Depth-1 expanders on the unitary group and applications

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^† 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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BibTeX RIS

BibTeX

@misc{anshu2026depth1,
  title = {{Depth-1 expanders on the unitary group and applications}},
  author = {Anurag Anshu and Shankar Balasubramanian and Jonas Haferkamp and Aram W. Harrow and Xinyu Tan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01605},
  eprint = {2609.01605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01605},
}

RIS

TY  - UNPB
TI  - Depth-1 expanders on the unitary group and applications
AU  - Anurag Anshu
AU  - Shankar Balasubramanian
AU  - Jonas Haferkamp
AU  - Aram W. Harrow
AU  - Xinyu Tan
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01605
UR  - https://arxiv.org/abs/2609.01605
AB  - 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^† 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.
ER  -

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