# Dynamics of reverse annealing for the fully-connected $p$-spin model

## TL;DR

This paper investigates the dynamics of reverse annealing in the fully-connected $p$-spin model, demonstrating exponential speedup over traditional quantum annealing and comparing classical and quantum approaches.

## Contribution

It provides a numerical analysis of reverse annealing dynamics, confirms equilibrium predictions, and compares different reverse annealing protocols for the $p$-spin model.

## Key findings

- Reverse annealing offers exponential speedup over conventional quantum annealing.
- Iterated reverse annealing is ineffective for the $p$-spin model.
- Adiabatic reverse annealing with $h$-gain may improve performance.

## Abstract

Reverse annealing is a relatively new variant of quantum annealing, in which one starts from a classical state and increases and then decreases the amplitude of the transverse field, in the hope of finding a better classical state than the initial state for a given optimization problem. We numerically study the unitary quantum dynamics of reverse annealing for the mean-field-type $p$-spin model and show that the results are consistent with the predictions of equilibrium statistical mechanics. In particular, we corroborate the equilibrium analysis prediction that reverse annealing provides an exponential speedup over conventional quantum annealing in terms of solving the $p$-spin model. This lends support to the expectation that equilibrium analyses are effective at revealing essential aspects of the dynamics of quantum annealing. We also compare the results of quantum dynamics with the corresponding classical dynamics, to reveal their similarities and differences. We distinguish between two reverse annealing protocols we call adiabatic and iterated reverse annealing. We further show that iterated reverse annealing, as has been realized in the D-Wave device, is ineffective in the case of the $p$-spin model, but note that a recently-introduced protocol ("$h$-gain"), which implements adiabatic reverse annealing, may lead to improved performance.