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[论文解读] Diagnostics of mixed-state topological order and breakdown of quantum memory

Ruihua Fan, Yimu Bao|arXiv (Cornell University)|Jan 13, 2023
Quantum Computing Algorithms and Architecture参考文献 68被引用 9
一句话总结

简述：论文定义了局部错误下拓扑量子记忆崩溃的内在诊断，显示三种信息论量度（量子相对熵、相干信息、拓扑纠缠负度数）在二维 Toric 码中经历共同转变，并将它们映射到统计力学框架，界定并达到解码阈值。

ABSTRACT

Topological quantum memory can protect information against local errors up to finite error thresholds. Such thresholds are usually determined based on the success of decoding algorithms rather than the intrinsic properties of the mixed states describing corrupted memories. Here we provide an intrinsic characterization of the breakdown of topological quantum memory, which both gives a bound on the performance of decoding algorithms and provides examples of topologically distinct mixed states. We employ three information-theoretical quantities that can be regarded as generalizations of the diagnostics of ground-state topological order, and serve as a definition for topological order in error-corrupted mixed states. We consider the topological contribution to entanglement negativity and two other metrics based on quantum relative entropy and coherent information. In the concrete example of the 2D Toric code with local bit-flip and phase errors, we map three quantities to observables in 2D classical spin models and analytically show they all undergo a transition at the same error threshold. This threshold is an upper bound on that achieved in any decoding algorithm and is indeed saturated by that in the optimal decoding algorithm for the Toric code.

研究动机与目标

  • 在局部错误下对拓扑量子记忆的崩溃进行内在表征。
  • 利用内在诊断识别混态拓扑序中的误差率驱动转变。
  • 建立对解码性能的上界并将其与内在态性质相关联。
  • 在具体模型（二维 Toric 码）中展示三种诊断的一致性。
  • 展示到经典自旋模型的映射和对偶性，连接到最优解码阈值。

提出的方法

  • 通过对拓扑有序的基态应用局部噪声通道，定义受噪声污染的混态。
  • 引入三种诊断：受损状态与任意子模态创建状态之间的量子相对熵、相干信息以及拓扑纠缠负度。
  • 使用这些诊断的 Rényi 泛化，并将它们的 n 次矩映射到 (n-1) 味道的伊辛自旋统计力学模型。
  • 展示诊断对应于铁磁序的不同探针，并在同一临界误差率处经历转变。
  • 证明映射的统计模型与控制解码转变的随机结合伊辛模型存在对偶关系，暗示解码阈值被最优解码所达到。
  • 讨论 n→1 极限与在纠错背景下对部分迹和熵的标准概念之间的关系。
Figure 1: Physical observables verses information quantities in error corrupted states. Each error corrupted state can be obtained from applying local unitaries to the system (topological order) plus ancilla qubits (trivial product state). Thus, physical observables must be smooth functions of the e
Figure 1: Physical observables verses information quantities in error corrupted states. Each error corrupted state can be obtained from applying local unitaries to the system (topological order) plus ancilla qubits (trivial product state). Thus, physical observables must be smooth functions of the e

实验结果

研究问题

  • RQ1在局部错误存在的情况下，三种信息论诊断（相对熵、相干信息、拓扑负度）是否表现出相同的临界误差率？
  • RQ2如何对受损混态进行内在表征，以界定解码阈值并揭示拓扑不同的混态？
  • RQ3这些诊断的 n 次 Rényi 版本是否可以映射到一个二维 (n-1) 味道的伊辛自旋模型，这对相变意味着什么？
  • RQ4解码阈值是否达到这些诊断所隐含的内在上界，并且这是否与最优解码相关？

主要发现

  • 三个诊断（D^(n)、I_c^(n)、E_A^(2n)）在二维 Toric 码中对不可相干比特翻转和相位错误下表现出在有限误差率的转变。
  • 每个诊断映射到一个 (n-1) 呀味道的伊辛模型中的观测量，并经历同时的顺磁—铁磁转变。
  • 相应的统计力学模型与控制解码转变的随机耦合伊辛模型互为对偶，暗示解码阈值被最优解码所达到。
  • 该转变代表混态拓扑序的崩溃，内在临界误差率为算法解码性能提供上界。
  • 在 n→1 极限下，结果回归关于量子记忆保护和纠错的标准相干/相对熵视角。
Figure 2: Critical error rates for various Rényi index $n$ . $p_{c}^{(2)}\approx 0.178$ and $p_{c}^{(3)}\approx 0.211$ are determined by the exact solution (blue diamonds). For $n\geqslant 4$ , $p_{c}^{(n)}$ is determined by calculating the crossing of the Binder ratio for various system sizes via M
Figure 2: Critical error rates for various Rényi index $n$ . $p_{c}^{(2)}\approx 0.178$ and $p_{c}^{(3)}\approx 0.211$ are determined by the exact solution (blue diamonds). For $n\geqslant 4$ , $p_{c}^{(n)}$ is determined by calculating the crossing of the Binder ratio for various system sizes via M

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