论文标题
基态能量以外的翻译不变问题的复杂性
The Complexity of Translationally Invariant Problems beyond Ground State Energies
论文作者
论文摘要
众所周知，有关当地汉密尔顿人的三个基本问题 - 近似于基态能量（当地的汉密尔顿问题），模拟地面空间上的当地测量值（APX-SIM），并确定低能量空间是否具有能源壁垒（GSCON） - 是$ \ MATHSF {QMA} $ - HARTS $ - 硬， $ \ mathsf {p}^{\ mathsf {qma} [log]} $ - 硬和$ \ mathsf {qcma} $ - 硬，这意味着即使在量子计算机上，它们也可能是棘手的。然而，尽管众所周知，当地的哈密顿量问题的硬度即使对于翻译不变的系统也存在，但尚不清楚APX-SIM和GSCON在这样的“简单”系统中是否仍然很难。在这项工作中，我们表明APX-SIM和GSCON的翻译不变版本仍然棘手，即$ \ MathSf {p}^{\ MathSf {QMA} _ {\ MathSf {\ Mathsf {Exp}}}}}}}} $ - and $ \ \ \ \ \ \ \ \ \ \ \ \ {qcma} $} $}这些结果都是通过给出各自的通用“提升定理”来产生硬度结果来实现的。例如，对于APX-SIM，我们为“举起”任何抽象的本地电路映射到$ h $（满意温和的假设）的框架对APX-SIM的硬度对$ h $生产的汉密尔顿家族的硬度，同时保留$ h $ $ h $的结构和几何属性（e.g. h $ translation translation translation invariance invariance invariance invariance，locemence，locement，locement，locementry localiance，locement，locement，locepery of locemence，locement，等等）。每个结果还利用我们的构造的违反直觉属性：对于APX-SIM，我们将多个并行查询的答案“压缩”到QMA Oracle到单个量子的答案。对于GSCON，我们对高度非本地单位的硬度构造具有坚固的稳定性，即，即使对手在每个步骤中都对系统中除了一个Qudit都作用。
It is known that three fundamental questions regarding local Hamiltonians -- approximating the ground state energy (the Local Hamiltonian problem), simulating local measurements on the ground space (APX-SIM), and deciding if the low energy space has an energy barrier (GSCON) -- are $\mathsf{QMA}$-hard, $\mathsf{P}^{\mathsf{QMA}[log]}$-hard and $\mathsf{QCMA}$-hard, respectively, meaning they are likely intractable even on a quantum computer. Yet while hardness for the Local Hamiltonian problem is known to hold even for translationally-invariant systems, it is not yet known whether APX-SIM and GSCON remain hard in such "simple" systems. In this work, we show that the translationally invariant versions of both APX-SIM and GSCON remain intractable, namely are $\mathsf{P}^{\mathsf{QMA}_{\mathsf{EXP}}}$- and $\mathsf{QCMA}_{\mathsf{EXP}}$-complete, respectively. Each of these results is attained by giving a respective generic "lifting theorem" for producing hardness results. For APX-SIM, for example, we give a framework for "lifting" any abstract local circuit-to-Hamiltonian mapping $H$ (satisfying mild assumptions) to hardness of APX-SIM on the family of Hamiltonians produced by $H$, while preserving the structural and geometric properties of $H$ (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we "compress" the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we give a hardness construction robust against highly non-local unitaries, i.e. even if the adversary acts on all but one qudit in the system in each step.
