[论文解读] The Complexity of Relating Quantum Channels to Master Equations

Toby S. Cubitt, Jens Eisert+1 more|arXiv (Cornell University)|Aug 17, 2009

Quantum Computing Algorithms and Architecture参考文献 33被引用 2

一句话总结

本文证明了判断一个给定的量子通道是否源自马尔可夫主方程（即马尔可夫性问题）是NP难的，这意味着除非P=NP，否则不存在高效的判别准则。作者将问题约化为整数半定规划，并表明当系统维度固定时，仅需单个采样点即可高效求解，为当前量子过程层析实验提供了实际应用价值。

ABSTRACT

Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find it's form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations. We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P=NP: any efficiently computable criterion for Markovianity would imply P=NP; whereas a proof that P=NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description. However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case. Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.

研究动机与目标

• 解决长期悬而未决的开放问题：即给定的完全正且迹保持（CPT）映射是否可由林德布拉德主方程生成。
• 确定量子开放系统中马尔可夫性问题的计算复杂度。
• 将复杂性理论分析扩展至经典类比：随机矩阵的嵌入问题。
• 为系统维度固定时的马尔可夫性问题提供高效算法，与当前实验设置相关。
• 阐明NP难性对物理推理的影响，以及从实验数据中识别潜在马尔可夫动力学的可行性。

提出的方法

• 将马尔可夫性问题约化为整数半定规划，这是一个已知的NP完全问题。
• 构建1-in-3SAT问题的多项式时间可计算编码，映射到具有特定谱性质的量子CPT映射。
• 利用微扰理论确保仅有效的林德布拉德生成元对应于编码后的逻辑可满足性。
• 应用对数矩阵表示和弱成员资格形式化，将CPT映射与其生成元候选者关联起来。
• 将量子约化技术适配至随机矩阵的古典嵌入问题。
• 利用由林德布拉德生成元生成的CPT映射集合具有非空内部的性质，从而支持微扰分析。

实验结果

研究问题

• RQ1 是否存在CPT映射可由林德布拉德主方程生成的充要条件？
• RQ2 判断给定CPT映射是否为马尔可夫性的计算复杂度是多少？
• RQ3 随机矩阵的嵌入问题能否约化为已知的NP难问题？
• RQ4 马尔可夫性问题的复杂度是否依赖于系统维度？若依赖，其依赖关系如何？
• RQ5 是否存在一种高效算法，可从单个量子过程层析采样点重建林德布拉德生成元？

主要发现

• CPT映射的马尔可夫性问题是NP难的，这意味着除非P=NP，否则不存在高效的可计算判别准则。
• 判断随机矩阵是否可嵌入连续时间马尔可夫过程的问题同样是NP难的。
• 提出了一种显式算法，将马尔可夫性问题约化为整数半定规划，为固定系统维度提供了构造性解法。
• 对于固定维度的系统，该算法在所需精度下具有高效可扩展性，支持从单个采样点实现实际重建。
• 在有限维空间中，马尔可夫性与非马尔可夫性CPT映射的集合均具有非空内部和非零测度。
• 经典嵌入问题与量子马尔可夫性问题在复杂度上等价，且两者均等价于求解P=NP。