发表机构
Harvard University; California Institute of Technology; Ruhr-University Bochum; Massachusetts Institute of Technology（哈佛大学; 加州理工学院; 波鸿鲁尔大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了深度1的量子扩展器及酉群上的对应扩展器，将其应用于无挫折哈密顿量、纠缠态检测协议，改进了Bourgain等人关于酉群带间隙游走的前期工作。
AI 中文摘要
我们构造了一个针对n个量子比特的常数度数、常数间隙量子扩展器，其中每个酉操作可通过深度为1的一维Pauli或CNOT门电路实现。我们给出该扩展器的两个应用：其一，利用它构造了一族无挫折的一维哈密顿量，其基态满足纠缠-间隙关系S=Θ(Δ⁻¹/²)，该关系被认为是最优的，但此前尚未实现；其二，利用它提供了一种流式协议，用于检测与一类一维体积律纠缠态的接近程度。此外，我们将该量子扩展器扩展为酉群上的常数度数、常数间隙扩展器，其中每个酉操作为单个T门、单个T†门或深度为1的Clifford电路。这意味着，来自该扩展器的酉操作随机序列会在酉群的稠密子群上产生带间隙的游走，这改进了Bourgain与Gamburd的前期工作，该工作未控制间隙对维度的依赖关系。
英文摘要
We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = Θ(Δ^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.
Comments36 pages, 2 figures
