# Possible Ergodic-nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model and quantum annealing

## TL;DR

This study investigates the ergodic and nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model, revealing how quantum fluctuations influence ergodicity and annealing efficiency at different temperatures.

## Contribution

It identifies and characterizes the finite-temperature ergodic region in the quantum spin glass model using numerical methods, highlighting its impact on quantum annealing.

## Key findings

- Existence of a low-temperature ergodic region dominated by quantum fluctuations.
- System size independence of annealing time in the ergodic region.
- Nonergodic behavior persists at high temperatures with non-vanishing order parameter distribution.

## Abstract

We explore the behavior of order parameter distribution of quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamil- tonian at finite temperatures and that at zero temperature obtained using exact diagonalization method. Our numerical results indicate the existence of low but finite temperature quantum fluc- tuation dominated ergodic region along with the classical fluctuation dominated high temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases. In the other region, the Parisi order distribution function has non-vanishing value every- where in thermodynamic limit, indicating nonergodicity. We also show, that the average annealing time for convergence (to a low energy level of the model; within a small error range) becomes system size independent for annealing down through the (quantum fluctuation dominated) ergodic region. It becomes strongly system size dependent for annealing through the nonergodic region. Possible finite size scaling type behavior for the extent of the ergodic region is also addressed.