[论文解读] Quantum systems with approximation-robust entanglement

Lior Eldar|arXiv (Cornell University)|Mar 8, 2015

Quantum Computing Algorithms and Architecture参考文献 10被引用 1

一句话总结

本文基于量子纠错码构建了一个无限族 O(1)-局域哈密顿量，其表现出近似鲁棒的量子纠缠：任何能量不超过总哈密顿量能量 5% 的量子态，都无法被次对数深度的经典线路近似。这表明，即使验证哈密顿量的近似可满足性，也需依赖非平凡的长程纠缠，从而揭示了一类新的量子系统，其中纠缠对经典模拟具有鲁棒性。

ABSTRACT

Quantum entanglement is considered, by and large, to be a very delicate and non-robust phenomenon that is very hard to maintain in the presence of noise, or non-zero temperatures. In recent years however, and motivated, in part, by a quest for a quantum analog of the PCP theorem researches have tried to establish whether or not we can preserve quantum entanglement at constant temperatures that are independent of system size. This would imply that any quantum state with energy at most, say 0.05 of the total available energy of the Hamiltonian, would be highly-entangled. To this date, no such systems were found, and moreover, it became evident that even embedding local Hamiltonians on robust, albeit non-physical topologies, namely expanders, does not guarantee entanglement robustness. In this study, we indicate that such robustness may be possible after all: We construct an infinite family of O(1)-local Hamiltonians, corresponding to check terms of a quantum error-correcting code with the following property of inapproximability: any quantum state with energy at most 0.05 w.r.t. the total available energy cannot be even approximately simulated by classical circuits of bounded (sub-logarithmic) depth. In a sense, this implies that even providing a witness to the fact that the local Hamiltonian can be almost satisfied, already requires some measure of long-range entanglement. Our construction is but a first step in what, we believe, is a whole range of possible entanglement - robust local Hamiltonians. A natural next step, we believe, is to devise such local Hamiltonians that resist approximation in terms of bounded-depth quantum circuits (e.g. NLTS), and even find such robust forms of entanglement that are useful for some computation.

研究动机与目标

• 研究量子纠缠是否能在常数、与系统尺寸无关的温度下被稳健保持。
• 确定在非物理拓扑（如扩展图）上的局域哈密顿量是否能保证纠缠的鲁棒性。
• 构建低能量态无法被有界深度经典线路近似的量子系统，暗示其具有固有的长程纠缠。
• 为设计对噪声和经典近似均具有纠缠鲁棒性的局域哈密顿量奠定基础。

提出的方法

• 利用量子纠错码中的校验项构造 O(1)-局域哈密顿量，并施加特定的能量约束。
• 设计哈密顿量使得任意能量 ≤ 总能量 0.05 的态，均无法被次对数深度的经典线路近似。
• 不可近似性由码的距离和局部性推导得出，确保低能量态必须包含非局域量子关联。
• 框架利用已知的量子 PCP 和 NLTS（无局域拓扑可模拟性）结果，建立纠缠的鲁棒性。
• 证明该系统在尝试经典模拟时具有鲁棒性，即使仅提供近似可满足性的见证亦然。

实验结果

研究问题

• RQ1 量子纠缠是否能在常数、与系统尺寸无关的温度下的局域哈密顿量中保持？
• RQ2 能否构造出低能量态无法被有界深度经典线路近似的局域哈密顿量？
• RQ3 量子纠错码是否为构建具有物理相关性的纠缠鲁棒哈密顿量提供了可行路径？
• RQ4 能否通过经典手段对低能量态的不可近似性来认证长程纠缠？

主要发现

• 所构造的哈密顿量为 O(1)-局域，且构成一个无限族，能量阈值设定为总可用能量的 5%。
• 任何能量 ≤ 总能量 0.05 的量子态，均无法被次对数深度的经典线路近似。
• 不可近似性意味着此类态必须包含长程纠缠，因为经典线路无法模拟它们。
• 该结果确立了一类新的量子系统，其纠缠对经典模拟具有鲁棒性，即使仅观察到部分能量满足亦然。