Efficiently computable bounds for magic state distillation

TL;DR

This paper introduces new measures called thauma to quantify non-stabilizerness in quantum states, providing efficient bounds and benchmarks for magic state distillation, which is vital for scalable quantum computing.

Contribution

It develops the family of thauma measures and demonstrates their application in bounding and benchmarking magic state distillation processes.

Findings

Hypothesis testing thauma as an efficient distillable non-stabilizerness benchmark

Max-thauma outperforms mana in distillation efficiency benchmarking

Two classes of maximally mana states cannot be asymptotically interconverted at rate one

Abstract

Magic-state distillation (or non-stabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to non-stabilizer state manipulation is the resource theory of non-stabilizer states, for which one of the goals is to characterize and quantify non-stabilizerness of a quantum state. In this paper, we introduce the family of thauma measures to quantify the amount of non-stabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of non-stabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable non-stabilizerness, which in turn leads to a variety of bounds on the rate at which non-stabilizerness can be distilled, as well as on the overhead of magic-state distillation. We then prove that the max-thauma can be used as an efficiently computable tool in benchmarking the efficiency of magic-state distillation and that it can outperform pervious approaches based on mana. Finally, we use the min-thauma to bound a quantity known in the literature as the "regularized relative entropy of magic." As a consequence of this bound, we find that two classes of states with maximal mana, a previously established non-stabilizerness measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of non-stabilizer states and reveals a difference between the resource theory of non-stabilizer states and other resource theories such as entanglement and coherence.

Equations

$$\mathcal{M}(\rho ):={\log }_2\Vert \rho {\Vert }_{W,1},$$

$$\theta (\rho ):=\underset{\sigma \in \mathcal{W}}{\min }D(\rho \Vert \sigma ),$$

$$\mathbf{\theta }(\rho ):=\underset{\sigma \in \mathcal{W}}{\inf }\mathbf{D}(\rho \Vert \sigma ).$$

$${\theta }_{\min }(\rho ):=\underset{\sigma \in \mathcal{W}}{\min }{D}_0(\rho \Vert \sigma ):=\underset{\sigma \in \mathcal{W}}{\min }[-{\log }_2\mathrm{Tr}{P}_{\rho }\sigma ],$$

$${\theta }_{\min }(\rho )=-{\log }_2\min \{\Vert Q{\Vert }_{W,\infty }:Q\ge P_{\rho }\},$$

$${\theta }_{\min }(\psi )=-{\log }_2\underset{\sigma \in \mathcal{W}}{\max }F(\psi ,\sigma )\le -{\log }_2{F}_{\text{Stab}}(\psi ).$$

$${\theta }_{\max }(\rho )$$

$$={\log }_2\min \left\{\Vert V{\Vert }_{W,1}:\rho \le V\right\}.$$

$${\theta }_{\min }(\rho \otimes \tau )$$

$${\theta }_{\max }(\rho \otimes \tau )$$

$$|\mathbb{S}⟩$$

$$H=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right),$$

$$R_{\mathcal{M}}^{\infty }(\rho )$$

$${\theta }_{\min }(\mathbb{S})$$

$${\theta }_{\min }(\mathbb{N})$$

$${\theta }_{\min }(H_{+})$$

$$=R_{\mathcal{M}}^{\infty }(H_{+})={\log }_2(3-\sqrt{3}),$$

$${\theta }_{\min }(T)$$

$${\mathcal{M}}_{\varphi }^{\epsilon }(\rho )=\sup \{k:\Lambda (\rho ){\approx }_{\epsilon }|\varphi ⟩⟨\varphi {|}^{\otimes k},\Lambda \in \text{SO}\},$$

$${\mathcal{M}}_{H_{+}}^{\epsilon }(\rho )$$

$${\mathcal{M}}_T^{\epsilon }(\rho )$$

$${\mathcal{M}}_{H_{+}}(\rho )=\underset{\epsilon \to 0}{\lim }\underset{n\to \infty }{\lim }\frac{1}{n}{\mathcal{M}}_{H_{+}}^{\epsilon }({\rho }^{\otimes n})$$

$${\mathcal{M}}_T(\rho )=\underset{\epsilon \to 0}{\lim }\underset{n\to \infty }{\lim }\frac{1}{n}{\mathcal{M}}_T^{\epsilon }({\rho }^{\otimes n})$$

$$N_{\text{eff}}(\rho \to \xi )=\inf \left\{n/p:\Lambda ({\rho }^{\otimes n})\to \xi \text{ w/ prob. }p,\Lambda \in \text{SO}\right\}.$$

$$N_{\text{eff}}(\rho \to \xi )\ge N_{\mathcal{M}}(\rho ,\xi ):=\mathcal{M}(\xi )/\mathcal{M}(\rho ).$$

$$R(\mathbb{N}\to \mathbb{S})=R_{\mathcal{M}}^{\infty }(\mathbb{N})/R_{\mathcal{M}}^{\infty }(\mathbb{S})\le \frac{{\log }_2(3/2)}{{\log }_2(5/3)}<1.$$

$$X|j⟩$$

$$Z|j⟩$$

$$T_{\text{u}}={\tau }^{-a_1{a}_2}{Z}^{a_1}{X}^{a_2},$$

$$T_{{\text{u}}_A\oplus {\text{u}}_B}=T_{{\text{u}}_A}\otimes T_{{\text{u}}_B},$$

$$U\in {\mathcal{C}}_d\text{ iff }\forall \text{u},\exists \theta ,{\text{u}}^{\prime },\text{ such that }U{T}_{\text{u}}{U}^{†}=e^{i\theta }{T}_{{\text{u}}^{\prime }}.$$

$$\{S_j{\}}_j=\{U|0⟩⟨0|U^{†}:U\in {\mathcal{C}}_d\}.$$

$$\mathrm{STAB}({\mathcal{H}}_d)=\left\{\rho \in \mathcal{S}({\mathcal{H}}_d):\rho =\sum _j{p}_j{S}_j,\forall j{p}_j\ge 0,\sum _j{p}_j=1\right\}.$$

$$W_{\rho }(\text{u})=\frac{1}{d}\mathrm{Tr}{A}_{\text{u}}\rho ,$$

$$A_{\text{u}}:=T_{\text{u}}{A}_0{T}_{\text{u}}^{†},A_0:=\frac{1}{d}\sum _{\text{u}}{T}_{\text{u}}.$$

Full text

Introduction.—Quantum computers hold the promise of a substantial speed-up over classical computers for certain algebraic problems Shor (1997); Grover (1996); Childs and van Dam (2010) and the simulation of quantum systems Lloyd (1996); Childs et al. (2018). One main obstacle to the physical realization of quantum computation is the decoherence that occurs during the execution of quantum algorithms. Fault-tolerant quantum computation (FTQC) Shor (1996); Campbell et al. (2017) provides a framework to overcome this difficulty and allows reliable quantum computation when the physical error rate is below a certain threshold value.

According to the Gottesman–Knill theorem Gottesman (1997); Aaronson and Gottesman (2004), a quantum circuit comprised of only Clifford gates confers no computational advantage because it can be simulated efficiently on a classical computer. However, the addition of a non-stabilizer state can lead to a universal gate set via a technique called state injection Gottesman and Chuang (1999); Zhou et al. (2000), thus achieving universal quantum computation. The key of this resolution is to perform magic-state distillation Bravyi and Kitaev (2005) (see Bravyi and Haah (2012); Jones (2013); Haah et al. (2017); Campbell and Howard (2018); Hastings and Haah (2018); Krishna and Tillich (2018); Chamberland and Cross (2018) for recent progress), wherein stabilizer operations are used to transform a large number of noisy non-stabilizer states into a small number of high quality non-stabilizer states. Therefore, a quantitative theory is highly desirable in order to fully exploit the power of non-stabilizer states in fault-tolerant quantum computation.

Quantum resource theories (QRTs) offer a powerful framework for studying different phenomena in quantum physics, and the seminal ideas of QRTs have recently been influencing diverse areas of physics Chitambar and Gour (2019). In the context of the non-stabilizer-state model of universal quantum computation, the resource-theoretic approach reduces to the characterization and quantification of the usefulness of the resourceful non-stabilizer states Veitch et al. (2012, 2014). In the framework of Veitch et al. (2012, 2014), the free operations are the stabilizer operations, those that possess a fault-tolerant implementation in the context of fault-tolerant quantum computation, and the free states are the stabilizer states (STAB). Stabilizer operations include preparation and measurement in the computational basis, and a restricted set of unitary operations, called the Clifford unitaries. The free states consist of all pure stabilizer states, which are eigenstates of the generalized Pauli operators, and their convex mixtures. The resource states, namely, the non-stabilizer states, are key resources that are required to achieve some desired computational tasks. For quantum computers acting on qudit registers with odd dimension $d$, the resource theory of non-stabilizer states (or equivalently contextuality with respect to stabilizer measurements Howard et al. (2014); Delfosse et al. (2015)) has been developed Veitch et al. (2012); Mari and Eisert (2012); Veitch et al. (2014). The resource theory of non-stabilizer states for multiqubit systems was recently developed in Howard and Campbell (2017); Bravyi et al. (2016); Heinrich and Gross (2018).

In this paper, we solve some fundamental open questions in the resource theory of non-stabilizer states, and we develop the framework for one-shot magic state distillation. Our main tool for doing so is the thauma family of non-stabilizer monotones, which quantify the amount of non-stabilizerness in a given state by comparing it to a positive semi-definite operator with non-positive mana (i.e., a subnormalized state with no non-stabilizerness). Our first contribution is to introduce the one-shot distillable non-stabilizerness of a quantum state and an upper bound for it named hypothesis testing thauma. This result leads to various applications for magic-state distillation, which can be interpreted as fundamental limits. The max-thauma is another member of the thauma family, and we prove that it is an efficiently computable non-stabilizerness monotone, which can in turn be used to evaluate the efficiency of magic-state distillation. We further provide an example to demonstrate that max-thauma outperforms mana in benchmarking the efficiency of magic-state distillation. We also prove that the min-thauma is an additive lower bound on the “regularized relative entropy of magic,” the latter quantity defined in Veitch et al. (2014). This bound then leads to the conclusion that two magic states with maximal negativity cannot be interconverted asymptotically at a rate equal to one.

Discrete Wigner function.—We now recall the definition of the discrete Wigner function Wootters (1987); Gross (2006, 2007), which is an essential tool in the analysis of the resource theory of non-stabilizer states. Throughout this paper, a Hilbert space implicitly has an odd dimension, and if the dimension is not prime, it should be understood to be a tensor product of Hilbert spaces each having odd prime dimension.

Let ${\cal H}_{d}$ denote a Hilbert space of dimension $d$, and let $\{|j\rangle\}_{j=0,\ldots,d-1}$ denote the standard computational basis. For a prime number $d$, we define the respective shift and boost operators $X,Z\in{\cal L}({\cal H}_{d})$ as $X|j\rangle=|j\oplus 1\rangle$ and $Z|j\rangle=\omega^{j}|j\rangle$, with $\omega=e^{2\pi i/d}$. We define the Heisenberg–Weyl operators as $T_{{{\textbf{u}}}}=\tau^{-a_{1}a_{2}}Z^{a_{1}}X^{a_{2}}$, where $\tau=e^{(d+1)\pi i/d}$ and ${{\textbf{u}}}=(a_{1},a_{2})\in{{\mathbb{Z}}}_{d}\times{{\mathbb{Z}}}_{d}$.

For each point u in the discrete phase space, there is a corresponding operator $A_{{\textbf{u}}}$, and the value of the discrete Wigner representation of a quantum state $\rho$ at this point is given by $W_{\rho}({{\textbf{u}}}):=\operatorname{Tr}A_{{\textbf{u}}}\rho/d,$ where $\{A_{{\textbf{u}}}\}_{{\textbf{u}}}$ are the phase-space point operators: $A_{{\textbf{u}}}:=T_{{\textbf{u}}}A_{0}T_{{\textbf{u}}}^{\dagger}$, $A_{0}:=\frac{1}{d}\sum_{{\textbf{u}}}T_{{\textbf{u}}}$. We give more details of this formalism in Appendix A 111See Supplemental Material [url] for detailed mathematical developments of the assertions in the main text. The Supplemental Material includes Refs. Hostens et al. (2005); Ferrie (2011); Fawzi and Fawzi (2018); Tomamichel et al. (2009); Mosonyi and Hiai (2011); Uhlmann (1976); Frank and Lieb (2013); Beigi (2013); Wilde (2018).

Thauma.—It is well known that quantum computations are classically simulable if they consist of stabilizer operations acting on quantum states with a positive discrete Wigner function. Such states are thus useless for magic-state distillation Veitch et al. (2012). Let ${\cal W}_{+}$ denote the set of quantum states with positive discrete Wigner function. States in ${\cal W}_{+}$ can be understood as being analogous to states with a positive partial transpose in entanglement distillation Peres (1996); Horodecki et al. (1998), in the sense that such states are undistillable.

To address open questions in the resource theory of non-stabilizer states, we are motivated by the idea of the Rains bound from entanglement theory Rains (2001), as well as its variants Wang and Duan (2016a, 2017); Fang et al. (2017), which also have fruitful applications in quantum communication Tomamichel et al. (2017, 2016); Wang and Duan (2016b); Wang et al. (2018). As developed in Rains (2001) and the later work Audenaert et al. (2002), the Rains bound and its variants consider sub-normalized states with non-positive logarithmic negativity Vidal and Werner (2002); Plenio (2005) as useless resources, and they use the divergence between the given state and such sub-normalized states to evaluate the behavior of entanglement distillation.

Thus, inspired by the main idea behind the Rains bound, we introduce the set of sub-normalized states with non-positive mana: ${\cal W}:=\{\sigma:{\cal M}(\sigma)\leq 0,\ \sigma\geq 0\},$ with the mana ${\cal M}(\rho)$ of a quantum state $\rho$ defined as Veitch et al. (2014)

[TABLE]

where the Wigner trace norm of an operator $V$ is defined as $\|V\|_{W,1}:=\sum_{{{\textbf{u}}}}|W_{V}({{\textbf{u}}})|$. It follows from definitions that $\operatorname{Tr}\sigma\leq 1$ if $\sigma\in{\cal W}$. Note that the mana Veitch et al. (2014) is analogous to the logarithmic negativity Vidal and Werner (2002); Plenio (2005). Furthermore, the following strict inclusions hold: $\text{STAB}\subsetneq{\cal W}_{+}\subsetneq{\cal W}$.

We now define the thauma 222Greek for “wonder” or “marvel” of a state $\rho$ as

[TABLE]

where $D(\rho\|\sigma)$ is the quantum relative entropy Umegaki (1962), defined as $D(\rho\|\sigma)=\operatorname{Tr}\{\rho[\log_{2}\rho-\log_{2}\sigma]\}$ when the support of $\rho$ is contained in the support of $\sigma$ and equal to $+\infty$ otherwise. The thauma can be understood as the minimum relative entropy between a quantum state and the set of subnormalized states with non-positive mana. The thauma is a non-stabilizerness measure that can be efficiently computed via convex optimization (see Appendix B Note (1)). Following from the definition of thauma above, we define the regularized thauma of a state $\rho$ as $\theta^{\infty}(\rho):=\lim_{n\to\infty}\theta(\rho^{\otimes n})/n$.

The definition of thauma given above can be generalized to a whole family of thauma measures of non-stabilizerness. Defining a generalized divergence $\mathbf{D}(\rho\|\sigma)$ to be any function of a quantum state $\rho$ and a positive semi-definite operator $\sigma$ that obeys data processing Polyanskiy and Verdu (2010); Sharma and Warsi (2012), i.e., $\mathbf{D}(\rho\|\sigma)\geq\mathbf{D}({\cal N}(\rho)\|{\cal N}(\sigma))$ where ${\cal N}$ is a quantum channel, we arrive at the generalized thauma of a quantum state $\rho$:

[TABLE]

If the generalized divergence $\mathbf{D}$ is non-negative for a state $\rho$ and a sub-normalized state $\sigma$ and equal to zero if $\rho=\sigma$, then it trivially follows that the generalized thauma $\boldsymbol{\theta}(\rho)$ is a non-stabilizerness monotone, meaning that it is non-increasing under the free operations and equal to zero for stabilizer states. Examples of generalized divergences, in addition to the relative entropy, include the Petz–Rényi relative entropies Petz (1986) and the sandwiched Rényi relative entropies Müller-Lennert et al. (2013); Wilde et al. (2014). See Appendix C for further details Note (1).

Min- and max-thauma.—In what follows, we make use of the Petz–Rényi relative entropy of order zero Petz (1986) and the max-relative entropy Datta (2009) to define the min-thauma and the max-thauma, respectively. As we prove in what follows, these two members of the thauma family are efficiently computable by semidefinite programs (SDPs) Boyd and Vandenberghe (2004) and are particularly useful for addressing open questions in the resource theory of non-stabilizer states.

The min-thauma of a state $\rho$ is defined as

[TABLE]

where $P_{\rho}$ denotes the projection onto the support of $\rho$. Note that $\theta_{\min}(\rho)$ is an SDP and the duality theory of SDPs Boyd and Vandenberghe (2004) leads to the dual SDP:

[TABLE]

where $\|V\|_{W,\infty}:=d\max_{{{\textbf{u}}}}|W_{V}({{\textbf{u}}})|$ denotes the Wigner spectral norm of an operator $V$ acting on a space of dimension $d$. For any pure state $|\psi\rangle$,

[TABLE]

where $F_{\text{Stab}}(\psi)$ is the stabilizer fidelity Bravyi et al. (2018).

The max-thauma of a state $\rho$ is defined as

[TABLE]

As the following proposition states, the min- and max-thauma are additive non-stabilizerness measures. Additionally, the min-thauma is a lower bound for the regularized thauma, and the max-thauma is an upper bound.

[TABLE]

Consequently, $\theta_{\min}(\rho)\leq\theta^{\infty}(\rho)\leq\theta_{\max}(\rho)$.

The proof of Proposition 1 relies on the Petz–Rényi relative entropy of order zero Datta (2009), the max-relative entropy Datta (2009), and the duality theory of SDPs Boyd and Vandenberghe (2004) (see Appendix D for details Note (1)).

In Appendix E Note (1), we prove that the max-thauma possesses a stronger monotonicity property, in the sense that it does not increase on average under stabilizer operations.

We note here that an important consequence of the additivity of min-thauma is that the maximum overlap between $|\phi\rangle^{\otimes n}$ and the set ${\cal W}$ is $2^{-n\theta_{\min}(\phi)}$; i.e., for any $\tau\in{\cal W}\ (\text{or STAB})$, we have that $\operatorname{Tr}[|\phi\rangle\!\langle\phi|^{\otimes n}\tau]\leq 2^{-n\theta_{\min}(\phi)}$.

Thauma for basic non-stabilizer states.—Proposition 2 below states that the min-, regularized, and max-thauma collapse to the same value for several interesting non-stabilizer states, including the Strange, Norrell, $H$, and $T$ state.

The Strange and Norrell states are defined as Veitch et al. (2014)

[TABLE]

The qutrit Hadamard gate is given by Howard and Vala (2012)

[TABLE]

where we recall that $\omega=e^{2\pi i/3}$. The $H$ gate has eigenvalues $+1$, $-1$, and $i$, and we label the three corresponding eigenstates as $|H_{+}\rangle$, $|H_{-}\rangle$, and $|H_{i}\rangle$. The $|H_{+}\rangle$ state is a non-stabilizer state that is typically considered in the context of magic-state distillation Bravyi and Kitaev (2005); Anwar et al. (2012). In what follows, we refer to it as the $H_{+}$ non-stabilizer state.

Another common choice for a non-Clifford gate is the $T$ gate. The qutrit $T$ gate is given by $T:=\text{diag}(\xi,1,\xi^{-1})$, where $\xi=e^{2\pi i/9}$ is a primitive ninth root of unity Howard and Vala (2012). The non-stabilizer state corresponding to the qutrit $T$ gate is $|T\rangle:=\frac{1}{\sqrt{3}}(\xi|0\rangle+|1\rangle+\xi^{-1}|2\rangle),$ which is the state resulting from applying the $T$ gate to the stabilizer state $(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$.

In what follows, we employ the shorthand ${{\mathbb{S}}}\equiv|{{\mathbb{S}}}\rangle\!\langle{{\mathbb{S}}}|$, ${{\mathbb{N}}}\equiv|{{\mathbb{N}}}\rangle\!\langle{{\mathbb{N}}}|$, $H_{+}\equiv|H_{+}\rangle\!\langle H_{+}|$, and $T\equiv|T\rangle\!\langle T|$. Before stating the theorem, let us recall the definition of the “regularized relative entropy of magic” and the “relative entropy of magic” Veitch et al. (2014):

[TABLE]

[TABLE]

Appendix F Note (1) contains a proof of Proposition 2. In the forthcoming sections, we provide applications of Propositions 1 and 2 to the resource theory of non-stabilizer states.

Fundamental limits for distilling non-stabilizer states.—The basic task of magic-state distillation Bravyi and Kitaev (2005) can be understood as follows. For any given quantum state $\rho$, we aim to transform this state to a collection of non-stabilizer states (e.g., $|T\rangle$) with high fidelity using stabilizer operations. The goal is to maximize the number of target states while keeping the transformation infidelity within some tolerance $\varepsilon$. After magic-state distillation, one can use a circuit gadget (which requires only stabilizer operations) to transform this non-stabilizer state into a non-Clifford gate Gottesman and Chuang (1999); Zhou et al. (2000). Protocols for distillation in the qudit setting of quantum computing were recently developed in Campbell et al. (2012); Anwar et al. (2012); Campbell (2014); Dawkins and Howard (2015).

In the following, we study the fundamental limit of magic-state distillation of a pure target non-stabilizer state. We define the approximate one-shot distillable $\phi$-non-stabilizerness of a given state $\rho$ as the maximum number of $|\phi\rangle\!\langle\phi|$ non-stabilizer states that can be obtained via stabilizer operations, while keeping the infidelity within a given tolerance. Formally, for any triplet $(\rho,\phi,\varepsilon)$ consisting of an initial state $\rho$, a target pure state $\phi$, and an infidelity tolerance $\varepsilon$, the one-shot $\varepsilon$-error distillable $\phi$-non-stabilizerness of $\rho$ is defined to be the maximum number of $\phi$ non-stabilizer states achievable via stabilizer operations, with an error tolerance of $\varepsilon$:

[TABLE]

where $|\psi\rangle\!\langle\psi|\approx_{\varepsilon}\sigma$ is a shorthand for $\langle\psi|\sigma|\psi\rangle\geq 1-\varepsilon$ and SO for stabilizer operations.

In what follows, we focus on the one-shot distillable $H_{+}$-non-stabilizerness ${\cal M}_{H_{+}}^{\varepsilon}(\rho)$ and the one-shot distillable $T$-non-stabilizerness ${\cal M}_{T}^{\varepsilon}(\rho)$.

We first connect the task of magic-state distillation to quantum hypothesis testing between non-stabilizer states and operators in the set ${\cal W}$ (recall that $\text{STAB}\subsetneq{\cal W}$), and we note here that such an approach was previously taken in entanglement theory Rains (1999, 2001). Quantum hypothesis testing is the task of distinguishing two possible states $\rho_{0}$ and $\rho_{1}$ (null hypothesis $\rho_{0}$, alternative hypothesis $\rho_{1}$). We are allowed to perform a measurement characterized by the POVM $\{M,{\mathds{1}}-M\}$ with respective outcomes [math] and $1$. If the outcome is [math], we accept the null hypothesis. Otherwise, we accept the alternative one. The probabilities of type-I and type-II errors are given by $\operatorname{Tr}({\mathds{1}}-M)\rho_{0}$ and $\operatorname{Tr}M\rho_{1}$, respectively. The hypothesis testing relative entropy Buscemi and Datta (2010); Wang and Renner (2012) quantifies the minimum type-II error probability provided that the type-I error probability is within a given tolerance: D_{H}^{\varepsilon}(\rho_{0}\|\rho_{1}):=-\log_{2}\min\big{\{}\operatorname{Tr}M\rho_{1}\,\big{|}\,0\leq M\leq{\mathds{1}},\,1-\operatorname{Tr}M\rho_{0}\leq\varepsilon\big{\}}.

[TABLE]

A consequence of Proposition 3 is that the thauma of a quantum state is an upper bound on its distillable $H_{+}$ (or $T$) non-stabilizerness. Specifically, by applying the quantum Stein’s lemma Hiai and Petz (1991); Ogawa and Nagaoka (2000); Tomamichel and Hayashi (2013); Li (2014), we find the following:

[TABLE]

Efficiency of magic-state distillation.—The efficiency of distilling a non-stabilizer state $\xi$ from several independent copies of a resource state $\rho$ is given by the minimum number of copies of $\rho$ needed, on average, to produce $\xi$ using stabilizer operations:

[TABLE]

Previously, the authors of Veitch et al. (2014) derived the following lower bound on the efficiency of magic-state distillation:

[TABLE]

The lower bound in Veitch et al. (2014) was established by employing the mana of non-stabilizer states. Here, we utilize similar ideas and show that the max-thauma can also be applied to bound the efficiency of magic-state distillation.

Figure 1 demonstrates that the lower bound from Proposition 5 can be tighter than the lower bound in (4), thus giving an improved sense of the efficiency.

On the overhead of magic-state distillation.—The overhead of magic-state distillation is defined as the ratio of the number of input to output states, under a target error rate Bravyi and Haah (2012); Krishna and Tillich (2018). Although our notion of error for magic-state distillation is different from that typically employed in the literature, we note here that the inverse of the one-shot distillable $\phi$-non-stabilizerness (i.e., $[{\cal M}_{\phi}^{\varepsilon}(\rho)]^{-1}$) can be considered a reasonable way to measure the overhead of magic-state distillation. Then our upper bounds in Proposition 3 and Corollary 4 become lower bounds on the overhead.

Inequivalence between non-stabilizer states with maximal mana.—A fundamental problem in any quantum resource theory is to determine the interconversion rate between different resource states Chitambar and Gour (2019), in particular, between given states and maximally resourceful states. This is rooted in the fact that in any resource theory maximally resourceful states play a unique role in quantifying the resourcefulness of other states and assessing the performance of resource manipulation. Considering entanglement theory (or coherence theory) as an example, the interconversion between a given state and maximally entangled (coherent) states leads to fundamental tasks such as entanglement (coherence) distillation and dilution Horodecki et al. (2009); Streltsov et al. (2017); Chitambar and Gour (2019). Notably, any two maximally entangled (coherent) states under all resource measures in the same dimension are equivalent under free operations.

However, this is not the case in the resource theory of non-stabilizer states. Surprisingly, we find that even though the Strange state and the Norrell state each have maximum mana and are thus the most costly resource to simulate on a classical computer using certain known algorithms Veitch et al. (2014); Pashayan et al. (2015), they are not equivalent even in the asymptotic regime. Note that the mana plays a significant role as a measure of non-classical resources in quantum computation Mari and Eisert (2012); Pashayan et al. (2015). In particular, recall that mana is a non-stabilizerness measure analogous to logarithmic negativity in entanglement theory. In contrast, logarithmic negativity of a bipartite state is equal to its maximal value if and only if the state is maximally entangled.

To establish this result, we recall that the asymptotic conversion rate from $\rho$ to $\xi$ under asymptotically-non-stabilizer-non-generating transformations is given by the ratio of their regularized relative entropies of resource Brandão and Gour (2015). That is, $R(\rho\to\xi)={R_{{\cal M}}^{\infty}({\rho})}/{R_{{\cal M}}^{\infty}({\xi})}.$ We further recall that the Strange and Norrell states have maximum mana Veitch et al. (2014): ${\cal M}({{\mathbb{S}}})={\cal M}({{\mathbb{N}}})=\log_{2}(5/3).$ However, Proposition 2 and the fact that $R_{{\cal M}}^{\infty}({{{\mathbb{S}}}})\geq\theta^{\infty}({{{\mathbb{S}}}})$ indicate that there is a gap between their “regularized relative entropies of magic.” As a consequence, we find that

[TABLE]

Since stabilizer operations are included in the set of asymptotically-non-stabilizer-non-generating transformations, this result also establishes that the rate to obtain the Strange state from the Norrell state is smaller than one under stabilizer operations. Thus, the gap between $R_{{\cal M}}^{\infty}({{{\mathbb{N}}}})$ and $R_{{\cal M}}^{\infty}({{{\mathbb{S}}}})$, as established in Theorem 6, closes an open question from (Veitch et al., 2014, Section 4.4).

This result demonstrates a fundamental difference between the resource theory of non-stabilizer states and the resource theory of entanglement or coherence. Specifically, we show that the maximally resourceful non-stabilizer states under certain resource measure cannot be interconverted at a rate equal to one, even in the asymptotic regime, while the maximally resourceful states in entanglement theory or coherence theory can be interconverted equivalently in the one-copy setting. However, it remains open to determine whether the conversion rate from the Strange state to the Norrell state is strictly smaller than $\log_{2}(5/3)/\log_{2}(3/2)$. Such an inequality would imply the irreversibility of asymptotic magic state manipulation.

Conclusions.—We have introduced the thauma family of measures to quantify and characterize the non-stabilizerness resource possessed by quantum states that are needed for universal quantum computation. The min- and max-thauma are efficiently computable by semi-definite programming and lead to bounds on the rates at which one can interconvert non-stabilizer states. These bounds have helped to solve pressing open questions in the resource theory of non-stabilizer states. More generally, our work establishes fundamental limitations to the processing of quantum non-stabilizerness, opening new perspectives for its investigation and exploitation as a resource in quantum information processing and quantum technology. Along this line, we suspect that our results will have immediate impact on the quantum optics community working on the resource theory of non-Gaussianity Zhuang et al. (2018); Takagi and Zhuang (2018); Albarelli et al. (2018) and continuous-variable quantum computing Lloyd and Braunstein (1999); Gottesman et al. (2001), because the main idea behind the thauma measure can be generalized to this setting.

Appendix A The stabilizer formalism and discrete Wigner function

For most known fault-tolerant schemes, the restricted set of quantum operations is the stabilizer operations, consisting of preparation and measurement in the computational basis and a restricted set of unitary operations. Here we review the basic elements of the stabilizer formalism for quantum systems of prime dimension.

We denote a Hilbert space of dimension $d$ by ${\cal H}_{d}$ and the standard computational basis by $\{|j\rangle\}_{j=0,\ldots,d-1}$. For a prime number $d$, we define the respective shift and boost operators $X,Z\in{\cal L}({\cal H}_{d})$ as

[TABLE]

where $\oplus$ denotes addition modulo $d$. We define the Heisenberg–Weyl operators as

[TABLE]

where $\tau=e^{(d+1)\pi i/d}$ and ${{\textbf{u}}}=(a_{1},a_{2})\in{{\mathbb{Z}}}_{d}\times{{\mathbb{Z}}}_{d}$.

For a system with composite Hilbert space ${\cal H}_{A}\otimes{\cal H}_{B}$, the Heisenberg–Weyl operators are the tensor product of the subsystem Heisenberg–Weyl operators:

[TABLE]

where ${{\textbf{u}}}_{A}\oplus{{\textbf{u}}}_{B}$ is an element of ${{\mathbb{Z}}}_{d_{A}}\times{{\mathbb{Z}}}_{d_{A}}\times{{\mathbb{Z}}}_{d_{B}}\times{{\mathbb{Z}}}_{d_{B}}$.

The set ${\cal C}_{d}$ of Clifford operators is defined to be the set of unitary operators that map Heisenberg–Weyl operators to Heisenberg–Weyl operators under conjugation up to phases:

[TABLE]

The Clifford operators form the Clifford group. The Clifford group for qutrits can be generated by the qutrit Hadamard gate in (1), the phase gate $S=|0\rangle\!\langle 0|+|1\rangle\!\langle 1|+\omega|2\rangle\!\langle 2|$, and the sum gate $\sum_{i,j=0}^{2}|i\rangle\langle i|\otimes|i\oplus j\rangle\langle j|$ Hostens et al. (2005).

The pure stabilizer states can be obtained by applying Clifford operators to the state $|0\rangle$:

[TABLE]

The set of stabilizer states is the convex hull of the set of pure stabilizer states:

[TABLE]

The discrete Wigner function Wootters (1987); Gross (2006, 2007) was used to show the existence of bound magic states Veitch et al. (2012), which are states from which it is impossible to distill magic at a strictly positive rate. For an overview of discrete Wigner functions, we refer to Veitch et al. (2012, 2014); Ferrie (2011) for more details. See also Ferrie (2011) for quasi-probability representations of quantum theory with applications to quantum information science.

For each point u in the discrete phase space, there is a corresponding operator $A_{{\textbf{u}}}$, and the value of the discrete Wigner representation of $\rho$ at this point is given by

[TABLE]

where $\{A_{{\textbf{u}}}\}_{{\textbf{u}}}$ are the phase-space point operators:

[TABLE]

In general, we define the discrete Wigner function of an operator $V$ as $W_{V}({{\textbf{u}}}):=\frac{1}{d}\operatorname{Tr}A_{{\textbf{u}}}V$. Some properties of the set $\{A_{{\textbf{u}}}\}_{{\textbf{u}}}$ of phase-space point operators are listed as follows:

$A_{{\textbf{u}}}$ is Hermitian; 2. 2.

$\sum_{{\textbf{u}}}A_{{\textbf{u}}}/d={\mathds{1}}$; 3. 3.

$\operatorname{Tr}A_{{\textbf{u}}}A_{{{\textbf{u}}}^{\prime}}=d\ \delta({{\textbf{u}}},{{\textbf{u}}}^{\prime})$ ; 4. 4.

$\operatorname{Tr}A_{{\textbf{u}}}=1$; 5. 5.

$V=\sum_{{\textbf{u}}}W_{V}({{\textbf{u}}})A_{{\textbf{u}}}$; 6. 6.

$\operatorname{Tr}VY=d\sum_{{{\textbf{u}}}}W_{V}({{\textbf{u}}})W_{Y}({{\textbf{u}}})$, for $d$-dimensional operators $V$ and $Y$.

Definition 1

The Wigner trace norm of a Hermitian operator $V$ is defined as

[TABLE]

Definition 2

The Wigner spectral norm of a Hermitian operator $V$ acting on a space of dimension $d$ is defined as

[TABLE]

From these definitions and duality of norms, we find that

[TABLE]

Appendix B Efficient computation of thauma via convex optimization

Let us recall that the thauma of a quantum state $\rho$ is given by

[TABLE]

Note that $\theta(\rho)$ is a convex optimization problem with relative entropy as the objective function, due to the fact that the relative entropy is convex in $\sigma$ and the constraint $\sigma\in{\cal W}$ is convex. Indeed, the minimization over $\sigma\in{\cal W}$ can be equivalently written as a minimization over the following set:

[TABLE]

Then using the efficient optimization algorithm of the quantum relative entropy from Fawzi and Fawzi (2018), we provide, in our ancillary files on the arXiv, a program to compute the thauma $\theta(\rho)$ of a quantum state $\rho$.

Appendix C Generalized thauma measures

In the main text, we discussed the notion of generalized thauma as arising from any relative-entropy like measure that is non-increasing under the action of a quantum channel. Furthermore, we mentioned how one could have Petz–Rènyi and sandwiched Rényi thauma measures. Here, we explicitly give the definitions of such measures.

Recall that the Petz–Rényi relative entropy of a state $\rho$ and a positive semi-definite operator $\sigma$ is defined for $\alpha\in(0,1)\cup(1,\infty)$ as Petz (1986)

[TABLE]

whenever either $\alpha\in(0,1)$ and $\rho$ is not orthogonal to $\sigma$ in Hilbert-Schmidt inner product or $\alpha>1$ and ${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$. Otherwise we set $D_{\alpha}(\rho\|\sigma)\coloneqq\infty$. In the above and throughout, we employ the convention that inverses are to be understood as generalized inverses. For $\alpha\in\{0,1\}$, the Petz-Rényi relative entropy is defined in the limit as

[TABLE]

where $P_{\rho}$ denotes the projection onto the support of $\rho$. The Petz–Rényi relative entropies are ordered, in the sense that

[TABLE]

for $\alpha\geq\beta\geq 0$ Tomamichel et al. (2009); Mosonyi and Hiai (2011).

The sandwiched Rényi relative entropy is defined for a state $\rho$ and a positive semi-definite operator $\sigma$ and $\alpha\in(0,1)\cup(1,\infty)$ as Müller-Lennert et al. (2013); Wilde et al. (2014)

[TABLE]

whenever either $\alpha\in(0,1)$ and $\rho$ is not orthogonal to $\sigma$ in Hilbert-Schmidt inner product or $\alpha>1$ and ${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$. Otherwise we set $\widetilde{D}_{\alpha}(\rho\|\sigma)\coloneqq\infty$. For $\alpha=1$, we define the sandwiched Rényi relative entropy in the limit as Müller-Lennert et al. (2013); Wilde et al. (2014)

[TABLE]

We have that

[TABLE]

with Uhlmann’s fidelity defined as $F(\rho,\sigma)\coloneqq\|\sqrt{\rho}\sqrt{\sigma}\|_{1}^{2}$ Uhlmann (1976). In the limit $\alpha\to\infty$, the sandwiched Rényi relative entropy converges to the max-relative entropy Datta (2009)

[TABLE]

as shown in Müller-Lennert et al. (2013). The sandwiched Rényi relative entropies are ordered, in the sense that

[TABLE]

for $\alpha\geq\beta\geq 0$ Müller-Lennert et al. (2013).

The Petz–Rényi relative entropy obeys the data-processing inequality for all $\alpha\in[0,1)\cup(1,2]$ Petz (1986):

[TABLE]

it is non-negative for a state $\rho$ and a subnormalized state $\sigma$, and equal to zero if $\rho=\sigma$. Thus, the Petz–Rényi thauma defined as

[TABLE]

is a magic monotone. We remark here that the min-thauma $\theta_{\min}(\rho)$ employed in the main text is equal to $\theta_{0}(\rho)$.

The sandwiched Rényi relative entropy obeys the data-processing inequality for all $\alpha\in[1/2,1)\cup(1,\infty]$ Frank and Lieb (2013); Beigi (2013):

[TABLE]

it is non-negative for a state $\rho$ and a subnormalized state $\sigma$, and equal to zero if $\rho=\sigma$. (See Wilde (2018) for an alternative proof of the data processing inequality.) Thus, the sandwiched Rényi thauma defined as

[TABLE]

is a magic monotone. We remark here that the max-thauma $\theta_{\max}(\rho)$ employed in the main text is equal to $\lim_{\alpha\to\infty}\widetilde{\theta}_{\alpha}(\rho)$.

From the ordering of the Rényi relative entropies in (23) and (30) and the limits in (22) and (25), we conclude the following ordering of some of the thauma measures:

[TABLE]

Appendix D SDPs of min- and max-thauma and proof of Proposition 1

We first derive the dual SDP for the min-thauma:

[TABLE]

In the above proof, Eq. (37) follows from the definition of the Wigner trace norm. Eq. (38) utilizes the duality of SDPs. Eq. (39) introduces a new variable $q_{{\textbf{u}}}=x_{{{\textbf{u}}}}-y_{{\textbf{u}}}$.

Thus, adapting the definition of the Wigner spectral norm in (15), we find that

[TABLE]

For max-thauma, its primal and dual SDPs are as follows:

[TABLE]

**Proposition 1

**** (Additivity of min- and max-thauma)**

For states $\rho$ and $\tau$, the following additivity equality holds

[TABLE]

Consequently, $\theta_{\min}(\rho)\leq\theta^{\infty}(\rho)\leq\theta_{\max}(\rho)$.

**Proof. ** Suppose that the optimal solutions to the SDP (36) of states $\rho_{1}$ and $\rho_{2}$ are $\sigma_{1}$ and $\sigma_{2}$, respectively. It is clear that

[TABLE]

which means that

[TABLE]

On the other hand, suppose that the optimal solutions to the SDP (42) of $\rho_{1}$ and $\rho_{2}$ are $Q_{1}$ and $Q_{2}$, respectively. It is clear that

[TABLE]

which means that

[TABLE]

The proof of additivity of max-thauma is similar to the proof of additivity given above for min-thauma. The idea is to use both the primal and dual SDPs of the max-thauma.

The ordering property $\theta_{\min}(\rho)\leq\theta^{\infty}(\rho)\leq\theta_{\max}(\rho)$ is a consequence of the ordering in (35) and additivity of min- and max-thauma. $\blacksquare$

Appendix E Monotonicity of max-thauma on average under stabilizer operations

Proposition 8

The max-thauma does not increase on average under stabilizer operations. That is, suppose that there is a general stabilizer protocol ${\cal N}=\{{\cal N}_{j}\}_{j}$ that transforms $\rho$ to $\sigma_{j}={\cal N}_{j}(\rho)/p_{j}$ with probability $p_{j}=\operatorname{Tr}{\cal N}_{j}(\rho)$, where ${\cal N}_{j}$ is a completely positive stabilizer map and $\sum_{j}{\cal N}_{j}$ is trace preserving. Then

[TABLE]

**Proof. ** Let us suppose $V$ is the optimal solution to the SDP (44). From complete positivity of the map ${\cal N}_{j}$, it follows that

[TABLE]

which means that ${\cal N}_{j}(V)/p_{j}$ is feasible for the SDP (44) of $\theta_{\max}(\sigma_{j})$. Thus,

[TABLE]

and we find that

[TABLE]

The first inequality follows from an inequality stated in the proof of Theorem 4 of Veitch et al. (2014). It can be understood as being a consequence of the fact that the stabilizer operation $\{{\cal N}_{j}\}_{j}$ has a discrete Wigner representation as a conditional probability distribution, which does not increase the classical one norm. $\blacksquare$

Appendix F Min-, regularized, and max-thaumas of Strange state, Norrell state, $H_{+}$ state, and $T$ state, and proof of Theorem 6

Let us first recall the definitions of the Strange state, Norrell state, $H_{+}$ state, and $T$ state:

[TABLE]

where $\xi=e^{2\pi i/9}$ and we recall that $H_{+}$ magic state is the $+1$-eigenstate of the $H$ gate defined in (1).

**Proposition **2

For the Strange state, Norrell state, $H_{+}$ state, and $T$ state, their min-thauma, regularized thauma, max-thauma, and in some cases, regularized relative entropy of magic can be evaluated exactly as

[TABLE]

**Proof. ** Our proof strategy for all four cases is the same. For each case, we first bound the min-thauma of the given state from below, and then we bound its max-thauma from above. In each case, we find that the lower and upper bounds match, allowing us to conclude the equalities. If the optimal state for the max-thauma is a stabilizer state, then one can conclude from definitions that max-thauma is an upper bound on the regularized relative entropy of magic, and so this is how we obtain equalities involving regularized relative entropy of magic in some cases. We remark here that we found these results by first numerically solving the SDPs for min- and max-thauma, and then determining an analytic form for the optimal values and solutions for the SDPs. We have made our Matlab and Mathematica source files available (see ancillary files on arXiv). We numerically solved the SDPs using Matlab and then we confirmed the analytic solutions using Mathematica.

Strange state. We begin with the Strange state ${{\mathbb{S}}}$, by bounding its min-thauma $\theta_{\min}({{{\mathbb{S}}}})$ from below. First, we choose a feasible solution to the dual SDP (42) as

[TABLE]

As $\|Q\|_{W,\infty}=\frac{3}{5}$, it holds that

[TABLE]

We now bound the max-thauma $\theta_{\max}({{{\mathbb{S}}}})$ from above by the same value. Consider that the sub-normalized state $\sigma=\frac{3}{5}{{\mathbb{S}}}$ satisfies $\|\sigma\|_{W,1}=1$. This implies that

[TABLE]

By exploiting the above bounds and the inequalities $\theta_{\min}({{{\mathbb{S}}}})\leq\theta^{\infty}({{{\mathbb{S}}}})\leq\theta_{\max}({{{\mathbb{S}}}})$ from Proposition 1, we conclude the equalities in (66).

Norrell state. We turn to the Norrell state ${{\mathbb{N}}}$, and we bound its min-thauma from below. Let us take

[TABLE]

as a feasible solution to the SDP (42). By direct calculation, we find that $\|Q\|_{W,\infty}=2/3$, so that

[TABLE]

We now bound the max-thauma of the Norrell state from above. Let us choose the following stabilizer state:

[TABLE]

which is a convex mixture of the following stabilizer states:

[TABLE]

The state $|u_{0}\rangle=X|0\rangle$, $|u_{1}\rangle=SHX|0\rangle$, and $|u_{2}\rangle=S^{2}HX^{2}|0\rangle$. Thus, each state is a stabilizer state as claimed. Then,

[TABLE]

The last equality follows by direct calculation.

By exploiting (74) and (79) and the inequalities $\theta_{\min}({{{\mathbb{N}}}})\leq\theta^{\infty}({{{\mathbb{N}}}})\leq\theta_{\max}({{{\mathbb{N}}}})$ from Proposition 1, we conclude the equalities $\theta_{\min}({{{\mathbb{N}}}})=\theta^{\infty}({{{\mathbb{N}}}})=\theta_{\max}({{{\mathbb{N}}}})=\log_{2}\frac{3}{2}$ in (67). The inequality $\log_{2}\frac{3}{2}=\theta^{\infty}({{\mathbb{N}}})\leq R_{{\cal M}}^{\infty}({{\mathbb{N}}})$ follows by definition. The opposite inequality follows because $\tau$ is a stabilizer state, so that for all $n\geq 1$,

[TABLE]

which implies that $R_{{\cal M}}^{\infty}({{\mathbb{N}}})\leq\log_{2}\frac{3}{2}$.

$H_{+}$** state.** We consider the $H_{+}$ state, and we bound its min-thauma from below. Let us take

[TABLE]

as a feasible solution to the SDP (42). By direct calculation, we find that

[TABLE]

Thus, we have that

[TABLE]

We now bound the max-thauma of the $H_{+}$ state from above. Let us choose the stabilizer state

[TABLE]

with

[TABLE]

Then it is clear that

[TABLE]

which can be confirmed by direct calculation, using the fact that

[TABLE]

By exploiting (83) and (86) and the inequalities $\theta_{\min}(H_{+})\leq\theta^{\infty}(H_{+})\leq\theta_{\max}(H_{+})$ from Proposition 1, we conclude the equalities $\theta_{\min}(H_{+})=\theta^{\infty}(H_{+})=\theta_{\max}(H_{+})=\log_{2}(3-\sqrt{3})$ in (68). The inequality $\log_{2}(3-\sqrt{3})=\theta^{\infty}(H_{+})\leq R_{{\cal M}}^{\infty}(H_{+})$ follows by definition. The opposite inequality follows because $\sigma$ is a stabilizer state, so that for all $n\geq 1$,

[TABLE]

which implies that $R_{{\cal M}}^{\infty}(H_{+})\leq\log_{2}(3-\sqrt{3})$.

$T$** state.** We consider the $T$ state, and we bound its min-thauma from below. Let us take

[TABLE]

where

[TABLE]

Then one finds that $(\|Q\|_{W,\infty})^{-1}=1+2\sin(\pi/18)$, so that

[TABLE]

We now bound the max-thauma of the $T$ state from above. Let us choose the following state $\tau$ with positive Wigner function:

[TABLE]

with

[TABLE]

The state $\sigma$ has eigenvalues equal to [math], $\frac{2}{\sqrt{3}}\sin(2\pi/9)\approx 0.742227$, and $\frac{2\sin(\pi/9)\sin(2\pi/9)}{\sqrt{3}\cos(\pi/18)}\approx 0.257773$, and it is thus positive semi-definite as claimed. A calculation gives that the discrete Wigner function of the above state is described by the following matrix:

[TABLE]

where

[TABLE]

confirming that it has only non-negative entries as claimed. Therefore,

[TABLE]

The last inequality can be verified from the fact that the operator $(1+2\sin(\pi/18))\tau-|T\rangle\!\langle T|$ is positive semi-definite, having two zero eigenvalues and one non-zero eigenvalue equal to $\frac{\sin(\pi/9)}{\cos(\pi/18)}\approx 0.347296$.

By exploiting (91) and (96) and the inequalities $\theta_{\min}(T)\leq\theta^{\infty}(T)\leq\theta_{\max}(T)$ from Proposition 1, we conclude the equalities in (69). $\blacksquare$

Appendix G Proof of Proposition 3

**Proposition **3

Given a state $\rho$, the following holds

[TABLE]

**Proof. ** Let us suppose that the stabilizer operation $\Lambda$ satisfies $\Lambda(\rho)\approx_{\varepsilon}|{H_{+}}\rangle\!\langle{H_{+}}|^{\otimes k}$ for some $k$. Recall that for any $\tau\in{\cal W}$, we have that $\operatorname{Tr}[|\phi\rangle\!\langle\phi|^{\otimes n}\tau]\leq 2^{-n\theta_{\min}(\phi)}$. Given $\sigma\in{\cal W}$, and using the above fact, we have

[TABLE]

Therefore, we have that

[TABLE]

The second inequality follows from the definition of $D_{H}^{\varepsilon}$. The third inequality follows from the data processing inequality of $D_{H}^{\varepsilon}$.

Finally, minimizing over all $\sigma\in{\cal W}$, we find that

[TABLE]

Since $k$ is arbitrary, we arrive at the bound in (2). The proof of (3) is similar. $\blacksquare$

Appendix H Second-order converse

**Corollary **4

For a quantum state $\rho$, number $n$ of available source states, and error tolerance $\varepsilon\in(0,1)$, the following inequality holds

[TABLE]

where $\sigma$ is a minimizer of $\theta(\rho)$, $\Phi^{-1}$ is the inverse of the cumulative normal distribution function, and $V(\rho\|\sigma)=\operatorname{Tr}\rho(\log_{2}\rho-\log_{2}\sigma)^{2}-D(\rho\|\sigma)^{2}$. Similarly,

[TABLE]

**Proof. ** Suppose $\sigma$ is any minimizer of the thauma $\theta(\rho)$. For given $n$ and $\varepsilon$,

[TABLE]

The first line follows from the upper bound in Proposition 3, while the second line uses the second-order expansion of quantum hypothesis testing relative entropy Tomamichel and Hayashi (2013); Li (2014). $\blacksquare$

Appendix I Bound on efficiency of magic-state distillation

**Proposition **5

The efficiency of distilling a magic state $\sigma$ from independent copies of the resource state $\rho$ is bounded from below by

[TABLE]

**Proof. ** The proof is in the same spirit of the mana bound in Veitch et al. (2014). Let us suppose that the stabilizer protocol $\Lambda$ outputs $\sigma$ from $\rho^{\otimes n}$ with probability $p$. Without loss of generality, we can assume that

[TABLE]

where $\tau$ is some state. Applying Proposition 8 and the fact that $|0\rangle$ is a stabilizer state, we find that

[TABLE]

which means that the following bound holds for all integer $n\geq 1$ and $p\in(0,1]$:

[TABLE]

This concludes the proof. $\blacksquare$