Depth-1 expanders on the unitary group and applications

Anurag Anshu; Shankar Balasubramanian; Jonas Haferkamp; Aram W. Harrow; Xinyu Tan

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arXiv · 2609.01605

Depth-1 expanders on the unitary group and applications

Anurag Anshu·Shankar Balasubramanian·Jonas Haferkamp·Aram W. Harrow·Xinyu Tan

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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 $S = \Theta(\Delta^{-1/2})$; 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.

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Anurag Anshu, Shankar Balasubramanian, Jonas Haferkamp, Aram W. Harrow, Xinyu Tan. 2026-09-01. Depth-1 expanders on the unitary group and applications. https://arxiv.org/abs/2609.01605

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