[论文解读] Insights from Quantum Information into Fundamental Physics

Terry Farrelly|arXiv (Cornell University)|Aug 29, 2017

Quantum many-body systems参考文献 22被引用 10

一句话总结

本论文将量子信息概念应用于基础物理，证明了离散时间量子行走和量子细胞自动机可在连续极限下模拟具有相对论动力学的相对论性量子系统。一个关键成果是，在二维量子行走中可避免费米子加倍问题（该问题在格点场论中具有挑战性），且在离散时空上的因果费米子系统可通过量子细胞自动机高效模拟。

ABSTRACT

This thesis is split into two parts, which are united in the sense that they involve applying ideas from quantum information to fundamental physics. The first part is focused on examining discrete-time models in quantum computation (discrete-time quantum walks and quantum cellular automata) as discretized models of relativistic systems. One of the results here is a theorem demonstrating that a large class of discrete-time quantum walks have relativistic dynamics in the continuum limit. Additionally, the problem of fermion doubling for these models is investigated, and it is seen that the problem can be circumvented in two dimensional space. This was already known for one dimensional systems. Another result involves taking the limits of causal free field theories in discrete spacetime to recover continuum field theories, something that is not straightforward because of the nontrivial nature of the vacuum in quantum field theory. Additionally, it is shown that general systems of fermions evolving causally in discrete spacetime can be represented by quantum cellular automata, which makes them efficiently simulable by quantum computers. A related result is that quantum cellular automata composed of fermions are equivalent to regular quantum cellular automata. In the second part of this thesis, the focus is on the foundations of statistical physics. The main result of this part is a general bound on the time it takes a quantum system to effectively reach equilibrium. The discussion also includes a practical definition of equilibration that takes our measurement capabilities into account. Finally, the nature of the equilibrium state is also discussed, with a focus on initial state independence, which relates to the important question of when the equilibrium state is a Gibbs state.

研究动机与目标

• 探讨离散时间量子计算模型是否可作为相对论性量子场论的可行离散化表示。
• 利用量子行走框架解决格点场论中长期存在的费米子加倍问题。
• 研究利用量子细胞自动机在量子计算机上模拟因果费米子系统的可能性。
• 推导在考虑物理测量限制条件下的量子系统均衡时间的一般界限。
• 考察在存在局部相互作用时，量子系统平衡态为何种吉布斯态的条件。

提出的方法

• 使用李-特罗特乘积公式从离散量子行走近似连续动力学，实现向相对论场论的收敛。
• 应用约旦-维尔纳变换将自旋模型映射到费米子系统，从而在离散模型中研究费米子自由度。
• 采用二次量子化技术定义离散费米子场，并分析其在格点模型中的真空结构。
• 使用量子细胞自动机（QCAs）在离散时空格点上模拟费米子的因果演化，确保幺正性和局域性。
• 提出一种基于受限测量集合下可区分性的物理动机均衡定义，而非基于迹距离。
• 应用量子信息理论工具——如态可区分性和能量滤波——以界定均衡时间并分析平衡性质。

实验结果

研究问题

• RQ1 二维离散时间量子行走是否能在连续极限下再现相对论性粒子的动力学？
• RQ2 量子细胞自动机在多大程度上可模拟离散时空格点上的因果、局域费米子场论？
• RQ3 二维量子行走中费米子加倍问题是否可避免？与一维情况相比有何差异？
• RQ4 当测量能力受到物理限制时，量子系统的均衡时间是否存在一般界限？
• RQ5 在何种条件下，量子系统的平衡态会与吉布斯态一致，特别是对于局域哈密顿量？

主要发现

• 一大类二维离散时间量子行走表现出连续极限下的相对论性动力学，其有效连续理论为狄拉克方程。
• 在二维量子行走中成功避免了费米子加倍问题，将一维中已知结果推广至更高维度。
• 在离散时空上具有因果性和局域性的费米子系统可用量子细胞自动机表示，从而实现高效的量子模拟。
• 费米子量子细胞自动机等价于标准量子细胞自动机，简化了其理论处理。
• 可利用基于物理动机的可区分性度量界定量子系统的均衡时间，表明即使在测量分辨率有限的情况下，均衡仍可实现。
• 当能级满足本征态热化假说时，系统的平衡态极有可能为吉布斯态，尽管其依赖于哈密顿量的局部性质。