}}$such that333The statement with fixed input state is only briefly stated towardsthe end of their paper, as that claim is in fact easier to prove than theirmain theorem for the worst-case entanglement fidelity.

[TABLE]

and thus

[TABLE]

where $P(\sigma,\rho)=\sqrt{1-F^{2}(\sigma,\rho)}$ denotes the “purifieddistance” or “root infidelity” between the twostates Tomamichel et al. (2010); Tomamichel (2012). Now, using knowninequalities between this distance measure and the tracedistance Tomamichel et al. (2010), we have

[TABLE]

which in combination with \tagform@.16proves \tagform@.12a. The first partof \tagform@.12a trivially follows from the fact that$\epsilon_{\mathrm{worst}}(\cdot)=\max_{\lvert{\phi}\rangle}\epsilon_{\lvert{\phi}\rangle}(\cdot)$.From \tagform@.29, and using the fact that the purified distancecannot increase under partial trace, we find with$\rho_{E^{\prime}}=\widehat{\mathcal{N}\circ\mathcal{E}}(\phi_{L})$,

[TABLE]

By triangle inequality, and using again the known inequality between tracedistance and purified distance, we obtain

[TABLE]

Combining this with \tagform@.17proves \tagform@.12b.Now we further assume that$\widehat{\mathcal{N}\circ\mathcal{E}}=\sum q_{\alpha}\,\widehat{\mathcal{N}_{\alpha}\circ\mathcal{E}}$ for some set of $\alpha$’s and aprobability distribution $\{{q_{\alpha}}\}$. Then as above, invoking Bény andOreshkov for each $\alpha$ with corresponding optimal states $\zeta_{E^{\prime}}^{\alpha}$,we have

[TABLE]

using the joint convexity of the trace distance and defining$\zeta^{\prime}_{E^{\prime}}=\sum q_{\alpha}\zeta_{E^{\prime}}^{\alpha}$. Directlyinvoking \tagform@.16 then proves thefirst bound in \tagform@.15. We also have$\text{\hyperref@@ii[eq:ioiugrehbkjhois]{{\tagform@{\ref*{eq:ioiugrehbkjhois}}}}}\geqslant\delta\left({\rho_{E^{\prime}},\zeta^{\prime}_{E^{\prime}}}\right)$, and hence by triangle inequality

[TABLE]

Combining with \tagform@.17 thenyields the second bound in \tagform@.15.∎

We may now combine these two lemmas to finally proveIV.

• Proof of section IV.  Thanks to Appendix A thereexists $Z_{C^{\prime}E}$ and $\nu^{\prime}\in\mathbb{R}$ such that

[TABLE]

We may directly plug this observable intoAppendix A to deduce thatthe bound \tagform@.12a applies to our approximatelycovariant code. We now need to compute the form of the bound for theparticular quantities $\epsilon_{\mathrm{e}}(\mathcal{N}\circ\mathcal{E})$,$\langle{\epsilon_{\mathrm{e}}(\mathcal{N}^{\alpha}\circ\mathcal{E})}\rangle_{\alpha}$ and$\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})$.First, let $\lvert{\phi}\rangle_{LR}=\lvert{\hat{\phi}}\rangle_{LR}$ be the maximally entangledstate between $L$ and $R\simeq L$. Then by definition, and recalling thealternative expressionin \tagform@.23 for$C_{\phi_{L},T_{L}}$ with a maximization, we have

[TABLE]

Let $\mu(T_{L})$ denote a median eigenvalue of $T_{L}$ counted with multiplicity,which implies the following. Let $\{{\lvert{k}\rangle_{L}}\}$ for $k=1,\ldots,d_{L}$ be aneigenbasis of $T_{L}$ with its elements arranged such that the eigenvalues of$T_{L}$ are nonincreasing in $k$,$\langle{1}\hskip 0.86108pt|\hskip 0.86108pt{T_{L}}\hskip 0.86108pt|\hskip 0.86108pt{1}\rangle\geqslant\langle{2}\hskip 0.86108pt|\hskip 0.86108pt{T_{L}}\hskip 0.86108pt|\hskip 0.86108pt{2}\rangle\geqslant\cdots\geqslant\langle{d_{L}}\hskip 0.86108pt|\hskip 0.86108pt{T_{L}}\hskip 0.86108pt|\hskip 0.86108pt{d_{L}}\rangle$. Let

[TABLE]

noting that the two projectors are orthogonal and that$\operatorname{rank}(P_{+})=\operatorname{tr}(P_{+})=\operatorname{tr}(P_{-})=\operatorname{rank}(P_{-})$. That is, we divide all basisvectors into two sets of equal size, corresponding to the smallest eigenvaluesand the largest eigenvalues respectively, possibly leaving out the middlebasis vector if the space dimension is odd. Then, the eigenvaluescorresponding to the eigenbasis vectors included in $P_{+}$ (respectively,$P_{-}$) are all greater than or equal to (respectively less than or equal to)$\mu(T_{L})$. If $d_{L}$ is odd, then the basis vector that was left outcorresponds to the eigenvalue $\mu(T_{L})$.Now set $X=P_{+}-P_{-}$, satisfying $\lVert{X}\rVert_{\infty}\leqslant 1$. We have$\lVert{T_{L}-\mu(T_{L})\mathds{1}}\rVert_{1}=\operatorname{tr}\bBigg@{1.2}[{X(T_{L}-\mu(T_{L})\mathds{1})}\bBigg@{1.2}]$: Indeed,the one-norm is equal to the sum of the absolute values of the eigenvalues ofits argument, which is precisely taken care of by our careful choice of $X$.Then$\lVert{T_{L}-\mu(T_{L})\mathds{1}}\rVert_{1}=\operatorname{tr}\bBigg@{1.2}({XT_{L}}\bBigg@{1.2})-\mu(T_{L})\operatorname{tr}(X)=\operatorname{tr}\bBigg@{1.2}({XT_{L}}\bBigg@{1.2})$ because $\operatorname{tr}(X)=0$ by construction. Now because both $\mu(T_{L})$ and $X$are optimization candidates in \tagform@.38, we have

[TABLE]

which implies that$C_{\mathds{1}/d_{L},T_{L}}=d_{L}^{-1}\,\bBigg@{1.2}\lVert{T_{L}-\mu(T_{L})\mathds{1}}\bBigg@{1.2}\rVert_{1}$.Appendix A states that$\bBigg@{1.2}[{C_{\mathds{1}/d_{L},T_{L}}-(\delta+\eta)}\bBigg@{1.2}]/(2\lVert{Z_{C^{\prime}E}}\rVert_{\infty})$ is alower bound both to $\epsilon_{\mathrm{e}}(\mathcal{N}\circ\mathcal{E})$ andto $\langle{\epsilon_{\mathrm{e}}(\mathcal{N}^{\alpha}\circ\mathcal{E})}\rangle_{\alpha}$,which proves \tagform@30c as we recall theproperty \tagform@.37b.That the norm term in \tagform@30c can be replacedby $\lVert{T_{L}-\operatorname{tr}(T_{L})\mathds{1}_{L}/d_{L}}\rVert_{1}/(2d_{L})$ follows from the alternativebound in Appendix A, statingthat$\bBigg@{1.2}[{(C^{0}_{\mathds{1}/d_{L},T_{L}}/2)-(\delta+\eta)}\bBigg@{1.2}]/(2\lVert{Z_{C^{\prime}E}}\rVert_{\infty})$[cf. \tagform@.10a] is also a lower bound toboth error parameters considered in \tagform@30c.For $\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})$, we get to pick$\lvert{\phi}\rangle_{LR}$ freely and this will yield a valid bound. Let$\lvert{\psi^{\pm}}\rangle_{L}$ be eigenstates of $T_{L}$ corresponding to the maximal andminimal eigenvalues $T_{L}$, respectively, with$\langle{\psi^{+}}\hskip 0.86108pt|\hskip 0.86108pt{T_{L}}\hskip 0.86108pt|\hskip 0.86108pt{\psi^{+}}\rangle-\langle{\psi^{-}}\hskip 0.86108pt|\hskip 0.86108pt{T_{L}}\hskip 0.86108pt|\hskip 0.86108pt{\psi^{-}}\rangle=\Delta T_{L}$. Now choosetwo arbitrary orthogonal states $\lvert{\pm}\rangle_{R}$ on $R$ and set

[TABLE]

with $\phi_{L}=\Pi_{L}/2$, where we write$\Pi_{L}=\lvert{\psi^{+}}\rangle\langle{\psi^{+}}\rvert+\lvert{\psi^{-}}\rangle\langle{\psi^{-}}\rvert$. Recall the alternative expressionin \tagform@.23 for$C_{\phi_{L},T_{L}}$ with a maximization. We can choose as candidate$X_{L}=\lvert{\psi^{+}}\rangle\langle{\psi^{+}}\rvert_{L}-\lvert{\psi^{-}}\rangle\langle{\psi^{-}}\rvert_{L}$, since we have indeed$\operatorname{tr}({\phi_{L}X_{L}})=0$ and $\lVert{X_{L}}\rVert_{\infty}\leqslant 1$, and we obtain

[TABLE]

Appendix A then asserts that

[TABLE]

where we recall \tagform@.37b. Thisproves \tagform@30d.∎

At this point we comment onCondition \tagform@29 in the statementof IV. It may look a bit awkward, but its meaningis intuitively simple: First, we need to shift the charge values to center themat zero for each $\alpha$ for our proof. Second, we need to make sure that ifwe project any codeword into the given range of physical charge values for each$\alpha$, then the total error we make when attempting to determine theexpectation value of the actual (possibly unbounded) charge observable $T_{A}$ issmall. In practice, this just means that the part of the codewords outside ofthe given range of charge values only has a small contribution to the totalexpectation value of charge. For convenience we may use the followingsimplified criterion, where we simply fix a charge cut-off value $t$:

Proposition 8**.**

Consider $V_{L\to A}$, $T_{L}$, $K$ and $T_{A}=\sum_{\alpha\in K}T_{\alpha}$ as inIV. Let $t>0$. Set$t_{\alpha}^{+}=-t_{\alpha}^{-}=t$ and define $\Pi_{\alpha},\Pi_{\alpha}^{\perp}$ asin the statement of IV. Let$\{\lvert{\phi_{\alpha}^{t^{\prime},j}}\rangle\}$ be an eigenbasis of $T_{\alpha}$ correspondingto eigenvalues $t^{\prime}$ with a possible degeneracy index $j$.Suppose that there is an $\eta^{\prime}\geqslant 0$ such that for any logical state$\psi_{L}$ and for any $\alpha$,

[TABLE]

where we write $\rho_{\alpha}=\operatorname{tr}_{A\setminus A_{\alpha}}(V\psi_{L}V^{\dagger})$and where the sum ranges over the eigenstate labels $(t^{\prime},j)$ such that$\lvert{t^{\prime}}\rvert>t$.Then, condition \tagform@29 issatisfied with $\eta=\lvert{K}\rvert\,\eta^{\prime}$, and furthermore $\Delta T_{\alpha}=2t$for all $\alpha$.

• Proof of Appendix A.  We have $t_{\alpha}=(t_{\alpha}^{+}+t_{\alpha}^{-})/2=0$.For any $\psi_{L}$, calculate

[TABLE]

Note by the way that the left hand sideof \tagform@.44 is exactly$\operatorname{tr}(\Pi_{\alpha}^{\perp}\,\lvert{T_{\alpha}}\rvert V\psi_{L}V^{\dagger})$.∎

Appendix B Correlation functions and bounds

In this section we present an alternative strategy for proving thebound \tagform@28, by studying the connected correlation functionsbetween the physical subsystems and the logical information.The covariance of the codes can be seen as a linear constraint, which can beeasily employed to obtain a second order constraints. To start, we again assumethe simpler case of isometric encoding. We construct the state corresponding tothe encoding isometry $V_{L\rightarrow A}$ by injecting a maximally entangledstate $\lvert{\hat{\phi}}\rangle_{LR}$ to $V_{L\rightarrow A}$ (Fig. 7):

[TABLE]

We have$T_{A}\lvert{\Psi}\rangle_{LA}=T_{A}V\lvert{\hat{\phi}}\rangle_{LR}=V(T_{L}-\nu\mathds{1}_{L})\lvert{\hat{\phi}}\rangle_{LR}$ for some constant $\nu$. Define$T_{R}=(T_{L}-\nu\mathds{1}_{L})^{T}$ where the transpose is taken as a matrix ignoringthe Hilbert space label; this ensures that$(T_{L}-\nu\mathds{1}_{L})\lvert{\hat{\phi}}\rangle_{LR}=T_{R}\lvert{\hat{\phi}}\rangle_{LR}$. Therefore,the covariance of $V$ translates to the invariance of $\lvert{\Psi}\rangle$:

[TABLE]

We define the connected correlator between two operators $A,B$ as

[TABLE]

Consider an arbitrary operator $X_{R}$. It be seenfrom \tagform@.47 that

[TABLE]

Using the triangle inequality, we obtain

[TABLE]

Although the derivation of \tagform@.49 is very simple, itprovides a general lower bound to the amount of correlations between thereference system and the physical subsystems, from which we can draw physicalconsequences.The correlation functions measure how close the state $\Psi_{RA_{i}}$ is to theproduct state $\Psi_{R}\otimes\Psi_{A_{i}}$:

[TABLE]

where we used Hölder’s inequality. We can replace$T_{A_{i}}\to T_{A_{i}}-t\mathds{1}$ in \tagform@B without changing theleft hand side of the inequality as$\langle X_{R},T_{A_{i}}-t\mathds{1}\rangle=\langle X_{R},T_{A_{i}}\rangle$:

[TABLE]

where the second line follows by a suitable choice of $t$, and where$\Delta T_{A_{i}}$ is the difference between the maximal and minimal eigenvalue of$T_{A_{i}}$.The accuracy to which the code $V$ can correct against errors is preciselydetermined by how close $\Psi_{RA_{i}}$ is to a product state. Indeed, considerthe noise channel $\mathcal{N}^{i}_{A\to A}$in \tagform@10b that erases the system $A_{i}$. ByBény and Oreshkov \tagform@11a, we have

[TABLE]

where $\widehat{\mathcal{N}^{i}\circ\mathcal{E}}(\hat{\phi}_{LR})=\Psi_{RA_{i}}$ and$\Psi_{R}=\mathds{1}_{R}/d_{L}$, and where we have used the known relation$\delta(\cdot,\cdot)\leqslant\sqrt{1-F^{2}(\cdot,\cdot)}$ between the tracedistance and the fidelity. Because the trace distance cannot increase under thepartial trace, and if we set $\zeta_{A_{i}}$ to be the optimal state in theexpression above, also have$(1/2)\lVert{\Psi_{A_{i}}-\zeta_{A_{i}}}\rVert_{1}\leqslant\epsilon_{\mathrm{e}}(\mathcal{N}^{i}\circ\mathcal{E})$ and thus by triangleinequality,

[TABLE]

It remains to combine \tagform@.53with \tagform@B and \tagform@.49 and to choosethe best possible $X_{R}$ to get our final result.

Theorem 9**.**

The individual entanglement fidelities of recovery of a covariant code$\mathcal{E}(\cdot)=V(\cdot)V^{\dagger}$ against single erasures at knownlocations satisfy the following inequality:

[TABLE]

Furthermore, this can be used to show that

[TABLE]

Note that $T_{L}-\operatorname{tr}(T_{L})\mathds{1}/d_{L}$ is just a shift of $T_{L}$ by a multiple ofidentity to make it traceless. Therefore, $\lVert{T_{L}-\operatorname{tr}(T_{L})\mathds{1}/d_{L}}\rVert_{1}$is a 1-norm measure for the spread of eigenvalues of $T_{L}$. The bounds ofAppendix B and Eq. 28 have a very similar nature.

• Proof of Appendix B.  We start with the correlator in the left hand sideof \tagform@.49:

[TABLE]

Now, choose the optimal $X_{R}$ such that $\lVert{X_{R}}\rVert_{\infty}\leqslant 1$ andthat$\lVert{T_{R}-\operatorname{tr}(T_{R})\mathds{1}/d_{L}}\rVert_{1}=\operatorname{tr}\left[{X_{R}\left({T_{R}-\operatorname{tr}(T_{R})\mathds{1}/d_{L}}\right)}\right]$.Plugging into \tagform@.49, and combiningwith \tagform@.51and \tagform@.53, immediatelygives \tagform@.54.Furthermore from \tagform@.54 we have

[TABLE]

By convexity of $x\mapsto x^{2}$, and by Appendix G, wehave

[TABLE]

Combining \tagform@.58 with \tagform@.57proves \tagform@.55.∎

Appendix C Criterion for approximate codes

When we come up with a new code, how can we show that it forms an$\epsilon$-approximate error-correcting code against erasures at knownlocations?Here we provide a criterion that, when it can be applied, certifies that a givencode performs well.Let $L$ be the logical space and $A$ be the physical space, and consider anencoding operation $\mathcal{E}_{L\to A}$ that can be any completely positive,trace-preserving map. Note that in the case of a more general noise model, $A$does not necessarily have to be composed of several subsystems. Consider acollection of noise channels $\{\mathcal{N}^{\alpha}\}$ and probabilities$\{q_{\alpha}\}$. We assume that the environment applies a random noise channelfrom this set with the corresponding probability, while providing a record ofwhich noise channel was applied in a separate register $C$. The overall noisechannel that is applied by the environment is then

[TABLE]

Given complementary channels $\widehat{\mathcal{N}^{\alpha}\circ\mathcal{E}}$ of$\mathcal{N}^{\alpha}\circ\mathcal{E}$, we can construct a complementary channelof $\mathcal{N}\circ\mathcal{E}$ as

[TABLE]

with an additional register $C^{\prime}$ and where the outputs of the individualcomplementary channels for each $\alpha$ are embedded into a system $E$.We fix any basis $\{\lvert{x}\rangle_{L}\}$ of $L$, and we define for each $\alpha$ theoperators

[TABLE]

Note that $\rho_{\alpha}^{x,x}$ is a quantum state for each $\alpha$ and for each$x$, but that $\rho_{\alpha}^{x,x^{\prime}}$ is not necessarily even Hermitian for $x\neq x^{\prime}$.For an isometric encoding $\mathcal{E}$, and in the noise $\mathcal{N}$acts by erasing a collection of subsystems labeled by $\alpha$ and chosen withprobability $q_{\alpha}$, the operators $\rho_{\alpha}^{x,x^{\prime}}$ are simply thereduced operators on the sites labeled by $\alpha$ of the logical operator$\lvert{x}\rangle\langle{x^{\prime}}\rvert$:

[TABLE]

Proposition 10**.**

Assume that there exists $\nu,\epsilon^{\prime}\geqslant 0$, and that there exists aquantum state $\zeta_{\alpha}$ for each $\alpha$, such that for all $\alpha$,

[TABLE]

Then $\mathcal{E}_{L\to A}$ is an approximate error-correcting code againstthe noise $\mathcal{N}$, with approximation parameter

[TABLE]

where $d_{L}$ is the dimension of the logical system $L$.

• Proof of Appendix C.  Let

[TABLE]

Using the Bény-Oreshkov property \tagform@11, the proofstrategy is to find a lower bound to the entanglement fidelity of the channel$\widehat{\mathcal{N}\circ\mathcal{E}}$ to the constant channel$\mathcal{T}_{\zeta}$ outputting the state $\zeta_{C^{\prime}E}$ defined above.Consider a reference system $R\simeq L$, and let $\{\lvert{x}\rangle_{R}\}$ be any fixed basisof $R$. Let $\lvert{\Phi}\rangle_{L:R}=\sum_{x}\lvert{x}\rangle_{L}\otimes\lvert{x}\rangle_{R}$.For any state $\lvert{\sigma}\rangle_{LR}$, there exists a complex matrix $B_{R}$ suchthat $\lvert{\sigma}\rangle_{LR}=B_{R}\,\lvert{\Phi}\rangle_{L:R}$ and$\sigma_{R}=\operatorname{tr}_{L}(\sigma_{LR})=B_{R}B_{R}^{\dagger}$ (choose$B_{R}=\sum_{x,x^{\prime}}\langle{x,x^{\prime}}\hskip 0.86108pt|\hskip 0.86108pt{\sigma}\rangle_{LR}\lvert{x^{\prime}}\rangle\langle{x}\rvert_{R}$). Note that$\lVert{B_{R}B_{R}^{\dagger}}\rVert_{\infty}=\lVert{B_{R}^{\dagger}B_{R}}\rVert_{\infty}\leqslant 1$.We have

[TABLE]

where we have defined for each $\alpha$ the positive semidefinite operator

[TABLE]

While the $\rho_{ER}^{\alpha}$’s are positive semidefinite, they are notnormalized to unit trace as proper quantum states. Recalling that thefidelity is jointly concave, we have

[TABLE]

At this point, we define for each $\alpha$ the positive semidefiniteoperator

[TABLE]

Note that $B_{R}\tilde{\rho}_{ER}^{\alpha}B_{R}^{\dagger}$ is a quantum state, because$\operatorname{tr}(B_{R}\tilde{\rho}_{ER}^{\alpha}B_{R}^{\dagger})=\sum_{x}\operatorname{tr}(B_{R}\lvert{x}\rangle\langle{x}\rvert B_{R}^{\dagger})=\operatorname{tr}(B_{R}B_{R}^{\dagger})=1$. In fact, the quantum states$B_{R}\tilde{\rho}_{ER}^{\alpha}B_{R}^{\dagger}$ and $B_{R}\rho_{ER}^{\alpha}B_{R}^{\dagger}$are close in trace distance:

[TABLE]

using ourassumption \tagform@.63b, andnoting that$\lVert{B_{R}\,\lvert{x}\rangle\langle{x^{\prime}}\rvert\,B_{R}^{\dagger}}\rVert_{1}\leqslant\lVert{B_{R}\lvert{x}\rangle}\rVert_{1}\,\lVert{\langle{x^{\prime}}\rvert B_{R}^{\dagger}}\rVert_{1}=\lVert{\langle{x}\rvert B_{R}^{\dagger}}\rVert_{1}\,\lVert{\langle{x^{\prime}}\rvert B_{R}^{\dagger}}\rVert_{1}\leqslant 1$ because$\operatorname{tr}\sqrt{\langle{x}\hskip 0.86108pt|\hskip 0.86108pt{B_{R}^{\dagger}B_{R}}\hskip 0.86108pt|\hskip 0.86108pt{x}\rangle}\leqslant 1$. Recalling therelation$P(\cdot,\cdot)\leqslant\sqrt{2\delta(\cdot,\cdot)}=\sqrt{\lVert{(\cdot)-(\cdot)}\rVert_{1}}$ between the purified distance$P(\cdot,\cdot)=\sqrt{1-F^{2}(\cdot,\cdot)}$ and the trace distance, we have

[TABLE]

On the other hand, using again the joint concavity of the fidelity, we have

[TABLE]

recalling ourassumption \tagform@.63a andusing the fact that $\operatorname{tr}(B^{\dagger}B)=\operatorname{tr}(BB^{\dagger})=1$; hence

[TABLE]

By triangle inequality for the purified distance, we have

[TABLE]

Returning to \tagform@.68, we now have$F\bBigg@{1.2}({B_{R}\rho^{\alpha}_{BR}B_{R}^{\dagger},\zeta_{\alpha}\otimes\sigma_{R}}\bBigg@{1.2})\geqslant\sqrt{1-(d_{L}\sqrt{\nu}+\epsilon^{\prime})^{2}}$ and hence

[TABLE]

As this holds for any $\lvert{\sigma}\rangle_{LR}$, we deduce that

[TABLE]

which implies

[TABLE]

Appendix D Calculations for covariant code examples

.1 Three-rotor secret-sharing code

Sharp cutoff.

We complete the exposition in the main text inVI.1 by calculating the approximationparameter $\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E}^{(m)})$ of theconstructed code.The strategy is to apply Appendix C. First write the operators \tagform@.61 in our situation,

[TABLE]

We need to show that $\rho_{i}^{x,x}$ is approximately constant of $x$ and that$\rho_{i}^{x,x^{\prime}}$ is very small for $x\neq x^{\prime}$. The latter condition turns out tobe simple: for any $i$ and for any $x\neq x^{\prime}$, we will see that$\rho_{i}^{x,x^{\prime}}=0$; hence we may take $\nu=0$ inAppendix C.For each $i$, we would like to show that there exists a state $\zeta_{i}$ suchthat $\rho_{i}^{x,x}$ is close to $\zeta_{i}$ in fidelity distance for each $x$. Wechoose to work with the trace distance instead, and deduce that the states areclose in fidelity using the relation$F(\cdot,\cdot)\geqslant\sqrt{1-2\delta(\cdot,\cdot)}$ between the fidelityand the trace distance. We bound the trace distance as follows. For each $i$,we find a positive semidefinite operator $\tau_{i}$ with the property tht$\rho_{i}^{x,x}\geqslant\tau_{i}$ for all $x$. This implies that$\rho_{i}^{x,x}=\tau_{i}+\Delta_{i}^{x}$ for some positive semidefinite operators$\Delta_{i}^{x}$ with $\operatorname{tr}(\Delta_{i}^{x})=1-\operatorname{tr}(\tau_{i})$. Define$\zeta_{i}=\tau_{i}+\xi_{i}$, for any freely chosen $\xi_{i}\geqslant 0$ with$\operatorname{tr}(\xi_{i})=1-\operatorname{tr}(\tau_{i})$. Then, we have$\rho_{i}^{x,x}-\zeta_{i}=\Delta_{i}^{x}-\xi_{i}$, and$\delta(\rho_{i}^{x,x},\zeta_{i})=(1/2)\lVert{\rho_{i}^{x,x}-\zeta_{i}}\rVert_{1}\leqslant(1/2)(\operatorname{tr}(\Delta_{i}^{x})+\operatorname{tr}(\xi_{i}))=1-\operatorname{tr}(\tau_{i})$. To summarize: If wefind, for each $i$, an operator $\tau_{i}\geqslant 0$ with$\rho_{i}^{x,x}\geqslant\tau_{i}$ for all $x$, then we can deduce that there arestates $\zeta_{i}$ such that

[TABLE]

where $\epsilon^{\prime}=\min_{i}\sqrt{2(1-\operatorname{tr}(\tau_{i}))}$.We may calculate the corresponding operators $\rho_{i}^{x,x^{\prime}}$, starting with$i=1$:

[TABLE]

since the two Kronecker deltas force $x^{\prime}=x$ and $y^{\prime}=y$.Similarly, we have

[TABLE]

First of all, for each of $i=1,2,3$ we have that $\rho_{i}^{x,x^{\prime}}=0$ if $x\neq x^{\prime}$.Then, we have that $\rho_{1}^{x,x}$ is already independent of $x$, so we maychoose $\tau_{1}=\rho_{1}^{1,1}=\rho_{1}^{x,x}\ \forall\;x$.Next, $\rho_{2}^{x,x}$ is diagonal, with constant diagonal elements $1/(2m+1)$ atstates ${-m-x},{-m-x+1},\ldots,{m-x}$. We may thus choose

[TABLE]

such that $\rho_{2}^{x,x}\geqslant\tau_{2}$ for all $x$ (Fig. 8).

Finally, $\rho_{3}^{x,x}$ is also diagonal with elements $1/(2m+1)$ at states$-2m+2x,-2m+2x+2,\ldots,2m+2x$. Similarly we may choose

[TABLE]

which guarantees that $\rho_{3}^{x,x}\geqslant\tau_{3}$ for each $x$. We have

[TABLE]

so we may set according to the above $\epsilon^{\prime}=\sqrt{4h/(2m+1)}$. Accordingto Appendix C, the code $V^{(m)}_{L\to A}$ is anapproximate quantum error-correcting code with

[TABLE]

We have $1/(2m+1)\approx 1/2m$, and to first order in $h/m$, we have

[TABLE]

So our codes become good in the limit $h/m\to 0$.To compare with our bound \tagform@27, we choose$q_{1}=q_{2}=q_{3}=1/3$ and note that $\delta=0$, $\eta=0$, and $\Delta T_{L}=2h$.Also, we have

[TABLE]

so, for $m\gg h$, we have$\max_{i}q_{i}^{-1}\Delta T_{i}\approx 18m$. Our bound then reads

[TABLE]

Smooth cutoff.

Again, we make use ofAppendix C.First, we compute the normalization factor as

[TABLE]

where $\vartheta_{3}(z,q)$ is Jacobi’s theta function.444See DLMF:http://dlmf.nist.gov/20. Our notation follows DLMF’s notation. Astraightforward observation is that $c_{w}\geqslant 1$ (the term $y=0$ in the sumis already equal to one).We need to determine the operators $\rho_{1,2,3}^{x,x^{\prime}}$. We have

[TABLE]

Similarly, for the second and third systems,

[TABLE]

Hence, we have$\lVert{\rho_{1}^{x,x^{\prime}}}\rVert_{1}=\lVert{\rho_{2}^{x,x^{\prime}}}\rVert_{1}=\lVert{\rho_{3}^{x,x^{\prime}}}\rVert_{1}=0$for all $x\neq x^{\prime}$, so theconditions \tagform@.63b aresatisfied with $\nu=0$.Now we need to verify theconditions \tagform@.63a. Forthe first system, $\rho_{1}^{x,x}$ doesn’t depend on $x$, so choosing$\zeta_{1}=\rho_{1}^{0,0}$ we have $P(\rho_{1}^{x,x},\zeta_{1})=0$ for all $x$.For the second system, we choose $\zeta_{2}=\rho_{2}^{0,0}$ and calculate

[TABLE]

where the calculation of the last inequality is carried out below in.1. Hence

[TABLE]

Now, we look at the third system. Defining $\zeta_{3}=\rho_{3}^{0,0}$, we have

[TABLE]

invoking again the calculation in.1. Hence

[TABLE]

We are now in position to apply our criterion.Appendix C tells us that

[TABLE]

Hence, our code’s performance scales as $(1/2)(h/w)$. For instance, it performswell in the limit $h/w\to 0$, for instance in the limit $w\to\infty$ with aconstant $h$.Let’s now see how our bound applies to our code (we need the more general bound,because we are dealing with infinite-dimensional systems with an unboundedcharge observable). We need to cut off tails of the codeword states on thephysical systems to make the range of charge values finite.Choose cut-offs $W_{1},W_{2},W_{3}\geqslant 0$ for each physical system. We wouldlike to compute an upper bound to$\sum\psi_{x}\psi_{x^{\prime}}^{*}\operatorname{tr}(\Pi_{i}^{\perp}\,\rho_{i}^{x,x^{\prime}})=\sum\lvert{\psi_{x}}\rvert^{2}\operatorname{tr}(\Pi_{i}^{\perp}\,\rho_{i}^{x,x})$, where $\Pi_{i}^{\perp}$ projects outside of thecut-off region. We have

[TABLE]

where the bound is calculated in.1 below. Then,

[TABLE]

Similarly,

[TABLE]

Hence, choosing $W_{1}=W_{2}=W_{3}=:W$ with $W\geqslant 2h$ and choosing forsimplicity $W$ as a multiple of $6$, we have$\lfloor W_{1}/3\rfloor=W/3\geqslant(1/3)(W-2h)$, as well as$W_{2}-h\geqslant W-2h$ and also $\lfloor W_{3}/2\rfloor-h=(1/2)(W-2h)$;furthermore $W-2h\geqslant(1/2)(W-2h)\geqslant(1/3)(W-2h)$. Then,

[TABLE]

recalling that $c_{w}\geqslant 1$. Also, $\Delta T_{i}=2W_{i}=2W$ byconstruction. Furthermore $\Delta T_{L}=2h$ and $\delta=0$. So, our boundreads (assuming that the noise erasure probabilities are $q_{1}=q_{2}=q_{3}=1/3$)

[TABLE]

considering the regime $W\gg h$, i.e., $W-2h\approx W$. Now, if we choose thecutoff $W=\beta w$ to be proportional to $w$, then we can write our bound as afunction of $h/w$:

[TABLE]

The second term is exponentially suppressed in $\beta$; so choosing $\beta$ onlyvery moderately large, we get a bound which is effectively proportional to $h/w$with a proportionality constant $1/(6\beta)$.Now we find a suitable $\beta$ to pluginto \tagform@.103 to get a bound in termsof $h/w$ only. If we attempt to minimize thebound \tagform@.103, we get as minimizationcondition

[TABLE]

Writing $z=\beta^{2}/4$ (i.e., $\beta=2\sqrt{z}$) we obtain$h/(6w)=e^{-z/2}\,(4+2z)/(\sqrt{z})$; the square of this equation gives

[TABLE]

To render this equation tractable, and since we only have to come up with anapproximate educated guess for $\beta$, we may simplify this equation by keepingthe leading term, expecting that $z$ should be moderately large, yielding

[TABLE]

The solution to the equation $x^{2}=ze^{-z}$ is given by the Lambert Wfunction555https://dlmf.nist.gov/4.13 with $z=-W({-x^{2}})$. Usingthe expansion of the negative branch $W_{\mathrm{m}}$ of the function near$z\to-\infty$, we have666https://dlmf.nist.gov/4.13.E11$-W_{\mathrm{m}}(-x^{2})\approx\ln(1/x^{2})$, and hence we may select$z\approx\ln\bBigg@{1.2}({(12w/h)^{2}}\bBigg@{1.2})=2\ln(12w/h)$. This in turn yields the educatedguess $\beta=2\sqrt{2\ln(12w/h)}$ to pluginto \tagform@.103, and the bound becomes

[TABLE]

using $\sqrt{\ln(12w/h)}=\sqrt{\ln(w/h)+\ln(12)}\approx\sqrt{\ln(w/h)}$.

Lemma 11**.**

We have for integer $x,h$, with $\lvert{x}\rvert\leqslant h$ and with $h$ even,

[TABLE]

• Proof of subsection .1.  First, we may assume without loss of generality that we have the “$+$” casein the exponent (or else simply send $x\to-x$).Completing the square, we have$(y+x)^{2}+y^{2}=2y^{2}+2xy+x^{2}=2(y+x/2)^{2}+x^{2}/2$, and hence

[TABLE]

At this point we need to distinguish the case where $x$ is even from the casewhere $x$ is odd. Assuming first that $x$ is even, we may redefine $y\to y+x/2$in the summation and we have

[TABLE]

recalling that $\lvert{x}\rvert\leqslant h$. In the case that $x$ is odd, we need towork a little bit more; we may redefine $y\to y+(x-1)/2$, and we have

[TABLE]

using another theta function corresponding to this type of summation..1 shows that$\vartheta_{2}\bBigg@{1.2}({0,e^{-1/(2w^{2})}}\bBigg@{1.2})\geqslant e^{-1/(8w^{2})}\,\vartheta_{3}\bBigg@{1.2}({0,e^{-1/(2w^{2})}}\bBigg@{1.2})$, and so we have

[TABLE]

where we have assumed that $h$ is even, and so $\lvert{x}\rvert+1\leqslant h$.∎

Lemma 12**.**

We have, for $W\geqslant 0$,

[TABLE]

• Proof of subsection .1.  Assume $x\geqslant 0$, or else redefine $x\to-x$. We have

[TABLE]

where the integral is necessarily an overestimation of the sum, as the sum canbe seen as an integral of a step function, where each step is specified at theright edge by the value of the integrand function; this step function liesbeneath the actual decreasing function $e^{-y^{2}/(2w^{2})}$. Setting$t=y/(w\sqrt{2})$,

[TABLE]

We use the known bound777See for instancehttp://dlmf.nist.gov/7.8.E4 orhttp://mathworld.wolfram.com/Erfc.html

[TABLE]

leading to

[TABLE]

Finally, we prove a property of the theta functions that we used above.

Lemma 13**.**

Let $0<q\leqslant 1$, and let $z\in\mathbb{C}$ with $\operatorname{Re}(z)=0$ and$\operatorname{Im}(z)\geqslant 0$. Then

[TABLE]

Furthermore, we have

[TABLE]

• Proof of subsection .1.  We start by proving \tagform@.117. Writing$q=e^{i\pi\tau}$ with $\operatorname{Re}(\tau)=0$ and $\operatorname{Im}(\tau)\geqslant 0$, wehave888See http://dlmf.nist.gov/20.5.E7, Eq. (20.5.7)

[TABLE]

We will show that the product is greater than $1$, by showing that$a_{n}\geqslant 1$ for each $n$. We have

[TABLE]

defining $a,b\geqslant 0$ as $a=(n-1/2)\,\pi\,\operatorname{Im}(\tau)$ and $b=\operatorname{Im}(z)$.Since $\cos\left({i\,\varphi}\right)=\cosh\left({\varphi}\right)$, we have

[TABLE]

(By the way, this is another way of seeing that $\vartheta_{3}(z,q)$ must bereal and positive, since all the $a_{n}$ are real positive and$\vartheta_{3}(0,q)$ is real positive as given by its series representation.Recall that $z$ is pure imaginary with $\operatorname{Im}(z)\geqslant 0$, and that$0<q\leqslant 1$.) With the usual properties of the hyperbolic functions, wehave

[TABLE]

using $\tanh(a)\leqslant 1$ and $1/\cosh^{2}(b)=1-\tanh^{2}(b)$. Hence finally,$a_{n}\geqslant 1$. This proves \tagform@.117.To prove \tagform@.118, we invoke the followingproperty of the theta functions,999Seehttp://dlmf.nist.gov/20.2.E12, Eq. (20.2.12) valid for any$q=e^{i\pi\tau}$,

[TABLE]

For $0<q\leqslant 1$, necessarily $\tau$ is pure imaginary with$\operatorname{Im}(\tau)\geqslant 0$; we may thusinvoke \tagform@.117, whichproves \tagform@.118.∎

.2 Five-rotor perfect code

The normalized encoding for this code is

[TABLE]

where $c_{w,x}$ is the normalization and $T^{(\infty)}$ is defined in Eq. 54.

Single erasure.

We first calculate $\rho_{\ell}^{x,x}$ for $\ell\in\{1,2,3,4,5\}$and then outline why $||\rho_{\ell}^{x,x^{\prime}\neq x}||_{1}=O(e^{-cw^{2}})$.By the cyclic permutation symmetry of the code, we only have to calculate$\rho_{1}^{x,x}$. Performing the partial trace and simplifying allKronecker delta functions leaves us with the diagonal reduced densitymatrix

[TABLE]

Now we apply the Poisson summation formula,

[TABLE]

to each of the three sums above. Typically, the $n=0$ term on the right-hand-side is dominant (i.e., the leading order contribution in the large-$w$ limit), and taking only this term is equivalent to approximating the sum with a Gaussian integral. Each of the remainingterms suppressed as $O(e^{-cw^{2}})$, where $c$ is a positive constantincreasing with $n$. Because $c$ increases with $n$, the $n+1$-thterm is subleading with respect to the $n$th term. Thus, the entiresum of exponentially suppressed terms can itself be bounded by anexponential (e.g., $e^{-2x}+e^{-3x}<e^{-x}$ for $x>1$). We omitthese corrections and focus on the dominant term $k=l=m=0$ after having applied Poisson summation to Eq. .125:

[TABLE]

Forcing $\text{Tr}\{\rho_{1}^{x,x}\}=1$ and once again approximatingthe resulting sum with an integral solves for the normalization $c_{w,x}$in the large $w$ limit. Plugging that back into the above equationand simplifying produces

[TABLE]

Now we calculate the fidelity of the above state to $\rho_{1}^{0,0}$.Using the fact that the states commute with each other, taking thesquare root of each entry in the resulting diagonal matrix, and applyingPoisson summation yields

[TABLE]

Plugging this into the infidelity yields the result \tagform@55.Returning to the $x\neq x^{\prime}$ case, we show why those casesdo not significantly contribute. The reduced density matrix is ofthe form

[TABLE]

where $\gamma_{j,k,l,m}^{x,x^{\prime}}$ is a product of a Gaussian in thevariables $k,l,m$ (just like the $x=x^{\prime}$ case above) and a phase$\propto 2\pi\Phi$ (which goes away when $x=x^{\prime}$). We first apply Poissonsummation to the internal three sums and evaluate the normalizations in thelarge-$w$ limit. In this case, the centers of the Gaussians in the $k,l,m$-sumdepend on $\Phi$ and the dominant term on the right-hand-side ofEq. .126 may not longer be the center-of-mass term $n=0$. We willhowever set $\Phi$ to be an irrational number close to zero from now on, i.e.,taking $\Phi\ll 1$ while making sure that $\Phi w\rightarrow\infty$. Thismakes sure that the center-of-mass mode is dominant. Writing the norm andapplying Poisson summation to the remaining sum reveals

[TABLE]

We see that the one-norm is exponentially suppressed in $w^{2}$ for the off-diagonal (i.e., $x\neq x^{\prime}$) reduced matrices.

Two erasures.

We first calculate $\rho_{\ell,\ell^{\prime}}^{x,x}$ for $\ell,\ell^{\prime}\in\{1,2,3,4,5\}$and then argue that $||\rho_{\ell,\ell^{\prime}}^{x,x^{\prime}\neq x}||_{1}=O(e^{-cw^{2}})$.Due to the cyclic permutation symmetry, we only need to calculate$\rho_{1,2}^{x,x}$ and $\rho_{1,3}^{x,x}$. Performing the partialtrace, simplifying the Kronecker delta functions, plugging in thenormalization, and applying Poisson summation yields

[TABLE]

In other words, both $\rho_{1,2}^{x,x}$ and $\rho_{1,3}^{x,x}$ are identical inthe large $w$ limit. Note that, unlike $\rho_{1}^{x,x}$, these matrices haveoff-diagonal elements that are exponentially suppressed in $w^{2}$. These elementshave been ignored above, but we mention them in the $x\neq x^{\prime}$ casebelow. Taking the fidelity between $\rho_{\ell,\ell^{\prime}}^{x,x}$ and$\rho_{\ell,\ell^{\prime}}^{0,0}$ as before yields the result$F^{2}(\rho_{1,2}^{x,x},\rho_{1,2}^{0,0})\geqslant e^{-\frac{h^{2}}{60w^{2}}}$ claimed in \tagform@56.The $x\neq x^{\prime}$ case is more difficult this time because theunapproximated reduced density matrix no longer has just one nonzero diagonal. Without any approximations, it is

[TABLE]

Applying Poisson summation to the internal two sums for $x\neq x^{\prime}$ reveals that all matrix elements are exponentially suppressed with $w^{2}$,

[TABLE]

However, there are particular values of $(j,j^{\prime})$ for which the function in the exponent above is minimized; we select those and show that the trace norm is exponentially suppressed in $w^{2}$. For even $x-x^{\prime}$, the band at $j^{\prime}=j-\frac{x-x^{\prime}}{2}$ decays the slowest. Ignoring all other bands and calculating the trace norm yields

[TABLE]

For odd $x-x^{\prime}$, there are two bands $j^{\prime}=j-\frac{x-x^{\prime}\pm 1}{2}$ whose entries decay the slowest. Calculating the square root of $\rho_{1,2}^{x,x^{\prime}}\rho_{1,2}^{x,x^{\prime}\dagger}$ is more difficult since the resulting matrix is tri-diagonal. However, ignoring the off-diagonal entries, taking the square root, and bounding the resulting integral still yields exponential scaling with $w^{2}$.

.3 Thermodynamic codes

Here, we carry out the calculations that are relevant forVI.3 of the main text.The operators $\rho_{d}^{m,m}$ (reduced states on $d$ consecutive sites) areprovided as:

[TABLE]

with

[TABLE]

The fidelity between two states which commute reduces to the Bhattacharyyacoefficient (the classical version of the fidelity):

[TABLE]

The complicated calculation is deferred to.3 below, which gives us:

[TABLE]

This, in turn, tells us that

[TABLE]

The “logical off-diagonal” terms $\rho_{d}^{m,m^{\prime}}$ for $m\neq m^{\prime}$ are exactlyzero, because we made sure to space out the codewords in magnetization by$2d+1$, following the construction of ref. Brandão et al. (2017).Hence, applying Appendix C, we see that our codeis an AQECC against the erasure of $d$ consecutive sites, with

[TABLE]

This matches exactly the scaling of our bound \tagform@27.

Lemma 14**.**

There exists a constant $D_{d,m}$ of $N$ such that (for constant $d,m$):

[TABLE]

• Proof of subsection .3.  We use Stirling’s formula up to order $1/N^{2}$:

[TABLE]

(noting that there is in fact no term of order $1/N^{2}$). Now, for any $x$,ignoring terms of order $O(N^{-3})$, we have:

[TABLE]

Using the expansions

[TABLE]

one continues, still keeping all the terms up to order $1/N^{2}$:

[TABLE]

Now we may apply this to calculate $\ln\left({\sqrt{K_{r,d,m}^{N}K_{r,d,0}^{N}}}\right)$,using the fact that$\ln(N-d)=\ln(N)+\ln(1-d/N)=\ln(N)-d/N-d^{2}/(2N^{2})+O(N^{-3})$:

[TABLE]

with

[TABLE]

Using $0\leqslant\lvert{r}\rvert\leqslant d$, write

[TABLE]

Then,

[TABLE]

Recall the identities

[TABLE]

We have$\exp\left\{{A_{m,r}/N}\right\}=1+r(m-r)/(2N)+r^{2}(m-r)^{2}/(8N^{2})+O(N^{-3})\geqslant{1+r(m-r)/(2N)+O(N^{-3})}$. Replacing the summation index $r$ by$k=(d+r)/2=0,1,\ldots,d$, we calculate

[TABLE]

and then

[TABLE]

Finally, plugging into \tagform@.146 gives us

[TABLE]

so we may define $D_{d,m}=C_{d,m}/4+d^{2}/8$, proving the claim.∎

Appendix E Proof of the approximate Eastin-Knill theorem

.1 Equivalence of the existence of a universal transversal gate setand the $\mathit{U}(d_{L})$-covariance property of the code

First, we show that the setting of the Eastin-Knill theorem is equivalent tostudying the $\mathit{U}(d_{L})$-covariance property of the corresponding code.More precisely, we show that given a code $V_{L\to A}$, if there exists amapping $u$ of logical unitaries $U_{L}$ to transversal physical unitaries$u(U_{L})=U_{1}\otimes\cdots\otimes U_{n}$ satisfying $V^{\dagger}u(U_{L})V=U_{L}$ forall $U_{L}$, where $u$ does not even have to be continuous, then the code$V_{L\to A}$ is necessarily covariant with respect to the full unitary group onthe logical space.The statement is pretty intuitive, because given any rule that maps logicalunitaries to physical transversal unitaries, we can compose the physicalunitaries corresponding to different logical unitaries, and presumably generatea bona fide representation by starting from a minimal generating set ofunitaries. This intuition proves correct, though it is not immediately clear ifthe mapping generated in this way is continuous. Here we provide a derivationthat smooths out these technical details.

Proposition 15**.**

Let $V_{L\to A}$ be any code, with $A=A_{1}\otimes\cdots\otimes A_{n}$. Supposethat for each unitary $U_{L}$ on $L$ there exists a transversal unitary$U_{A}=u(U_{L})=u_{1}(U_{L})\otimes\cdots\otimes u_{n}(U_{L})$ such that$V^{\dagger}u(U_{L})V=U_{L}$ for all $U_{L}$. Then there exists a mapping $u^{\prime}$that maps any $U_{L}$ to a transversal physical unitary$u^{\prime}(U_{L})=u_{1}^{\prime}(U_{L})\otimes\cdots\otimes u_{n}^{\prime}(U_{L})$ such that

• •

$u^{\prime}$* is continuous;*

• •

for all $U_{L}$, $V^{\dagger}u^{\prime}(U_{L})V=V^{\dagger}u(U_{L})V=U_{L}$; and

• •

for any $U_{L},U_{L}^{\prime}$, we have$u^{\prime}(U_{L}U_{L}^{\prime})=u^{\prime}(U_{L})\,u^{\prime}(U_{L}^{\prime})$.

• Proof of subsection .1.  Observe first that for all $U_{L}$, because $u(U_{L})$ implements a logicalunitary, it must fix the code space $\Pi=VV^{\dagger}$. Hence, we mustnecessarily have $[u(U_{L}),VV^{\dagger}]=0$ for all $U_{L}$.Let $T_{L}^{(j)}$ be a basis of the Lie algebra $\mathfrak{u}(d_{L})$ of $\mathit{U}(d_{L})$.Let $(\vartheta_{k})$ be a sequence of positive reals converging to zero, andfor each $j$, consider the sequence of transversal physical unitaries$\bBigg@{1.2}({u\bBigg@{1.2}({e^{-i\vartheta_{k}T_{L}^{(j)}}}\bBigg@{1.2})}\bBigg@{1.2})_{k}$. Let $\delta>0$. Since thesequence of unitaries is supported on a compact set, it admits a convergentsubsequence and hence, there exist $\vartheta^{\prime},\vartheta^{\prime\prime}$ such that$\lvert{\vartheta^{\prime}-\vartheta^{\prime\prime}}\rvert\leqslant\delta$ and$\bBigg@{1.2}\lVert{u\bBigg@{1.2}({e^{-i\vartheta^{\prime}T_{L}^{(j)}}}\bBigg@{1.2})-u\bBigg@{1.2}({e^{-i\vartheta^{\prime\prime}T_{L}^{(j)}}}\bBigg@{1.2})}\bBigg@{1.2}\rVert_{\infty}\leqslant\delta$. We define$\tilde{U}^{(j)}=u^{\dagger}\bBigg@{1.2}({e^{-i\vartheta^{\prime\prime}T_{L}^{(j)}}}\bBigg@{1.2})\,u\bBigg@{1.2}({e^{-i\vartheta^{\prime}T_{L}^{(j)}}}\bBigg@{1.2})$,which then satisfies

[TABLE]

We also have that$V^{\dagger}\,\tilde{U}^{(j)}V=V^{\dagger}\,u^{\dagger}\bBigg@{1.2}({e^{-i\vartheta^{\prime\prime}T_{L}^{(j)}}}\bBigg@{1.2})\,VV^{\dagger}\,u\bBigg@{1.2}({e^{-i\vartheta^{\prime}T_{L}^{(j)}}}\bBigg@{1.2})V=e^{i\vartheta^{\prime\prime}T_{L}^{(j)}}e^{-i\vartheta^{\prime}T_{L}^{(j)}}=e^{-i\alpha{}T_{L}^{(j)}}$, with $\alpha=\vartheta^{\prime}-\vartheta^{\prime\prime}$, where werecall that $[u(U_{L}),VV^{\dagger}]=0$ for any $U_{L}$. Define

[TABLE]

where $\chi^{(j)}_{i}$ is chosen such that there exists an eigenvector$\lvert{\chi^{(j)}_{i}}\rangle$ of $\tilde{U}_{i}^{(j)}$ with eigenvalue exactly equal toone. Choosing $\chi^{(j)}$ in $[{-\pi\mathclose{},\mathopen{}\pi}[$ such that$\chi^{(j)}=\sum_{i}\chi_{i}^{(j)}~{}\mathrm{mod}~{}2\pi$, we then have$\tilde{U}_{1}^{(j)}\otimes\cdots\otimes\tilde{U}_{n}^{(j)}=e^{-i\chi^{(j)}}\,\tilde{U}^{(j)}$.Recall that for any operator $X$, we have$\lVert{X}\rVert_{\infty}=\max_{\lvert{\phi}\rangle,\lvert{\psi}\rangle}\operatorname{Re}\operatorname{tr}\{{\lvert{\psi}\rangle\langle{\phi}\rvert\,X}\}$ wherethe optimization ranges over vectors satisfying$\lVert{\lvert{\phi}\rangle}\rVert,\lVert{\lvert{\psi}\rangle}\rVert_{\infty}\leqslant 1$.We have using \tagform@.149 and for a suitablychosen phase $e^{-i\xi}$ that

[TABLE]

where the phase $e^{-i\xi}$ is chosen such that$e^{-i\xi}\bBigg@{1.2}({e^{i\chi^{(j)}}-1}\bBigg@{1.2})$ is real positive. This implies that$\delta\geqslant\operatorname{Im}\{{e^{i\chi^{(j)}}}\}$ and$\operatorname{Re}\{{e^{i\chi^{(j)}}}\}\geqslant 1-\delta$, which implies in turn$\lvert{\chi^{(j)}}\rvert\leqslant\arcsin(\delta)\leqslant 2\delta$. This alsoimplies that$\bBigg@{1.2}\lVert{e^{-i\chi^{(j)}}\tilde{U}^{(j)}-\tilde{U}^{(j)}}\bBigg@{1.2}\rVert_{\infty}=\bBigg@{1.2}\lvert{e^{-i\chi^{(j)}}-1}\bBigg@{1.2}\rvert\bBigg@{1.2}\lVert{\tilde{U}^{(j)}}\bBigg@{1.2}\rVert_{\infty}\leqslant\delta$, and hence by triangle inequality

[TABLE]

Similarly, for each $i$, we have$\bBigg@{1.2}\lVert{\tilde{U}_{i}^{(j)}-\mathds{1}}\bBigg@{1.2}\rVert_{\infty}=\operatorname{Re}\operatorname{tr}\bBigg@{1.2}[{\lvert{\psi_{i}}\rangle\langle{\phi_{i}}\rvert\bBigg@{1.2}({\tilde{U}_{i}^{(j)}-\mathds{1}}\bBigg@{1.2})}\bBigg@{1.2}]$ for some$\lvert{\psi_{i}}\rangle$,$\lvert{\phi_{i}}\rangle$, and hence

[TABLE]

This implies that all eigenvalues of $\tilde{U}_{i}^{(j)}$ are $\delta$-close toone, and hence the corresponding phases are all close to zero; more precisely,every eigenvalue $e^{i\gamma}$ of $\tilde{U}_{i}^{(j)}$ satisfies$2\delta\geqslant\lvert{e^{i\gamma}-1}\rvert$; by the same reasoning as above, thestatements $2\delta\geqslant\operatorname{Im}\{{e^{i\gamma}}\}$ and$\operatorname{Re}\{{e^{i\gamma}}\}\geqslant 1-2\delta$ imply that$\lvert{\gamma}\rvert\leqslant\arcsin(2\delta)\leqslant 4\delta$, and hence

[TABLE]

Now we set $T_{A}^{(j)}=\sum_{i}T_{i}^{(j)}-({\chi^{(j)}/\alpha})\,\mathds{1}$,which is a sum of local terms. We then have$e^{-i\alpha T_{A}^{(j)}}=e^{i\chi^{(j)}}\,e^{-i\alpha{}T_{1}^{(j)}}\otimes\cdots\otimes e^{-i\alpha{}T_{n}^{(j)}}=e^{i\chi^{(j)}}\,\tilde{U}_{1}^{(j)}\otimes\cdots\otimes\tilde{U}_{n}^{(j)}=\tilde{U}^{(j)}$,with also $\bBigg@{1.2}\lVert{\alpha T_{A}^{(j)}}\bBigg@{1.2}\rVert_{\infty}\leqslant 4n\delta+2\delta$.For any $U_{L}$, the unitary $u(U_{L})$ commutes with the code space $\Pi$, andtherefore $[\tilde{U}^{(j)},\Pi]=0$. Furthermore, since$e^{-i\alpha T_{A}^{(j)}}$ and $T_{A}^{(j)}$ share the same eigenspaces, we alsohave that $[T_{A}^{(j)},\Pi]=[T_{A}^{(j)},VV^{\dagger}]=0$.We thus have$e^{-i\alpha{}T_{L}^{(j)}}=V^{\dagger}\tilde{U}^{(j)}V=V^{\dagger}e^{-i\alpha T_{A}^{(j)}}V=e^{-i\alpha V^{\dagger}T_{A}^{(j)}V}$, and thusby taking the logarithm, we obtain $V^{\dagger}T_{A}^{(j)}V=T_{L}^{(j)}$, whereno ambiguities arise from taking the logarithm since the operators$\alpha T_{A}^{(j)}$ and $\alpha T_{L}^{(j)}$ have small norm, controlled by asuitably small choice of $\delta$, and hence do not straddle the branch cut.We can then define the mapping $u^{\prime}(U_{L})$ as the Lie group representation of$\mathit{U}(d_{L})$ generated by the operators $T_{A}^{(j)}$ that span the correspondingLie algebra. More explicitly, for any $U_{L}$, we may write$i\log(U_{L})=\sum c_{j}T_{L}^{(j)}$ for some unique set of real coefficients$c_{j}=c_{j}(U_{L})$ (up the zero-measure set of unitaries that have an eigenvaluethat coincides with the logarithm branch cut), and we set

[TABLE]

where $c_{j}$ are the unique coefficients of the expansion of $i\log(U_{L})$ interms of the $T_{L}^{(j)}$ as defined above. We have that $u^{\prime}(U_{L})$ istransversal because each $T_{A}^{(j)}$ is a sum of local terms. Then we alsohave for any $U_{L}$ that$V^{\dagger}u^{\prime}(U_{L})V=V^{\dagger}\exp\bBigg@{1.2}({-i\sum c_{j}(U_{L})\,T_{A}^{(j)}}\bBigg@{1.2})V=\exp\bBigg@{1.2}({-i\sum c_{j}(U_{L})\,V^{\dagger}T_{A}^{(j)}V}\bBigg@{1.2})=\exp\bBigg@{1.2}({-i\sum c_{j}(U_{L})\,T_{L}^{(j)}}\bBigg@{1.2})=U_{L}$. Because $u^{\prime}$ is a Lie group representation, itis continuous and compatible with the group structure.∎

.2 Proof of the approximate Eastin-Knill bound

Recall that each irrep of $U(d_{L})$ is represented by a Young diagram$\lambda$, where $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{d_{L}})$, and$\lambda_{1}\geqslant\lambda_{2}\geqslant\cdots\geqslant\lambda_{d_{L}}=0$, and$\lambda_{i}\in\mathbb{Z}$. The dimension of each irrep is given by the Weyl dimension formula, which forthe $\mathit{U}(d_{L})$ group is equal to the Schur polynomial $S_{\lambda}$ evaluated atthe vector $(1,1,\cdots,1)$. More explicitly, it can be evaluated to

[TABLE]

To derive VII we need to first prove few intermediateresults. First, we will prove a bound on $D_{\lambda}$, based on $\lambda_{1}$.

Lemma 16**.**

The symmetric representation has the minimal dimension among therepresentations with fixed $\lambda_{1}$. More precisely, the followinginequality holds,

[TABLE]

where $\lambda$ is a representation of $\mathit{U}(d_{L})$ and$D_{\text{Sym}^{\lambda_{1}}}$ is the dimension of the symmetric representationwith the Young diagram $\lambda=(\lambda_{1},0,0,\cdots,0)$.

• Proof of subsection .2.  Suppose that $\lambda_{1}=l$. We use the dimension formulaEq. .156. Consider the logarithm of the dimension, which is (up to afixed constant) equal to:

[TABLE]

Note that we fix $\lambda_{1}=l$ and $\lambda_{d_{L}}=0$, so they do not appear asparameters of $f$. Also, the vector$\hat{\lambda}=(\lambda_{2},\cdots,\lambda_{d_{L}-1})$ is an integer vector inthe simplex $\Delta$ with $d_{L}$ extremal points $\hat{v}_{i}\in\mathbb{R}^{d_{L}-1}$,where $\hat{v}_{0}=(0,0,\cdots,0)$, $\hat{v}_{1}=(l,0,\cdots,0)$,$\cdots$, and$\hat{v}_{d_{L}-1}=(l,l,\cdots,l)$.We first extend the function $f$ to all of the real points in $\Delta$, andshow that $f$ is a concave function inside $\Delta$. This would show that theminimum of $f$ is attained at one of its extremal points.A direct computation of the Hessian of $f$, reveals that for$2\leqslant r,s\leqslant d_{L}-1$,

[TABLE]

where $K_{rs}=1/(\lambda_{s}-\lambda_{r}+r-s)^{2}$. One can see that if$w=\sum_{2\leqslant i\leqslant d_{L}-1}\alpha_{i}e_{i}$ is an arbitrary vector, then

[TABLE]

This is a negative number, and shows that $f$ is strictly concave. Therefore,the minimum of $f$ is attained on one of the extremal point $\hat{v}_{i}$,$0\leqslant i\leqslant d_{L}-1$. Using the Weyl-dimension formula we have,

[TABLE]

One can easily see that $f(\hat{v}_{i})$ is increasing for $i\leqslant(d_{L}-1)/2$ anddecreasing for $i\geqslant(d_{L}-1)/2$. Therefore, its minimum is attained at$f(\hat{v}_{0})=f(\hat{v}_{d_{L}-1})$.∎

Consider a fixed element in the Cartan subalgebra of $\mathfrak{su}(d_{L})$, a$d_{L}\times d_{L}$ matrix $T=\text{diag}(1,0,0,0,\cdots,-1)$, and $T_{\lambda}$,the corresponding generator in the representation given by the Young diagram$\lambda$. We have the following lemma:

Lemma 17**.**

It holds that$\lVert{T_{\lambda}}\rVert_{\infty}\leqslant\lambda_{1}$.

• Proof of subsection .2.  A basis for the representation $\lambda$ is given by different semi-standard fillings of the Young diagram $\lambda$ with numbers $1\cdots d_{L}$.If we indicate fillings of the $\lambda$ by $m_{\lambda}$, then $\{\lvert{m_{\lambda}}\rangle\}$ forms a basis for the representation $\lambda$.Although this is not an orthogonal basis, if the number content of $m_{\lambda}$ and $m^{\prime}_{\lambda}$ are different then $\lvert{m_{\lambda}}\rangle$ and $\lvert{m^{\prime}_{\lambda}}\rangle$ are orthogonal. This basis diagonalizes $T_{\lambda}$.In particular, if $\#_{i}m_{\lambda}$ indicates the number of times that $i$ appears in the filling $m_{\lambda}$, then $\langle{m_{\lambda}}\rvert T_{\lambda}\lvert{m_{\lambda}}\rangle=\#_{1}m_{\lambda}-\#_{d}m_{\lambda}$.This immediately leads to the conclusion that the eigenvalues of $T_{\lambda}$ are $\#_{1}m_{\lambda}-\#_{d}m_{\lambda}$, for different fillings $m_{\lambda}$.For any semi-standard filling of the Young diagrams, the numbers are strictly increasing in the columns. Therefore, $\#_{i}m_{\lambda}\leqslant\lambda_{1}$, as there are no repeats in the columns. So we showed that eigenvalues of $T_{\lambda}$ are between $-\lambda_{1}$ and $\lambda_{1}$, which completes the proof.∎

We are now in a position to prove VII. First,however, we prove a version of VII that provides astronger bound expressed as a binomial coefficient, which we will use to provethe bounds stated in VII.

Theorem 18**.**

Let $V_{L\to A}$ be an isometry that is covariant with respect to the full$\mathit{SU}(d_{L})$ group on the logical space, and write$\mathcal{E}(\cdot)=V(\cdot)V^{\dagger}$. Consider the single erasure noisemodel represented by $\mathcal{N}$ in \tagform@9 withequal erasure probabilities, $q_{i}=1/n$ for all $i$. Then

[TABLE]

In terms of the average entanglement fidelity measure, the bound reads instead

[TABLE]

The bound in .2 allows to deriveslightly stronger bounds than those obtained from the simplified expressions inVII. For instance, suppose that $d_{L}=d_{i}$ as inthe examples given in the main text. The binomial coefficient $\binom{a+b}{b}$is increasing in $b$, which can be seen using the recurrence relation$\binom{a+b+1}{b+1}=\frac{a+b+1}{b+1}\binom{a+b}{b}\geqslant\binom{a+b}{b}$.Also, the binomial coefficient $\binom{a+b}{b}$ for $b\geqslant 2$ satisfies$\binom{a+b}{b}\geqslant\binom{a+2}{2}=(a+2)(a+1)/2\geqslant a+2$ (assuming$a\geqslant 1$). Hence, if $d_{L}=d_{i}$, thencondition \tagform@.162 implies that$\lceil{(2n\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E}))^{-1}}\rceil\leqslant 1$, because otherwise we would have$\binom{d_{L}-1+\lceil{(2n\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E}))^{-1}}\rceil}{d_{L}-1}\geqslant\binom{d_{L}-1+2}{2}\geqslant d_{L}+1$. This impliesthat, for $d_{L}=d_{i}$, we must have$\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})\geqslant 1/(2n)$.

• Proof of subsection .2.  Combining .2 and .2, we get

[TABLE]

Now, we return to the original problem of approximate Eastin-Knill theorem,where the group $\mathit{SU}(d_{L})$ acts on physical subsystems. We fix the generator$T=\operatorname{diag}(1,0,0,0,\cdots,-1)$ of $\mathfrak{su}(d_{L})$, and let $T_{i}$ be thecorresponding generator acting on the subsystem $i$. Let$T_{i}=\bigoplus_{\lambda}T_{\lambda}$ be the decomposition of $T_{i}$ with respectto the decomposition of the representation on subsystem $i$, and assume that$\hat{\lambda}{(i)}$ is the Young diagram in this direct sum with thelargest $\lVert{T_{\lambda}}\rVert_{\infty}$. Therefore,$\lVert{T_{i}}\rVert_{\infty}=\bBigg@{1.2}\lVert{T_{\hat{\lambda}(i)}}\bBigg@{1.2}\rVert_{\infty}$, and we have:

[TABLE]

This implies

[TABLE]

Let $i^{\prime}$ denote the index of the subsystem that maximizes$\lVert{T_{i}}\rVert_{\infty}$, such that our bound \tagform@27 with$\Delta T_{L}=2$ and $\Delta T_{i}\leqslant 2\lVert{T_{i}}\rVert_{\infty}$ reads

[TABLE]

noting that$\max_{i}\Delta T_{i}\leqslant 2\max_{i}\lVert{T_{i}}\rVert_{\infty}=2\lVert{T_{i^{\prime}}}\rVert_{\infty}$, and writing$\epsilon_{\mathrm{worst}}=\epsilon_{\mathrm{worst}}(\mathcal{N}\circ\mathcal{E})$ as a shorthand.Therefore,$\lVert{T_{i^{\prime}}}\rVert_{\infty}\geqslant(2n\epsilon_{\mathrm{worst}})^{-1}$, and weobtain

[TABLE]

If we had used the bound \tagform@30c insteadof \tagform@27, we would have insteadof \tagform@.167 that

[TABLE]

and we can perform the replacement$\epsilon_{\mathrm{worst}}\to(d_{L}/2)\,\epsilon_{\mathrm{e}}$ in theargument above.∎

• Proof of section VII.  We use the following standard inequality of binomial coefficients. Forintegers $a,b>0$, we have the two lower bounds

[TABLE]

noting that $\binom{a+b}{b}=\binom{a+b}{a}$.Consider \tagform@.162, with $a=d_{L}-1$ and$b=\lceil(2n\epsilon_{\mathrm{worst}})^{-1}\rceil$. The first lower boundgives us

[TABLE]

and hence

[TABLE]

This proves \tagform@64a.Applying the second bound in \tagform@.170to \tagform@.162, we obtain

[TABLE]

We can rearrange \tagform@.173 to

[TABLE]

which in turn implies

[TABLE]

Henceforth we let $i$ denote the index of the physical subsystem with largestdimension, i.e., $d_{i}=\max_{i^{\prime}}d_{i^{\prime}}$.For large $d_{L}$, we have

[TABLE]

and thus

[TABLE]

which is the desired bound \tagform@63.The bound \tagform@64b followsfrom \tagform@.173 by noting that$\lceil(2n\epsilon_{\mathrm{worst}})^{-1}\rceil\geqslant(2n\epsilon_{\mathrm{worst}})^{-1}$ and that $\log(1+x)\geqslant\log(x)$.The alternative expressions for $\epsilon_{\mathrm{e}}$ follow from the use ofthe bound \tagform@.163, following the same steps asabove while effecting the replacement$\epsilon_{\mathrm{worst}}\to d_{L}\epsilon_{\mathrm{e}}/2$.∎

Appendix F Circumventing the Eastin-Knill theorem by randomized constructions

.1 Randomized constructions: Overview

The proof of VII.1 is technical, and relies on the recent developments in the representation theoryof $\mathit{U}(d)$, and new counting formulas for the Littlewood-Richardsoncoefficients. Here, we sketch the proof strategy, and refer the reader toAppendix F for the technical details.Although our randomized constructions do not properly work for producing good$\mathit{U}(2)$-covariant codes,101010More precisely, our techniques do not lead toproper lower bounds for the fidelity of recovery of random $\mathit{U}(2)$-covariantcodes, but this might only be caused by not lower bounding the fidelity ofrecovery with strong enough inequalities. for the $\mathit{U}(3)$ case we can findexplicit (non-asymptotic) bounds with a slightly different scaling. There, one can benefit from the fact that the fusion rules of $\mathit{U}(3)$ representation theory are known (Rassart, 2004, Section 5). We will not discuss$\mathit{U}(3)$ case further, and will focus on $d_{L}\geqslant 4$ for the rest of thissection.Consider codes that map logical information on the Hilbert space $\mathscr{H}_{L}$ to three physicalsubsystems $\mathscr{H}_{A}=\mathscr{H}_{A_{1}}\otimes\mathscr{H}_{A_{2}}\otimes\mathscr{H}_{A_{3}}$, and denote by $d_{i}$ the dimensionof $\mathscr{H}_{A_{i}}$. In order to precisely define what we mean by the random isometry $V$, consider the state corresponding to $V$ (similar to what we did in the analysis of correlationfunctions in Appendix B). The corresponding state, $\lvert{\Psi}\rangle$, liveson $\mathscr{H}_{R}\otimes\mathscr{H}_{A_{1}}\otimes\mathscr{H}_{A_{2}}\otimes\mathscr{H}_{A_{3}}$, where as before $\mathscr{H}_{R}\simeq\mathscr{H}_{L}$ is areference system. The covariance of $V$ translates to the invariance of$\lvert{\Psi}\rangle$:

[TABLE]

Therefore, $\Psi_{RA_{1}A_{2}A_{3}}$ lives on an invariant subspace of the unitarygroup. The projector to this invariant subspace is given by

[TABLE]

We denote by $d_{P}=\operatorname{tr}\bBigg@{1.2}({\Pi_{RA_{1}A_{2}A_{3}}}\bBigg@{1.2})$ the dimension of the invariantsubspace. Further, define$\Pi_{RA_{i}}:=\operatorname{tr}_{A\setminus A_{i}}\bBigg@{1.2}({\Pi_{RA_{1}A_{2}A_{3}}}\bBigg@{1.2})$ and$\Pi_{\widehat{RA_{i}}}:=\operatorname{tr}_{RA_{i}}\bBigg@{1.2}({\Pi_{RA_{1}A_{2}A_{3}}}\bBigg@{1.2})$.Now, we can chose the state $\lvert{\Psi}\rangle_{RA_{1}A_{2}A_{3}}$ randomly from $\Pi_{RA_{1}A_{2}A_{3}}$,and define $V$ to be the corresponding isometry, i.e.,$V_{L\rightarrow A_{1}A_{2}A_{3}}:=\langle{\Phi}\rvert_{LR}\lvert{\Psi}\rangle_{RA_{1}A_{2}A_{3}}$, where$\lvert{\Phi}\rangle=\sum\lvert{k}\rangle_{L}\lvert{k}\rangle_{R}$ for some standard choice of bases on $\mathscr{H}_{L}$and $\mathscr{H}_{R}$.As in Appendix B, we consider single erasures at known locations,i.e., the noise channel is given by$\mathcal{N}(\cdot)=\sum q_{i}\lvert{i}\rangle\langle{i}\rvert_{C}\otimes\mathcal{N}^{i}(\cdot)$, where$\mathcal{N}^{i}$ erases the $i$-th system asper \tagform@10. If the isometry $V$ is chosen at randomin the space of covariant isometries, then on average, the fidelity of recovery of the code defined by the isometry is lower bounded as follows.

Lemma 19**.**

Suppose that the covariant isometry $V$ is chosen randomly as above. Then, theinfidelity of the code after erasure of subsystem $i\in\{1,2,3\}$,averaged over all covariant isometries, satisfies the following inequality:

[TABLE]

Intuitively, .1 states that in order to get good quantumcodes we need to do the followings:

Control the constant offset,$\left\lVert{d_{P}^{-1}\Pi_{RA_{i}}-\mathds{1}_{RA_{i}}/(d_{R}d_{i})}\right\rVert_{1}$. This can beachieved by making sure that $\Pi_{RA_{i}}$ is close to a multiple of identity. 2. 2.

Control the fluctuations by minimizing$d_{p}^{-2}\operatorname{tr}\bBigg@{1.2}({\Pi_{\widehat{RA_{i}}}^{2}}\bBigg@{1.2})$. Note that this is the purity of density matrix $\Pi_{\widehat{RA_{i}}}/d_{p}$, so it would be small if $\Pi_{\widehat{RA_{i}}}$ is close to a multiple of a projector.

Section .1 is how far we can go without discussing the detailedrepresentation theory of $\mathit{U}(d_{L})$. From now on, we focus on analyzing$\Pi_{RA_{i}}$ and $\Pi_{\widehat{RA_{i}}}$.Without loss of generality assume that $i=1$. Also, suppose$\lambda,\mu,\nu$ are the Young diagrams defining the irreduciblerepresentations $r_{1}$, $r_{2}$ and $r_{3}$. Similarly, $r_{e_{1}}(U)=U$, where$e_{1}=(1,0,0,\cdots,0)$ is the Young diagram of the standardrepresentation. Now, we use representation theory techniques to explicitly compute$\Pi_{RA_{1}}$ and $\Pi_{\widehat{RA_{1}}}=\Pi_{A_{2}A_{3}}$.The degeneracies of fusion of different irreps of $\mathit{U}(d_{L})$ are known, andspecified by the so called Littlewood-Richardson coefficients$c^{\theta}_{\mu\nu}$:

[TABLE]

A specific case of this formula which is also applicable to our analysis is aversion Pieri formula (See Appendix A.1of Fulton and Harris (2013)): If $e_{i}$ is the $i$-th computation basisvector, then

[TABLE]

where $\mathcal{I}\in\{1,2,\cdots,d_{L}\}$ is the index set that $\lambda-e_{i}$ isa valid Young diagram, i.e., a non-increasing sequence. In particular, if $\lambda$is strictly decreasing then $\mathcal{I}=\{1,2,\cdots,d_{L}\}$. This relation can be derived by either directly applying the Littlewood-Richardson rule (Fulton and Harris, 2013, Appendix A.1), or starting from the standard Pieri formula and dualizing representations. With this, we have

[TABLE]

From the Schur orthogonality relations for compact groups (Peter-Weyl theorem), we have that $\int dU[\operatorname{tr}{\overline{r_{\beta}(U)}}]r_{\alpha}(U)=\delta_{\alpha\beta}I_{\alpha}/d_{\alpha}$. Applying this to Eq. .183 we get the following explicit relations:

[TABLE]

where $c_{\mu\nu\lambda}:=c^{\overline{\lambda}}_{\mu\nu}$, and$\overline{\lambda}$ is the dual of $\lambda$. Recall that in order for the randomcodes to perform well, we need that $\Pi_{RA_{1}}$ and $\Pi_{A_{2}A_{3}}$ to be closeto multiples of projectors. Equations .184 and .185 show that to achieve this weonly need $\frac{c_{\mu\nu(\lambda-e_{i})}}{d_{\lambda-e_{i}}}$ and$\frac{1}{d_{\lambda-e_{i}}}$ to be almost constants as $i$ varies. The followinglemma makes this observation quantitative:

Lemma 20**.**

Suppose that $0\leqslant\delta\leqslant 1/2$ is a real number such that for all $i\in\mathcal{I}$,

[TABLE]

then,

[TABLE]

Section .1 demonstrates that in order to get useful lower bounds on thefidelity of the codes, one has to show that $d_{\lambda}$ and$c_{\mu\nu\lambda}$ are stable under perturbations by $e_{i}$. Weconstruct our irreps such that they achieve this stability.Define $\lvert{\lambda}\rvert:=\sum_{i}{\lambda_{i}}$ for arbitrary Youngdiagram $\lambda$. It is known that if$\lvert{\mu}\rvert+\lvert{\nu}\rvert+\lvert{\lambda}\rvert\neq 0$, then $c_{\mu\nu\lambda}=0$. Now,the construction is as follows: Fix a triplet of Young diagrams$(\hat{\mu},\hat{\nu},\hat{\lambda})$ such that$\lvert{\hat{\mu}_{i}}\rvert+\lvert{\hat{\nu}_{i}}\rvert+\lvert{\hat{\lambda}_{i}}\rvert=0$ andset

[TABLE]

for large values of $N$. We used $N\hat{\mu}+e_{1}$ instead of$N\hat{\mu}$ is to ensure that $\lvert{\mu}\rvert+\lvert{\nu}\rvert+\lvert{\lambda-e_{i}}\rvert=0$ as weneed $c_{\mu\nu(\lambda-e_{i})}$ to be non-zero.Showing smoothness of $d_{\lambda}$ is much simpler, because by the Weyldimension formula (see (Fulton and Harris, 2013, Section 15.3)) it is polynomial in$\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{d_{L}})$. So by basic Taylorexpansion we have$d_{\lambda+e_{i}}=d_{\lambda}+\partial d_{\lambda}/\partial\lambda_{i}+1/2\,\partial^{2}d_{\lambda}/\partial\lambda_{i}^{2}+\cdots$. Note that the totaldegree of the terms in the sum decreases by differentiation. Hence, $d_{\lambda}$ is the dominant term in theexpansion of $d_{\lambda+e_{i}}$ and other termsare lower order in $N$. Therefore, there exist $N_{0}^{\prime}$ and $C_{0}^{\prime}$such that for $N\geqslant N_{0}^{\prime}$,

[TABLE]

The Littlewood-Richardson coefficients are much more complicated. They can becomputed using efficient algorithms, such as the Littlewood-Richardson rule, butthere is no explicit formula. In fact, they are specific cases of the calledKronecker coefficients whose computation is known to beNP-hard Ikenmeyer et al. (2017). However, a series of new developmentsin the representation theory of the unitary group has revealed interestingpolynomiality properties for the LR coefficients.It is known that $c_{\mu\nu\lambda}$ as a function of $\mu$, $\nu$ and $\lambda$ (inthe $3d_{L}-1$ dimensional subspace constrained by the condition$\lvert{\mu}\rvert+\lvert{\nu}\rvert+\lvert{\lambda}\rvert=0$) is non-zero if and only if $(\mu,\nu,\lambda)$ is in a particular convexcone. This cone, or chamber complex, is then divided to several sub-conesor chambers. In Ref. Rassart (2004) it is shown that$c_{\mu\nu\lambda}$ is a polynomial within each chamber. SeeFig. 5.We chose $(\hat{\mu},\hat{\nu},\hat{\lambda})$ such that it is in theinterior of one of the chambers, and $c_{N\hat{\mu},N\hat{\nu},N\hat{\lambda}}$is not constant. If $N$ is large enough, $c_{\mu\nu(\lambda-e_{i})}$ will remainin the interior of the same chamber for all $i$, and are described bythe same polynomial. Hence, similar to $d_{\lambda}$, we have:

[TABLE]

where $N\geqslant N_{0}^{\prime\prime}$ and for some $N_{0}^{\prime\prime}$ and $C_{0}^{\prime\prime}$. Clearly, these bounds the smoothness required for .1 to work, and therefore weget our main theorem. Their detailed proof will be inAppendix F.

Theorem 21**.**

Suppose that $d_{L}\geqslant 4$. There exist Young diagrams $\hat{\lambda}$,$\hat{\mu}$ and $\hat{\nu}$, an integer $N_{0}$, and a constant $C_{0}$, such thatif $(\mu,\nu,\lambda)=(N\hat{\mu}+e_{1},N\hat{\nu},N\hat{\lambda})$ and $V$ is arandom covariant isometry in the sense of \tagform@65, we have,

[TABLE]

For these constructions, we have,

[TABLE]

Finally, VII.1 follows immediately from .1.

.2 Randomized constructions: Detailed proofs

First, we prove .1.

• Proof of subsection .1.  First, we express the error-correcting accuracy of the code $V$ according tothe average entanglement fidelity in terms of the distance of the codewords toa maximally mixed state, including the reference system. ByBény/Oreshkov \tagform@11a (choosing$\zeta_{E}=\mathds{1}_{A_{i}}/d_{i}$), we have

[TABLE]

and hence

[TABLE]

where we recall the usual relations between trace distance and the fidelity.We denote by $\operatorname{\mathbb{E}}$ the averaging over all possible invariant states$\Psi_{RA_{1}A_{2}A_{3}}$. Taking an average over \tagform@.195 givesus

[TABLE]

where we write $\tau_{RA_{i}}=\mathds{1}_{RA_{i}}/(d_{L}d_{i})$. Applying triangleinequality, Cauchy-Schwarz inequality, and the concavity of square root givesus (see Ref. hayden2008decoupling for similar calculations),

[TABLE]

Now, consider the rank-$d_{P}$ projector to the invariant space $\Pi_{RA_{1}A_{2}A_{3}}$that we constructed in VII.1. Define$L:\mathbb{C}^{d_{P}}\rightarrow\mathscr{H}_{A}\otimes\mathscr{H}_{R}$ be the isometry mapping tothe invariant space, which satisfies

[TABLE]

We can define $\lvert{\Psi}\rangle_{RA}=L\lvert{\chi}\rangle$, where $\lvert{\chi}\rangle$ is a random statein $\mathbb{C}^{d_{p}}$. Then,

[TABLE]

where we used $\operatorname{\mathbb{E}}\chi=\mathds{1}/d_{P}$. For simplicity, we henceforth set $i=1$without loss of generality. If $\mathcal{F}_{A_{2}A_{3}}$ is flip operator swappingtwo copies of the Hilbert space $\mathscr{H}_{A_{2}A_{3}}=\mathscr{H}_{A_{2}}\otimes\mathscr{H}_{A_{3}}$, wehave

[TABLE]

Substituting into \tagform@ ‣ .2 and applying basic inequalities lead to,

[TABLE]

which is the desired formula.∎

Next, we prove .1.

• Proof of subsection .1.  For simplicity of exposition, define two probability distributions$p,q:\mathcal{I}\rightarrow\mathbb{R}_{\geqslant 0}$,

[TABLE]

From .1, we have,

[TABLE]

We would like to bound both terms on the right hand sideof \tagform@.203. We have

[TABLE]

Also,

[TABLE]

Now, the condition of the lemma can be written as

[TABLE]

By summing over $i$, we get,

[TABLE]

With some algebra, we obtain

[TABLE]

Similarly,$\bBigg@{1.2}\lvert{q_{i}-1/\lvert{\mathcal{I}}\rvert}\bBigg@{1.2}\rvert\leqslant{4\delta}/{\lvert{\mathcal{I}}\rvert}$.Therefore,

[TABLE]

On the other hand, $p_{i}\leqslant(1+\delta)\frac{c_{\mu\nu\lambda}}{d_{P}}$, and$1/q_{i}\leqslant\frac{d_{R}d_{1}}{d_{\lambda}(1-\delta)}$. Now, we get that$p_{i}/q_{i}\leqslant(1+\delta)^{2}/(1-\delta)^{2}$. So,

[TABLE]

Substituting in the formula for the fidelity completes the proof.∎

Next, we would like to prove our main theorem on random constructions,.1. Before that, we need to show that theLittlewood-Richardson coefficients can grow significantly with the size theYoung diagrams. This is the content of next lemma:

Lemma 22**.**

In the chamber complex of Littlewood-Richardson coefficients discussed inVII.1, there are chambers in which $c_{\mu\nu\lambda}$ isa polynomial of degree $\binom{d_{L}-1}{2}$

• Proof of subsection .2.  Consider the following relation for the Littlewood-Richardson coefficients,derived by comparing dimensions:

[TABLE]

Define the average of Littlewood-Richardson coefficients weighted by thedimension $d_{\lambda}$, i.e.,

[TABLE]

Also, assume that the number of $\lambda$’s where $c_{\mu\nu\lambda}\neq 0$ is$N_{\mu\nu}$ and the average dimension of $d_{\lambda}$, averaged over such$\lambda$’s is,

[TABLE]

Now \tagform@.211 becomes

[TABLE]

Consider the case where $\mu=N\mu_{0}$ and $\nu=N\nu_{0}$, for some fixed$\mu_{0}$ and $\nu_{0}$ and large $N$. It is known that the dimension of thechamber complex is $3d_{L}-1$, see, e.g., Proposition 1 in Knutson et al. (2004). Therefore, as two$d_{L}$ dimensional axis are fixed by $\mu$ and $\nu$, the section of the conecorresponding to $c_{\mu\nu\lambda}\neq 0$ is $d_{L}-1$ dimensional, andtherefore $N_{\mu\nu}=O(N^{d_{L}-1})$. From the Weyl dimension formula, it isknown that $d_{\mu}$, $d_{\nu}$ , and $\overline{d}$ are all$O\left(N^{d_{L}(d_{L}-1)/2}\right)$. So,

[TABLE]

This shows that there exists at least one chamber whose polynomial is at leastdegree $\binom{d_{L}-1}{2}$. On the other hand, it is known that degree of thepolynomials are bounded above by $\binom{d_{L}-1}{2}$ (see Corollary 4.2in Rassart (2004)). This completes the proof.∎

• Proof of subsection .1.  We start from Eqs. .190 and .191. If we set $C_{0}=\max(C_{0}^{\prime},C_{0}^{\prime\prime})$ and$N_{0}=\max(N_{0}^{\prime},N_{0}^{\prime\prime})$, we have

[TABLE]

Further, suppose that $\hat{\mu}$,$\hat{\nu}$, and $\hat{\lambda}$ where chosensuch that $c_{\mu\nu\lambda}$ grows superlinearly as a function of $N$. Thisis possible for $d_{L}\geqslant 4$ as a result of .2. Using thisfact and .1, we get

[TABLE]

In fact, the same relation holds for$\epsilon_{\mathrm{e}}(\mathcal{N}^{2}\circ\mathcal{E})$ and$\epsilon_{\mathrm{e}}(\mathcal{N}^{3}\circ\mathcal{E})$, and using the Markovinequality and the union bound we can show that there exists $\hat{\mu}$,$\hat{\nu}$, and $\hat{\lambda}$ for which

[TABLE]

As a consequence, and using Appendix G, weget \tagform@.192. The second equation, \tagform@.193,follows from \tagform@.192 using the Weyl dimension which indicates that$d_{i}=O\left(N^{d_{L}(d_{L}-1)/2}\right)$.∎

Appendix G Some general lemmas

A first lemma relates the correctability of the code to the environment’sability to distinguish two states in terms of the trace distance.

Lemma 23**.**

For any encoding channel $\mathcal{E}$ and noise channel $\mathcal{N}$, andfor any two logical states $\sigma_{L},\sigma^{\prime}_{L}$, and if$\widehat{\mathcal{N}\circ\mathcal{E}}$ is a complementary channel of$\mathcal{N}\circ\mathcal{E}$, we have that

[TABLE]

• Proof of Appendix G.  Let $\zeta$ be the state achieving the optimumin \tagform@11b. We have

[TABLE]

recalling that the trace distance obeys$\delta(\rho,\sigma)\leqslant\sqrt{1-F^{2}(\rho,\sigma)}$ (see,e.g., Tomamichel et al. (2010)). Choosing the optimizationcandidates $\sigma_{L}\otimes\lvert{0}\rangle\langle{0}\rvert_{R}$ and $\sigma^{\prime}_{L}\otimes\lvert{0}\rangle\langle{0}\rvert_{R}$ in thelast inequality, we obtain both

[TABLE]

Hence, by triangle inequality,

[TABLE]

The following lemma relates the global fidelity of the code to the fidelitiescorresponding to the correction of individual errors. Note that we do notnecessarily expect a similar result to hold for the worst-case entanglementfidelity, because the worst-case input state might be different for each erasureevent.

Lemma 24**.**

Let $\mathcal{N}_{A\to A}^{\alpha}$ and$\mathcal{N}_{A\to AC}(\cdot)=\sum q_{\alpha}\mathcal{N}^{\alpha}(\cdot)\otimes\lvert{\alpha}\rangle\langle{\alpha}\rvert_{C}$ correspond to a noise model of erasuresat known locations, as given in \tagform@10. Then, forany $\lvert{\phi}\rangle_{LR}$, the average entanglement fidelity of the code withrespect to $\lvert{\phi}\rangle_{LR}$ is directly related to the individual fidelities ofrecovery for each possible erasure:

[TABLE]

and consequently,

[TABLE]

• Proof of Appendix G.  The average entanglement fidelity associated with the different noisechannels can be written as:

[TABLE]

where the optimizations range over recovery channels $\mathcal{R}_{AC\to L}$and $\mathcal{R}_{A\to L}^{\alpha}$, respectively. We have

[TABLE]

showing that

[TABLE]

Physically, the reverse inequality follows because a global recovery strategyis to measure the register containing the record that indicates which erroroccurred, and to apply the optimal recovery strategy corresponding to thaterror. Specifically, if $\mathcal{R}^{\alpha}_{A\to L}$ are optimal choicesin \tagform@.225b for each $\alpha$, thenwe define

[TABLE]

Then,

[TABLE]

as claimed.∎

The following lemma is a technical consequence of the concavity of the fidelityfunction.

Lemma 25**.**

Let $\rho,\sigma$ be two (normalized) quantum states. Let $\tau\geqslant 0$with $\rho\geqslant\tau$. Then

[TABLE]

• Proof of Appendix G.  Since $\rho\geqslant\tau$, we have $\rho-\tau=:\Delta\geqslant 0$. Then$\rho=\tau+\Delta=\operatorname{tr}(\tau)\,\frac{\tau}{\operatorname{tr}(\tau)}+\operatorname{tr}(\Delta)\,\frac{\Delta}{\operatorname{tr}(\Delta)}$, and by concavity of the fidelity,

[TABLE]

The claim follows by noting that$\operatorname{tr}(\Delta)\,F\bBigg@{1.2}({\Delta/\operatorname{tr}(\Delta),\sigma}\bBigg@{1.2})\geqslant 0$.∎