Parameters of Pseudo-Random Quantum Circuits

TL;DR

This paper investigates how various design parameters influence the efficiency and convergence of pseudo-random quantum circuits, aiming to optimize their performance in generating states and operators that mimic Haar-random distributions.

Contribution

It provides a comprehensive analysis of how gate choices, topology, and other parameters affect convergence, using a Markov matrix approach to quantify asymptotic behavior.

Findings

Convergence rate depends on circuit size and topology.

Symmetric topologies improve convergence efficiency.

Results aid in optimizing pseudo-random circuit design.

Abstract

Pseudorandom circuits generate quantum states and unitary operators which are approximately distributed according to the unitarily invariant Haar measure. We explore how several design parameters affect the efficiency of pseudo-random circuits, with the goal of identifying relevant trade-offs and optimizing convergence. The parameters we explore include the choice of single- and two-qubit gates, the topology of the underlying physical qubit architecture, the probabilistic application of two-qubit gates, as well as circuit size, initialization, and the effect of control constraints. Building on the equivalence between pseudo-random circuits and approximate $t$-designs, a Markov matrix approach is employed to analyze asymptotic convergence properties of pseudo-random second-order moments to a 2-design. Quantitative results on the convergence rate as a function of the circuit size are presented for qubit topologies with a sufficient degree of symmetry. Our results may be theoretically and practically useful to optimize the efficiency of random state and operator generation.