Random unitary circuits with constant spectral gap

• URL: http://arxiv.org/abs/2607.20919v1
• Date: Thu, 23 Jul 2026 04:56:09 GMT
• Title: Random unitary circuits with constant spectral gap
• Abstract summary: We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $mathsfSU(2n)$ on $n$ qubits.<n> Importantly, the spectral gaps are independent of $n$ and apply for all finite dimensional unitary representations of $mathsfSU(2n)$ uniformly.
• Score: 0.34376560669160394
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $θ\in \mathbb R / 2π\mathbb Z$, 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 $\mathsf{SU}(4)$ 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 gaps 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 Clifford unitaries, which are indispensable for our result on Brickwork Random Unitary Circuit.

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