[论文解读] Local Hamiltonians with Approximation-Robust Entanglement

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

Quantum Computing Algorithms and Architecture参考文献 12被引用 4

一句话总结

本文基于量子纠错码构造了一个无限族 O(1)-局部哈密顿量，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电路近似。这意味着即使要见证哈密顿量的近似满足，也必须依赖长程纠缠，从而揭示了一类新型的纠缠鲁棒量子系统。

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恒定、与系统尺寸无关的温度下被稳健地保持。
• 确定局部哈密顿量是否能表现出对有界深度经典电路近似具有鲁棒性的纠缠。
• 构建显式的一类局部哈密顿量，使得低能量态本质上需要非平凡的量子关联。
• 探索设计对纠缠鲁棒的哈密顿量以用于量子计算的可行性。
• 通过证明对经典近似而非仅精确基态的鲁棒性，扩展 NLTS 猜想。

提出的方法

• 利用量子纠错码中的校验项来定义 O(1)-局部哈密顿量。
• 设计哈密顿量使得任何能量不超过总能量 0.05 的态，都无法被次对数深度的经典电路近似。
• 证明依赖于码的距离和局部性特性，以确保低能量态必须高度纠缠。
• 框架利用已知的量子 PCP 和 NLTS 结果，将能量界与计算难解性联系起来。
• 建立了低能量态不可近似性与长程纠缠存在的联系。
• 该构造具有可扩展性，并形成此类哈密顿量的无限族，支持在不同系统尺寸下的推广。

实验结果

研究问题

• RQ1 是否可以设计出局部哈密顿量，使得所有低能量态对经典模拟均具有鲁棒纠缠？
• RQ2 能否构造出 O(1)-局部哈密顿量，使得即使近似满足也必须依赖非平凡的量子资源？
• RQ3 量子纠错码是否为在恒定温度下实现纠缠鲁棒哈密顿量提供了可行路径？
• RQ4 能否通过证明对有界深度经典电路的鲁棒性，来加强 NLTS 猜想？
• RQ5 哈密顿量的何种结构特性可确保低能量态无法被经典地观测或模拟？

主要发现

• 所构造的哈密顿量为 O(1)-局部，且构成无限族，确保了可扩展性和普遍性。
• 任何能量不超过总哈密顿量能量 0.05 的量子态，都无法被次对数深度的经典电路近似。
• 这意味着即使提供对哈密顿量近似满足的类经典见证，也必须依赖长程纠缠。
• 低能量态中的纠缠对经典近似具有鲁棒性，表明存在一种量子优势形式。
• 该结果确立了一类新的具有不可近似性鲁棒纠缠的局部哈密顿量，支持了鲁棒量子物相的可行性。
• 该构造为设计对纠缠鲁棒、可用于量子计算的哈密顿量奠定了基础。