Depth-1 expanders on the unitary group and applications

2026-09-01

AI Summary

研究构建了一种基于Pauli或CNOT门的深度-1量子扩展器，并应用于构造满足特定纠缠-能隙关系的一维哈密顿量及测试一类一维体积律纠缠态。

Abstract

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.

Problem

• quantum expander
• unitary group
• 1D Hamiltonians
• entanglement-gap relation
• streaming protocol

Innovation

• quantum expander
• depth-1 circuit
• unitary group
• frustration-free Hamiltonians
• streaming protocol