Efficiency of Producing Random Unitary Matrices with Quantum Circuits

TL;DR

This paper analyzes how efficiently random unitary matrices can be generated using quantum circuits, demonstrating that several statistical properties converge quickly with a number of gates scaling as n_q log(n_q/ε).

Contribution

It provides a detailed numerical analysis of the convergence rates of various statistical properties of random unitary circuits towards the CUE, showing efficient reproducibility.

Findings

Statistical properties converge with gates scaling as n_q log(n_q/ε)

Efficient reproduction of matrix element distributions and moments

Quantities requiring exponential gates are more complex

Abstract

We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as $n_q\log (n_q/\epsilon)$ with the number of qubits $n_q$ for a given fixed precision $\epsilon$. This suggests that quantities which require an exponentially large number of gates are of more complex nature.