[论文解读] Performance and structure of bosonic codes

Victor V. Albert, Kyungjoo Noh+9 more|arXiv (Cornell University)|Aug 16, 2017

Quantum Computing Algorithms and Architecture被引用 3

一句话总结

本文通过保真度和哈希界，在光子损失通道下比较了费米子码（GKP、猫态、二项式码及数值优化码）的性能。GKP码在大多数损耗概率下表现最优，尤其在低损耗时，其性能优势源于保真度中的本质奇点；在小损耗时，二项式码优于猫态码；研究还建立了二项式码与自旋相干态之间的联系，并将该框架推广至高维系统。

ABSTRACT

The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce new codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss probabilities. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss probability. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat/binomial/GKP order of performance occurring at small loss probabilities, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss probability. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multi-qubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a new multi-qudit code.

研究动机与目标

• 评估并比较各类费米子码在统一错误模型——即纯损耗通道下的性能。
• 确定经过最优恢复后，每种码的纠缠保真度及可实现的通信速率（通过哈希界）。
• 分析尽管未专为纯损耗通道设计，GKP码为何仍显著优于其他码。
• 通过自旋相干态，建立二项式码与离散变量系统之间的解析联系。
• 通过扩展的自旋相干态形式化，将二项式码推广至高维系统。

提出的方法

• 针对纯损耗通道，数值优化新型费米子码以提升性能。
• 利用最优恢复操作，计算每种码的纠缠保真度。
• 通过计算哈希界，确定编码-错误-恢复通道的可实现通信速率。
• 使用自旋相干态表示法，表征单模与双模二项式码及多量子比特对称不变码。
• 基于自旋相干态框架，推导二项式码的校验算符与纠错程序。
• 通过推广自旋相干态形式化，构建新型多高维系统二项式码。

实验结果

研究问题

• RQ1 在纯损耗通道下，GKP码、猫态码、二项式码及数值优化费米子码在纠缠保真度上的表现如何比较？
• RQ2 尽管未专为纯损耗通道设计，为何GKP码显著优于其他码？
• RQ3 GKP码优越性能的解析根源是什么，特别是在损耗概率趋近于零的极限下？
• RQ4 如何通过自旋相干态，系统性地建立二项式码与离散变量量子系统之间的联系？
• RQ5 自旋相干态框架能否推广以构建新型多高维系统费米子码？

主要发现

• 在大多数损耗概率下，GKP码在纠缠保真度方面优于所有其他码，尤其在低损耗时优势显著。
• GKP码的纠缠保真度在损耗概率趋近于零的极限下表现出本质奇点，解释了其卓越性能。
• 在小损耗概率下，二项式码优于猫态码，而在中等损耗下两者性能相近。
• GKP码及部分二项式码的性能随平均光子数增加而单调上升。
• 本研究建立了二项式码与自旋相干态之间的正式联系，使得校验算符与纠错程序得以定义。
• 通过推广的自旋相干态形式化，可构建新型多高维系统二项式码，将该框架从量子比特系统扩展至高维系统。