Probes of chaos over the Clifford group and approach to Haar values

TL;DR

This paper investigates quantum chaos probes using Isospectral Twirling, analyzing transitions from stabilizer to Haar-random bases and comparing chaotic and non-chaotic systems.

Contribution

It introduces a method to analyze chaos probes over various ensembles, including random spectra and the Toric Code Hamiltonian, highlighting the transition to Haar randomness.

Findings

Probes approach Haar distribution values in chaotic regimes.

Transition from stabilizer to Haar-random bases observed with T-doped circuits.

Behavior over the Toric Code Hamiltonian indicates non-chaotic characteristics.

Abstract

Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian.