Random unitary circuits with constant spectral gap

2026/07/23 by Tim Baer, Jeongwan Haah · 1 citation
#quant-ph #math.PR

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Abstract

We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2ⁿ) on n qubits. (i) Random Pauli Rotation: choose an n-qubit Pauli operator P and an angle θ∈ \mathbb R / 2π\mathbb Z, 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 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 gaps are independent of n and apply for all finite dimensional unitary representations of SU(2ⁿ) uniformly, including those that appear in unitary t-designs. We also prove analogous constant gap results for Clifford unitaries, which are indispensable for our result on Brickwork Random Unitary Circuit.

Citations

• A complete theory of the Clifford commutant

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