Designing good bosonic quantum codes via creating destructive interference

TL;DR

This paper introduces a novel approach to designing bosonic quantum error-correcting codes by exploiting destructive interference through phase manipulation, leading to improved error suppression in continuous-variable quantum systems.

Contribution

It proposes using phase degrees of freedom in basis state amplitudes to create destructive interference, enhancing code performance against excitation loss and Kerr effects.

Findings

Phase manipulation significantly reduces decoherence in bosonic codes.

Tailored recovery strategies improve error correction effectiveness.

The approach applies to various bases and multi-qubit codes.

Abstract

Continuous-variable systems protected by bosonic quantum error-correcting codes have emerged as a promising platform for quantum information processing. To date, design of codewords has centered on optimizing the occupation of basis states in the error-relevant basis. Here, we propose utilizing the phase degree of freedom in basis state probability amplitudes to devise codes that feature destructive interference, and thus reduced overlap, between error codewords. To showcase, we first consider the correction of excitation loss using single-mode codes with Fock-space parity structure and show that, with a tailored "two-level" recovery, altering the signs of probability amplitudes can significantly suppress decoherence. We then study the joint channel of excitation loss and Kerr effect, and show the critical role of nontrivial phase for optimal quantum codes for such intricate yet important channels. The principle is extended to improve bosonic codes defined in other bases and multi-qubit codes, showing its wide applicability in quantum error correction.

Figures

Tables

Table 1. Table 1: Results from biconvex optimization under energy constraint n ¯ c = 2 subscript ¯ 𝑛 c 2 \bar{n}_{\mathrm{c}}=2 for single-mode RCQCs and CCQCs at γ = 0.1 𝛾 0.1 \gamma=0.1 and different K ​ t 𝐾 𝑡 Kt . Wigner functions of maximally mixed state 1 2 ​ P ^ 𝚌𝚘𝚍𝚎 1 2 subscript ^ 𝑃 𝚌𝚘𝚍𝚎 \frac{1}{2}\hat{P}_{\textnormal{$\mathtt{code}$}} are shown for the obtained optimal codes, respectively, with associated channel infidelities 1 − F o 1 superscript 𝐹 o 1-F^{\mathrm{o}} below them. The infidelities for 𝚋𝚒𝚗 ​ ( 2 , 2 ) 𝚋𝚒𝚗 2 2 \textnormal{$\mathtt{bin}$}(2,2) are shown for comparison.

$Kt$	0	0.5	1	1.5
RCQCs	6.1e-3	1.4e-2	2.5e-2	3.6e-2
CCQCs	6.1e-3	1.2e-2	2.3e-2	3.2e-2
$\mathrm{𝚋𝚒𝚗}(2,2)$	1.8e-2	2.1e-2	3.0e-2	4.5e-2

Equations

$$|{\mu }_{\sigma }⟩=\sum _{n\in S_{\sigma }}{c}_n^{\sigma }{e}^{i{\theta }_n^{\sigma }}|n⟩$$

$$⟨{\mu }_{\sigma }|{\hat{E}}_k^{†}{\hat{E}}_l|{\mu }_{{\sigma }^{\prime }}⟩$$

$$|0_L⟩$$

$$|1_L⟩$$

$$|0_{\text{bin}}/1_{\text{bin}}⟩$$

$$|0_{\text{sab}}⟩$$

$${\hat{U}}_{\mathrm{S}}|0_{\text{bin}}⟩=|0_{\text{sab}}⟩,{\hat{U}}_{\mathrm{S}}|1_{\text{bin}}⟩$$

$${\hat{R}}_k^{(2)}$$

$${\hat{R}}_{S+k}^{(2)}=\sum _{\sigma }(|{\mu }_{\bar{\sigma }}⟩⟨{\nu }_{\bar{\sigma }}^k|+{\hat{U}}_{S+k}^{\mathrm{res}}){\hat{P}}_{S+k}$$

$$F$$

$${\mathcal{N}}_{\gamma ,Kt}=e^{-\frac{1}{2}iKt[{\hat{n}}^2,\cdot ]-\ln (1-\gamma )(\hat{a}\cdot {\hat{a}}^{†}-\frac{1}{2}\{\hat{n},\cdot \})}.$$

$$|{\mu }_{\sigma }⟩=\sum _{s\in \mathbb{Z}}(-1{)}^{\sigma }|q=\alpha (\sigma +2s)⟩$$

Full text

Introduction

Quantum operations with continuous variables represent a promising alternative path towards scalable quantum computing and communication Knill et al. (2001); Braunstein and van Loock (2005); Gottesman et al. (2001); Ralph et al. (2003). Similar to qubit-based systems, a major challenge for faithful bosonic quantum information is to store, manipulate and communicate the encoded information in the presence of noise, such as excitation loss (aka amplitude damping, pure loss), phase-space drift, dephasing and cavity nonlinearities. To overcome excitation loss that fundamentally limits the cavity lifetime, multi-mode codes were first introduced Chuang et al. (1997); Leung et al. (1997); Ouyang (2014); Bergmann and van Loock (2016a); Niu et al. (2018); for phase-space drift, Gottesman, Kitaev, and Preskill ($\mathtt{gkp}$) codes Gottesman et al. (2001); Harrington and Preskill (2001) were proposed. More recently, motivated by the potential to utilize higher excitation states in a bosonic Hilbert space and perform hardware-efficient operations, single-mode codes for excitation loss such as $\mathtt{cat}$ codes Leghtas et al. (2013); Mirrahimi et al. (2014); Ofek et al. (2016); Bergmann and van Loock (2016b); Li et al. (2017) and binomial ($\mathtt{bin}$) codes Michael et al. (2016) were developed. Meanwhile, progress in superconducting circuit quantum electrodynamics (QED) Sun et al. (2014); Krastanov et al. (2015); Heeres et al. (2015) – e.g. FPGA adaptive control Ofek et al. (2016), readout of excitation parity Sun et al. (2014) and universal control of cavity states Krastanov et al. (2015); Heeres et al. (2015) – has opened up possibilities once thought unreachable, including implementing arbitrary quantum channels on a bosonic mode Shen et al. (2017); Lloyd and Viola (2001). With the advances, error-corrected $\mathtt{cat}$ and $\mathtt{bin}$ qubits and the associated universal quantum gate sets have been demonstrated, respectively Ofek et al. (2016); Heeres et al. (2017); Hu et al. . These capabilities are essential for higher-level operations such as distributing error-corrected entangled states Axline et al. (2018) and quantum gate teleportation Chou et al. (2018).

For the aforementioned codes, in the computational basis relevant to the error under consideration, i.e. Fock basis for \textnormal{\mathtt{cat}}/\textnormal{\mathtt{bin}} codes and position/momentum basis for square-lattice $\mathtt{gkp}$ codes, the codewords are spanned by distinct subsets of basis states with positive probability amplitudes. Recently, $\mathtt{gkp}$ codes have been found to protect against excitation loss extremely well, even though they were not originally designed for loss errors Albert et al. (2018); Noh et al. (2018). Similar to the newly discovered numerically optimized codes Michael et al. (2016); Albert et al. (2018), when expressed in the Fock basis, their codewords do not possess parity structure, yet feature negative probability amplitudes. The findings inspire us to better understand the recipe behind desirable quantum error-correction (QEC) capabilities and generalize the findings to devise more efficient codes.

In this Letter, we explore the conjugate degree of freedom to basis state occupation – the relative phases – and demonstrate its critical role for efficient quantum codes. Firstly, we show that tuning the phase degree of freedom in the codewords can improve code performance via making error codewords interfere destructively. As illustrations, concerning the correction of excitation loss, we modify $\mathtt{bin}$ and $\mathtt{cat}$ codes by periodically altering the signs of probability amplitudes in one codeword, and show that the sign alteration (SA) effectively reduces the overlap between selected error codewords. The periodic SA can be experimentally realized via adding a cavity Kerr to the encoding procedure. To capture the enhanced separation created by SA, we propose a two-level recovery that yields error protection close to optimal at loss rates of practical interest. The enhancement naturally leads to a question: Is phase degree of freedom necessary for $\mathit{optimal}$ code constructs? Using biconvex optimization of encoding and decoding Reimpell and Werner (2005); Kosut and Lidar (2009); Noh et al. (2018), we compute the optimal code given an error process and find that, although nontrivial phase (neither [math] nor $\pi$) is redundant for excitation loss, it is necessary for more complicated ones such as the joint channel of excitation loss and Kerr effect.

Extending the principle to codes defined in bases beyond Fock, we note that GKP codes over non-square lattices, which better overcome phase-space drift than square lattice GKP, can be generated from the later through SA in position-momentum basis Gottesman et al. (2001). In addition, noting that well-known multi-qubit CSS codes Shor (1995); Steane (1996); Kitaev (2003) share the same feature of all positive probability amplitudes with $\mathtt{bin}$ and $\mathtt{cat}$ codes, we modify the nine-qubit Shor code for enhanced protection against Pauli errors and amplitude damping error, demonstrating the wide applicability of the principle for enhanced quantum error correction.

Principle

We begin with introducing the principle that engineering the phase degree of freedom in codewords can create destructive interference between them after error, thus suppressing undesired overlap. Given a qudit code embedded in a Hilbert space $\mathcal{H}=\{|n\rangle\}$ where $n$ indexes computational basis states, normalized codewords in a chosen logical basis can be written as

[TABLE]

where $\sigma=0,1,...,d-1$ labels codewords, $c_{n}^{\sigma}>0$ and $S_{\sigma}$ is a subset of the computational basis chosen for each codeword. The error process that the code in Eq. (1) is devised to protect against can be expressed in the Kraus representation $\mathcal{E}(\rho)=\sum_{k}\hat{E}_{k}\rho\hat{E}_{k}^{\dagger}$, with each operator $\hat{E}_{k}$ associated with an error event. The decoherence under $\mathcal{E}$ is captured by the QEC matrix $M_{kl,\sigma\sigma^{\prime}}=\langle\mu_{\sigma}|\hat{E}_{k}^{\dagger}\hat{E}_{l}|\mu_{\sigma^{\prime}}\rangle$ Bennett et al. (1996); Knill and Laflamme (1997), i.e. overlap between codewords under errors, and

[TABLE]

We see that, depending on the nature of error and its relation to the computational basis, magnitude $c_{n}^{\sigma}$ and phase $\theta_{n}^{\sigma}$ are both critical in suppressing Eq. (2).

Figure 1 illustrates how phase allows codewords, after an error occurs, to remain distinguishable. Fig. 1(a) shows the case of positive-coefficient quantum code (PCQC), for which $\theta_{n}^{\sigma}=0$ and it relies only on $c_{n}^{\sigma}$ and a wise choice of $S_{\sigma}$ to ensure orthogonality between codewords and QEC capacity. Examples include $\mathtt{bin}$, $\mathtt{cat}$, multi-mode Chuang et al. (1997); Leung et al. (1997), square-lattice $\mathtt{gkp}$ codes for phase-space drift and multi-qubit CSS codes. Once error $\hat{E}_{k}$ takes $|\mu_{\sigma}\rangle$ to the subspace $|\mu_{\sigma^{\prime}}\rangle$ lies in, $|\mu_{\sigma^{\prime}}\rangle$ and error word $|\mu_{\sigma}^{k}\rangle\coloneqq\hat{E}_{k}|\mu_{\sigma}\rangle/\sqrt{\langle\mu_{\sigma}|\hat{E}_{k}^{\dagger}\hat{E}_{k}|\mu_{\sigma}\rangle}$ will overlap, inducing decoherence. However, if we consider a real-coefficient quantum code (RCQC) where $\theta_{n}^{\sigma}\in\{0,\,\pi\}$ [Fig. 1(b)], or even complex-coefficient quantum code (CCQC) where $\theta_{n}^{\sigma}\in[0,\,2\pi)$ [Fig. 1(c)], dependent on the error under consideration and selected computational basis, larger separation between $|\mu_{\sigma}^{k}\rangle$ and $|\mu_{\sigma^{\prime}}\rangle$ can potentially be realized due to destructive interference between basis states. Without losing generality, we focus on quantum codes that encode a qubit.

To see how better codes can emerge as a result, we begin with a neat example – the bosonic $\sqrt{17}$-code Michael et al. (2016) – that corrects a single loss without a parity structure key to other codes

[TABLE]

$\hat{a}|0_{L}\rangle$ ($\hat{a}$ is annihilation operator) and $|1_{L}\rangle$, spanned by different Fock states, do not overlap. Meanwhile, one can test that, due to destructive interference, $\hat{a}|1_{L}\rangle\propto\sqrt{9-\sqrt{17}}|0\rangle-2\sqrt{\sqrt{17}-3}|3\rangle$ is also orthogonal to $|0_{L}\rangle$, thus allowing the code to fully correct one excitation loss.

Sign-altered $\mathtt{bin}$ code for excitation loss

The energy decay of an oscillator is described by excitation loss channel $\mathcal{N}_{\gamma}$ ($\gamma$ is loss rate), for which Kraus operator $\hat{E}_{k}=\sqrt{\frac{\gamma^{k}}{k!}}(1-\gamma)^{\frac{\hat{n}}{2}}\hat{a}^{k}$ Chuang et al. (1997); Michael et al. (2016); Li et al. (2017) ($\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the excitation number operator) is associated with losing $k$ excitations. To correct the multi-loss events, based on \textnormal{\mathtt{bin}}(N,S) that corrects exactly $S-1$ losses for $N\geq S$

[TABLE]

we apply a periodic SA to |0_{\mathrm{\textnormal{\mathtt{bin}}}}\rangle while keeping those of |1_{\mathrm{\textnormal{\mathtt{bin}}}}\rangle unchanged, and obtain sign-altered binomial ($\mathtt{sab}$) code with

[TABLE]

and |1_{\textnormal{\mathtt{sab}}}\rangle=|1_{\mathrm{\textnormal{\mathtt{bin}}}}\rangle. With the same parity structure, \textnormal{\mathtt{sab}}(N,S) code also corrects $S-1$ losses perfectly as \textnormal{\mathtt{bin}}(N,S). The advantage emerges when we examine higher-order QEC matrix entries $\langle\mu_{\sigma}|\hat{E}_{k}^{\dagger}\hat{E}_{S+k}|\mu_{\sigma^{\prime}}\rangle\ (k=0,1,\ldots)$: Here, the $(-1)^{\frac{p}{2}}$ in Eq. (10) leads to destructive interference between the two error codewords, reducing the overlap responsible for logical-X error.

The encoding procedure of $\mathtt{sab}$ code builds on that of $\mathtt{bin}$ code with an additional Kerr unitary that imprints SA to |0_{\mathrm{\textnormal{\mathtt{bin}}}}\rangle while acting trivially on |1_{\mathrm{\textnormal{\mathtt{bin}}}}\rangle as a result of the parity structure. To see this, we first introduce Kerr unitary $\hat{U}_{\mathrm{Kr}}=e^{\frac{1}{2}iKt\hat{n}^{2}}$ with strength coefficient $K$. Denoting $\hat{U}_{\mathrm{S}}=\exp[i\pi\hat{n}^{2}/(2S)^{2}]$, we see that

[TABLE]

Note that $e^{i\frac{\pi}{4}}$ can be removed by redefining |1_{\mathrm{\textnormal{\mathtt{sab}}}}\rangle. Hence the encoding of $\mathtt{sab}$ is realized as \mathcal{S}_{\textnormal{\mathtt{sab}}}(\cdot)=\hat{U}_{\mathrm{S}}\mathcal{S}_{\textnormal{\mathtt{bin}}}(\cdot)\hat{U}_{\mathrm{S}}^{\dagger}.

Two-level recovery

A complete QEC process is described by the effective qubit channel $\mathcal{E}=\mathcal{S}^{-1}\circ\mathcal{R}\circ\mathcal{N}\circ\mathcal{S}$ that consists of encoding $\mathcal{S}$, error channel $\mathcal{N}$, recovery $\mathcal{R}$ and decoding $\mathcal{S}^{-1}$ Albert et al. (2018). Given $\mathcal{S}$ and $\mathcal{N}$, the code performance then depends on choice of $\mathcal{R}$ – in the case of $\mathtt{sab}$, we need to design an $\mathcal{R}$ that effectively captures the enhanced separation between error codewords.

We first recall the recovery proposed for equally-spaced codes with spacing $S$ Sun et al. (2014); Michael et al. (2016); Li et al. (2017). The recovery $\mathcal{R}^{(1)}=\{\hat{R}_{0}^{(1)},\hat{R}_{1}^{(1)},\ldots,\hat{R}_{S-1}^{(1)}\}$ and Kraus operator $\hat{R}_{i}^{(1)}=\hat{U}_{i}^{(1)}\hat{\Pi}_{i\bmod S}$ where $\hat{\Pi}_{i\bmod S}$ is the projection operator into the subspace with excitation number $i$ modulo $S$, and unitary $\hat{U}_{i}^{(1)}$ performs state transfer $|\mu_{\sigma}^{(S-i)\bmod S}\rangle\leftrightarrow|\mu_{\sigma}\rangle$. $\mathcal{R}^{(1)}$ makes use of the parity structure to correct the first $S-1$ excitation losses, and losses beyond the first $S-1$ will lead to bit-flip error. As such, here we call it “one-level” recovery.

To capture the enhanced error codeword separation, we propose a new recovery $\mathcal{R}^{(2)}$ that, in addition to correcting the first $S-1$ losses, exploits the component in error word $|\mu_{\bar{\sigma}}^{S+k}\rangle$ that is orthogonal to $|\mu_{\sigma}^{k}\rangle$ ($k=0,1,...,S-1$), i.e. $|\mu_{\bar{\sigma}}^{S+k}\rangle-\langle\mu_{\sigma}^{k}|\mu_{\bar{\sigma}}^{S+k}\rangle|\mu_{\sigma}^{k}\rangle$. Since events with $S$ to $2S-1$ losses are also partially corrected, we call $\mathcal{R}^{(2)}$ “two-level” recovery Albert et al. (2018). We note that $\mathcal{R}^{(2)}$ improves the performance of $\mathtt{bin}$ compared to $\mathcal{R}^{(1)}$ Albert et al. (2018), yet the enhancement is much more pronounced for $\mathtt{sab}$ due to the intentionally enlarged separation between $|\mu_{\bar{\sigma}}^{S+k}\rangle$ and $|\mu_{\sigma}^{k}\rangle$.

The $1^{\mathrm{st}}$ level of $\mathcal{R}^{(2)}$, similarly to $\mathcal{R}^{(1)}$ Michael et al. (2016), fully corrects the first $S-1$ losses. Each Kraus operator consists of a projection and a restoring unitary

[TABLE]

Here, $k=0,1,\ldots,S-1$, $\hat{P}_{k}=\sum_{\sigma}|\mu_{\sigma}^{k}\rangle\langle\mu_{\sigma}^{k}|$ projects to each error subspace, and $\hat{U}_{k}^{\mathrm{res}}$ finishes the unitary rotation in $\mathrm{Span}\{|\mu_{\sigma}\rangle,|\mu_{\sigma}^{k}\rangle\}$ – for $k\neq 0$, it is simply $|\mu_{\sigma}^{k}\rangle\langle\mu_{\sigma}|$.

Also, $\mathcal{R}^{(2)}$ has a $2^{\mathrm{nd}}$ level with $S$ Kraus operators

[TABLE]

where, for $k=0,1,\ldots,S-1$, the normalized $|\nu_{\bar{\sigma}}^{k}\rangle\propto|\mu_{\bar{\sigma}}^{S+k}\rangle-\langle\mu_{\sigma}^{k}|\mu_{\bar{\sigma}}^{S+k}\rangle|\mu_{\sigma}^{k}\rangle$ is recoverable and $\hat{P}_{S+k}=\sum_{\sigma}|\nu_{\bar{\sigma}}^{k}\rangle\langle\nu_{\bar{\sigma}}^{k}|$. To make $\mathcal{R}^{(2)}$ a CPTP map, we add $\hat{R}_{2S+1}^{(2)}=\hat{V}_{\mathrm{res}}(\hat{I}_{\mathcal{H}}-\sum_{i=0}^{2S}\hat{P}_{i})$ where $\hat{I}_{\mathcal{H}}$ is the identity operator on the entire Hilbert space and $\hat{V}_{\mathrm{res}}$ an arbitrary unitary acting on the complementary subspace of $\{|\mu_{\sigma}\rangle\}\cup\{|\mu_{\sigma}^{k}\rangle\}\cup\{|\nu_{\bar{\sigma}}^{k}\rangle\}$.

To quantify the performance of $\mathcal{E}$, we use channel fidelity (aka. entanglement fidelity) defined as

[TABLE]

where $\mathcal{I}_{A}$ is an identity channel on qubit A and $|\Psi\rangle=\frac{1}{\sqrt{2}}(|0_{A}0_{B}\rangle+|1_{A}1_{B}\rangle)$ is a maximally entangled state of qubits A and B.

In Fig. 2, we present the performance of $\mathcal{R}^{(2)}$ and enhanced QEC capability of $\mathtt{sab}$. At $\gamma=0.1$, we compute the channel infidelities for $\mathtt{bin}$ codes undergoing $\mathcal{N}_{\gamma}$ and optimal recovery $\mathcal{R}^{\mathrm{o}}$ (obtained from convex optimization Fletcher (2007); Albert et al. (2018)) and, in comparison, those for $\mathtt{sab}$ codes undergoing $\mathcal{R}^{(2)}$ and $\mathcal{R}^{\mathrm{o}}$, respectively, after $\mathcal{N}_{\gamma}$. We note $\mathcal{R}^{\mathrm{o}}$ is considered here as it enables the best code performance, yet it is only restricted to be completely positive, trace preserving (CPTP) and can lack physics intuition and ease of implementation.

We see from Fig. 2(a) that, at $\gamma=0.1$, desired $\mathtt{bin}$ codes are found along $S\approx 2N$ while the entire $N>S$ region features poor error protection. In comparison, as manifested in Fig. 2(b), $\mathtt{sab}$ codes with $\mathcal{R}^{(2)}$ achieve much lower infidelities overall and open up the $N>S$ region where destructive interference is pronounced. Fig. 2(c) shows the minimum channel infidelities for $\mathtt{sab}$ codes under $\mathcal{R}^{\mathrm{o}}$. Comparing Fig. 2(b) and (c), we see that $\mathcal{R}^{(2)}$ suffices to unleash the potential of $\mathtt{sab}$ at small $\gamma$, yielding infidelities close to optimal. The results also demystify $\mathcal{R}^{\mathrm{o}}$ for single-mode codes with parity – it is critical to capture the orthogonal component between partially overlapping error words. An alternative way to understand how SA helps suppress decoherence is to decompose effective QEC-protected qubit channel $\mathcal{E}$ into Kraus operators Sup . As shown in Fig. 2(d-f), the advantage of $\mathtt{sab}$ code becomes more significant at $\gamma=0.25$, with the emergence of a new desired regime instead of $S\approx 2N$. Since $\mathcal{R}^{(2)}$ only provides two layers of correction, it begins to perform sub-optimally in regions where $\bar{n}\gamma\apprge S$ (or $N\gamma\apprge 2$) at higher loss rate [Fig. 2(e)], indicating the need to resolve higher order error codewords or even consider the optimal recovery. Nonetheless, $\mathtt{sab}$ codes with $\mathcal{R}^{(2)}$ still outperform $\mathtt{bin}$ codes with $\mathcal{R}^{\mathrm{o}}$, offering a practically favorable and feasible QEC scheme.

$\mathtt{cat}$ code shares the parity structure with $\mathtt{bin}$ code and can be similarly enhanced (detailed in Sup ). Specifically, two-component $\mathtt{cat}$ code Mirrahimi et al. (2014), the simplest of the family which does not correct any excitation loss, will correct one loss approximately after SA is imposed. Same as $\mathtt{sab}$, the SA can be implemented by $\hat{U}_{\mathrm{S}}$ while now $2S$ is the number of coherent states in superposition – this explains why a small amount of cavity Kerr improves $\mathtt{cat}$’s performance (Albert et al. (2018), Fig. 9a).

Optimality vs. complex noise

As it is not obvious that periodic SA is the optimal modification for $\mathtt{bin}$, we relax the Kerr $\hat{U}_{\mathrm{K}}$ added to the encoding to be general (detailed in Sec. 3 of Sup ). At large $\gamma$ and $N\gg S$ regimes, indeed we are able to find modified $\mathtt{bin}$ codes that further suppress the infidelities. These results point to the potential benefit of going beyond RCQC to consider generic phases in codeword designs [Fig. 1(c)].

To find the optimal encoding schemes given an error process, we deploy the technique energy-constrained Noh et al. (2018) biconvex optimization of recovery and encoding for channel fidelity Reimpell and Werner (2005); Kosut and Lidar (2009). Given the encoding and error channel $\mathcal{N}\circ\mathcal{S}$, computing optimal recovery $\mathcal{S}^{-1}\circ\mathcal{R}$ to maximize channel fidelity is a convex problem, and vice versa. Using this technique, we can compute the optimal RCQCs and CCQCs given an error channel, respectively, to see if nontrivial phases are necessary in optimal code constructs glo .

For excitation loss, we find that optimal RCQCs and CCQCs are equivalent up to global rotations, indicating the redundancy of nontrivial phases Sup . The observation that neither channel imposes complex phases, reflected by their Kraus operators, confirms that the nature of error determines the role of phase in optimal code construction.

To see where nontrivial phases play a role, we consider the joint channel of excitation loss and Kerr effect. The channel is relevant (e.g. in circuit QED systems) due to the concurrence of cavity Kerr and intrinsic loss. Kerr alone is reversible, but as it does not commute with $\mathcal{N}_{\gamma}$, randomness in the timing of excitation loss leads to uncertainty in phase $\hat{U}_{\mathrm{Kr}}$ imprints and thus decoherence Hu et al. . The channel’s superoperator Ker is

[TABLE]

In Table 1 we show the Wigner functions for the maximally mixed state \frac{1}{2}\hat{P}_{\textnormal{\mathtt{code}}}, which contains full information of a code, and the associated infidelities $1-F^{\mathrm{o}}$ of the optimized RCQCs and CCQCs, respectively, at $\gamma=0.1$, $\bar{n}_{\mathrm{c}}=2$ and various $Kt$. As a comparison we show the infidelities for \textnormal{\mathtt{bin}}(2,2) with $\bar{n}=2$, which protects against pure excitation loss Albert et al. (2018). For all $Kt$, both scenarios yield codes better than \textnormal{\mathtt{bin}}(2,2), indicating the advantage of codes tailored to $\mathcal{N}_{\gamma,\,Kt}$. At $Kt=0$ (i.e. pure excitation loss) the optimized RCQC and CCQC are equivalent, as discussed earlier. For $Kt\neq 0$, nontrivial phases allow optimal CCQCs to offer better protection. The results exemplify the necessity of CQCCs for practical optimal QEC, as phase-imposing errors can often co-exist with other decoherence sources that they do not commute with.

Sign-altered $\mathtt{gkp}$ and multi-qubit codes

So far, we have focused on codes defined in the Fock basis, while the same idea of phase engineering should apply to code constructs in other bases. For example, one can modify $\mathtt{gkp}$ code, defined in phase space for drift error Gottesman et al. (2001), as follows

[TABLE]

where $\sigma=0,1$ and $2\alpha$ is the spacing between position eigenstates. Similar to $\mathtt{sab}$ and $\mathtt{sac}$, the SA can be imposed by following the original $\mathtt{gkp}$ encoding procedure with $e^{i\pi\hat{q}^{2}/4\alpha^{2}}$, which transforms the $\mathtt{gkp}$ stabilizer $\hat{S}_{1}=e^{-ip\alpha}$ to $\hat{S}^{\prime}_{1}=e^{-i(p-\pi q/2\alpha^{2})\alpha}$ while leaving $\hat{S}_{2}=e^{2\pi iq/\alpha}$ unchanged. The new stabilizers define a nonrectangular $\mathtt{gkp}$ lattice, and for $\alpha^{2}=\sqrt{3}\pi/2$ it is hexagonal – the optimal packing in two dimensions with a larger-in-size smallest uncorrectable shift Gottesman et al. (2001); Harrington and Preskill (2001).

The interference effects further extend into the multi-qubit regime. Consider Shor’s $[[9,1,3]]$ code that corrects arbitrary single-qubit Pauli errors with codewords |-_{\textnormal{\mathtt{shor}}}\rangle\propto|\tilde{1}\tilde{0}\tilde{0}\rangle+|\tilde{0}\tilde{1}\tilde{0}\rangle+|\tilde{0}\tilde{0}\tilde{1}\rangle+|\tilde{1}\tilde{1}\tilde{1}\rangle and |+_{\textnormal{\mathtt{shor}}}\rangle=\sigma_{x}^{\otimes 9}|-_{\textnormal{\mathtt{shor}}}\rangle, where $\tilde{i}=iii$ stands for blocks of three qubits. It detects weight-three $\sigma_{x}$ errors, except for $\sigma_{x}^{(i)}\sigma_{x}^{(i+1)}\sigma_{x}^{(i+2)}$ with $i=1,4,7$ as they are logical operators. We now consider a sign-altered variant with |-_{\textnormal{\mathtt{shor^{\prime}}}}\rangle\propto|\tilde{1}\tilde{0}\tilde{0}\rangle-|\tilde{0}\tilde{1}\tilde{0}\rangle+|\tilde{0}\tilde{0}\tilde{1}\rangle-|\tilde{1}\tilde{1}\tilde{1}\rangle and |+_{\textnormal{\mathtt{shor^{\prime}}}}\rangle=|+_{\textnormal{\mathtt{shor}}}\rangle. The new code detects $\sigma_{x}^{(i)}\sigma_{x}^{(i+1)}\sigma_{x}^{(i+2)}$ and thus all weight-three $\sigma_{x}$ errors. In addition, as detailed in Sup , it detects more weight-three hybrid $\sigma_{x}$ and $\sigma_{y}$ errors while offers the same protection over $\sigma_{z}$ as the original Shor code. A similar modification can improve Shor and Steane codes over qubit amplitude damping, which is a realistic concern for qubit systems Sup ; Darmawan and Poulin (2017).

Conclusion

In contrast to conventional designs of quantum codes whose error-correction capability comes from spanning codewords with distinct subsets of computational basis states, we explored the conjugate degree of freedom, the phases carried by basis states, to devise efficient quantum codes for various bosonic and qubit errors. The new codes can feature destructive interference and hence suppressed overlap between error codewords. To showcase the principle, we modify the codewords of bosonic binomial and cat codes that correct excitation loss errors by making certain probability amplitudes negative. With a quantum recovery that effectively captures the suppressed overlap, the modified codes demonstrate desired error-correction performance. For complex-valued noises, such as the joint channel of excitation loss and cavity Kerr, we show that considering complex-valued amplitudes in codewords is critical for optimal code constructs. The same principle also helps improve multi-qubit codes in overcoming noises such as Pauli errors and qubit amplitude damping. We expect the results developed here to deepen our understanding of quantum error correction and enable development of efficient quantum codes across a wide array of physical platforms for faithful quantum information processing.

Acknowledgements

We thank Philippe Faist, Mengzhen Zhang, Sisi Zhou and Yuhui Ouyang for helpful discussions. We acknowledge supports from the ARL-CDQI (W911NF-15-2-0067, W911NF-18-2-0237), ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO MURI (W911NF-16-1-0349), AFOSR MURI (FA9550-14-1-0052, FA9550-15-1-0015), DOE (DE-SC0019406), NSF (EFMA-1640959), and the Packard Foundation (2013-39273).