关于寻找无退相干子空间复杂性的研究
On the Complexity of Finding Decoherence Free Subspaces
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中文总结 AI 辅助
本文研究开放量子系统是否存在无退相干子空间的判定复杂性，通过引入k-局域Lindbladian问题，证明当k≥5时该问题对QMA困难，表明此判定对量子计算也难解。
中文摘要 AI 辅助
无退相干子空间是开放量子系统的一种稳态结构，它保持位于其中的状态之间的量子相干性，因此在量子信息科学与技术的各个领域都有广泛的应用。本文研究了判定一个开放量子系统是否存在无退相干子空间的计算复杂性。更具体地，我们在由时间无关的Lindblad主方程支配的马尔可夫开放量子系统的背景下研究这一问题。在此过程中，我们引入了$k$-局域Lindbladian问题，该问题刻画了在Lindbladian动力学下计算纯度衰减率的难度。我们证明了当局域性$k \geq 5$时，这两个问题对于复杂性类Quantum Merlin Arthur (QMA)都是困难的，其中第一个问题在完全完备性下是困难的，第二个问题则是QMA完全的。我们的困难性构造将Kitaev的时钟哈密顿量构造推广到开放量子系统设置，通过将量子电路的执行编码到包含纯和历史混合态的Lindbladian的稳态子空间中。然后根据编码电路的输出对该子空间进行混合。我们的结果表明，判定一个通用马尔可夫开放量子系统是否存在无退相干子空间，即使对于量子计算来说也是难以处理的。
英文摘要
Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the $k$-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality $k \geq 5$, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.
发表机构
- Center for Quantum Information and Control, University of New Mexico（新墨西哥大学量子信息与控制中心）
- Department of Physics and Astronomy, University of New Mexico（新墨西哥大学物理与天文学系）
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