Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: application to the $XXZ$ model on the square lattice

TL;DR

This paper investigates optimal bang-bang control protocols for variational quantum algorithms applied to the XXZ model, revealing topological phases characterized by pulse counts and geometric correlations, which could enhance large-scale quantum simulations.

Contribution

The study develops optimized Monte Carlo algorithms to identify topological and geometric patterns in optimal protocols for quantum ground state preparation.

Findings

Optimal protocols are characterized by a topological number of pulses.

A dynamical phase diagram with bifurcation transitions is established.

Protocols within the same phase exhibit geometric correlations.

Abstract

In this work, we address the challenge of uncovering patterns in variational optimal protocols for taking the system to ground states of many-body Hamiltonians, using variational quantum algorithms. We develop highly optimized classical Monte Carlo (MC) algorithms to find the optimal protocols for transformations between the ground states of the square-lattice XXZ model for finite systems sizes. The MC method obtains optimal bang-bang protocols, as predicted by Pontryagin's minimum principle. We identify the minimum time needed for reaching an acceptable error for different system sizes as a function of the initial and target states and uncover correlations between the total time and the wave-function overlap. We determine a dynamical phase diagram for the optimal protocols, with different phases characterized by a topological number, namely the number of on-pulses. Bifurcation transitions as a function of initial and final states, associated with new jumps in the optimal protocols, demarcate these different phases. The number of pulses correlates with the total evolution time. In addition to identifying the topological characteristic above, i.e., the number of pulses, we introduce a correlation function to characterize bang-bang protocols' quantitative geometric similarities. We find that protocols within one phase are indeed geometrically correlated. Identifying and extrapolating patterns in these protocols may inform efficient large-scale simulations on quantum devices.