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 $\hat{I}¸\in \mathbb{R}/2\ddot{I}€\mathbb{Z}$, both uniformly at random, and apply $e^{\mathrm{i}\hat{I}¸P}$. (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 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.

24 pages. Julia and Mathematica code