Local random quantum circuits are approximate polynomial-designs - numerical results

TL;DR

This paper numerically studies local random quantum circuits to evaluate their effectiveness as approximate polynomial-designs, revealing limitations of previous bounds and providing new spectral gap estimates for up to 20 qubits.

Contribution

It provides improved numerical estimates of spectral gaps for local random quantum circuits acting as approximate t-designs, challenging existing lower bounds and offering more accurate convergence insights.

Findings

Spectral gaps vary with number of qubits and degree t

Previous lower bounds may be too loose to be informative

Results suggest spectral gaps often close, indicating slower convergence

Abstract

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbour two-qubit gates form an approximate unitary poly(n)-design [F.G.S.L. Brandao et al., arXiv:1208.0692]. Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t-design are evaluated for a different number of qubits n (up to 20) and degrees t (t=2,3,4 and 5), improving previously known results for n=2 in the case of t=2 and 3. Their values lead to a conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells that a gap is closed. We compare our results to the another lower bounding technique, again showing that its results may not be tight.