[论文解读] Decoding the Entanglement Structure of Monitored Quantum Circuits

Beni Yoshida|arXiv (Cornell University)|Sep 17, 2021

Quantum Computing Algorithms and Architecture参考文献 56被引用 6

一句话总结

本文建立了受监控量子电路与经典纠错码之间的对偶性，使得可通过 Clifford 电路中的可恢复性显式计算纠缠度量。该研究提出了一种基于解码对偶经典码的确定性纠缠浓缩协议，揭示了体积定律纠缠源于算符增长，且在多项式码距离范围之外，与遥远的测量历史基本无关。

ABSTRACT

Given an output wavefunction of a monitored quantum circuit consisting of both unitary gates and projective measurements, we ask whether two complementary subsystems are entangled or not. For Clifford circuits, we find that this question can be mapped to a certain classical error-correction problem where various entanglement measures can be explicitly computed from the recoverability. The dual classical code is constructed from spacetime patterns of out-of-time ordered correlation functions among local operators and measured Pauli operators in the past, suggesting that the volume-law entanglement in a monitored circuit emerges from quantum information scrambling, namely the growth of local operators. We also present a method of verifying quantum entanglement by providing a simple deterministic entanglement distillation algorithm, which can be interpreted as decoding of the dual classical code. Discussions on coding properties of a monitored Clifford circuit, including explicit constructions of logical and stabilizer operators, are also presented. Applications of our framework to various physical questions, including non-Clifford systems, are discussed as well. Namely, we argue that the entanglement structure of a monitored quantum circuit in the volume-law phase is largely independent of the initial states and past measurement outcomes except recent ones, due to the decoupling phenomena from scrambling dynamics, up to a certain polynomial length scale which can be identified as the code distance of the circuit. We also derive a general relation between the code distance and the sub-leading contribution to the volume-law entanglement entropy. Applications of these results to black hole physics are discussed as well.

研究动机与目标

• 确定在存在投影测量的情况下，受监控量子电路中两个子系统是否纠缠。
• 开发一种实用方法，在无需完整掌握过去测量结果的情况下，验证并浓缩量子纠缠。
• 理解纠缠结构对初始态和测量历史的依赖性，特别是量子杂乱化的作用。
• 在受监控的 Clifford 电路中建立纠缠熵与码距离之间的联系，将量子信息与编码理论联系起来。

提出的方法

• 利用算符和测量的时空模式的反时间序相关函数，将受监控量子电路的纠缠结构映射为经典纠错问题。
• 从电路的测量和幺正演化历史构建对偶经典码，其中可恢复性对应于子系统之间的纠缠。
• 通过解码对偶经典码实现确定性纠缠浓缩算法，有效逆转测量过程。
• 使用稳定子形式推导逻辑算符和稳定子算符，通过码参数显式计算纠缠度量。
• 应用清洁引理和扩展字典形式，处理浓缩协议中反馈和测量依赖性。
• 将体积定律纠缠熵的次主导项与电路的码距离相关联，该码距离由经典码的最小距离导出。

实验结果

研究问题

• RQ1 如何从电路的测量和幺正演化历史中确定受监控量子电路中两个子系统之间的纠缠结构？
• RQ2 纠缠结构在多大程度上依赖于过去的测量结果和初始态，这种依赖性能否被界定？
• RQ3 是否存在一种确定性、物理可实现的协议，在无需完整经典记忆过去历史的情况下，验证并浓缩此类电路中的纠缠？
• RQ4 对偶经典码的码距离与体积定律纠缠熵的次主导项之间存在何种关系？
• RQ5 时空中的算符增长如何与受监控电路中体积定律纠缠的出现相关联？

主要发现

• 受监控 Clifford 电路中的纠缠结构完全由从算符和测量的时空模式构建的对偶经典纠错码的可恢复性决定。
• 通过解码对偶经典码的确定性算法，可在两个子系统之间实现纠缠浓缩，生成保真度高的 EPR 类态。
• 体积定律纠缠熵的次主导项被证明与对偶经典码的码距离成正比，关系为 $ \text{sub-leading term} \propto \text{code distance} $。
• 由于与杂乱动力学的退耦，纠缠结构在多项式长度尺度（即码距离）之外，对初始态和遥远测量结果的依赖性消失。
• 子系统 $ A $ 上的逻辑算符数量为 $ \log N_{I_A} = 2n_A - (\log N_{I_{\text{ext}}} - \log N_{{I_B}_{\text{ext}}}) $，将纠缠与码参数联系起来。
• 浓缩协议输出一个最大纠缠态 $ \frac{1}{N_{I_{\text{ext}}}} \sum_{P \in \mathcal{S}} |P\rangle\langle P| $，证实了码的稳定子群结构。