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arXiv:2607.20919

Random unitary circuits with constant spectral gap

Tim Baer, Jeongwan Haah

quant-phmath.PRSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations• Revised Sep 29, 2026 (v2)
v1 · Jul 23, 2026v2 · Sep 29, 2026
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Abstract

We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $θ\in [0,2π)$, both uniformly at random, and apply $e^{\mathrm i θP}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gap lower bounds are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs. We also prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit.

Comments
32 pages. 1 figrue. Julia and Mathematica code (v2) analogous results on orthogonal groups, simplified calculation using Gelfand pairs, and unitary complexity growth spelled out

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We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2n)\mathsf{SU}(2^n)SU(2n) on nnn qubits. (i) Random Pauli Rotation: choose an nnn-qubit Pauli operator PPP and an angle θ∈[0,2π)θ\in [0,2π)θ∈[0,2π), both uniformly at random, and apply eiθPe^{\mathrm i θP}eiθP. (ii) Brickwork Random Unitary Circuit: choose n−1n-1n−1 unitaries UiU_{i}Ui​ uniformly at random from SU(4)\mathsf{SU}(4)SU(4) independently, and apply U2j−1U_{2j-1}U2j−1​ on two qubits 2j−1,2j2j-1, 2j2j−1,2j and then U2jU_{2j}U2j​ on two qubits 2j,2j+12j, 2j+12j,2j+1. Importantly, the spectral gap lower bounds are independent of nnn and apply for all finite dimensional unitary representations of SU(2n)\mathsf{SU}(2^n)SU(2n) uniformly, including those that appear in unitary ttt-designs. We also prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit.
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