Box model of quantum annealing

TL;DR

This paper models quantum annealing using a particle-in-a-box approach, analyzing energy landscapes and diabatic transitions through numerical solutions of the Schrödinger equation.

Contribution

It introduces a novel box model for quantum annealing and investigates the effects of landscape roughness and annealing depth on residual energy and diabatic transitions.

Findings

Residual energy is largely independent of landscape roughness and annealing depth.

Diabatic transitions are prevalent during annealing, with discrepancies from Landau-Zener predictions.

Flat gaps in the energy spectrum may trap wave functions in local minima.

Abstract

A particle-in-a-box model of continuous space quantum annealing is proposed and studied numerically by solving the Schr\"odinger wave equation directly. Three types of energy landscapes with multiple local minima are considered, namely a sinusoidal wave modulated by a concave, a convex, or a flat envelope. Both static (energy spectrum) and dynamical (residual energy) behaviors are analyzed in detail, paying particular attention to the effects of landscape roughness and annealing depth. Simulation results show that the residual energy as a function of annealing speed is largely independent of these two factors. The prevalence of diabatic transitions during annealing is observed, and the discrepancy between our numerical results and the Landau-Zener formula is discussed. An interesting feature in the energy gap spectrum, which we call flat gaps, is examined. Based on it, we propose a mechanism to explain the trapping of wave function in local minima during diabatic transitions, widely observed in our data.