Random unitary circuits with constant spectral gap

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 $\theta \in [0,2\pi )$, both uniformly at random, and apply $e^{\mathrm{i}\theta P}$. (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.

arXiv comment32 pages. 1 figrue. Julia and Mathematica code (v2) analogous results on orthogonal groups, simplified calculation using Gelfand pairs, and unitary complexity growth spelled out