K-GRAPE: A Krylov Subspace approach for the efficient control of quantum many-body dynamics

TL;DR

K-GRAPE introduces a Krylov subspace method to enhance the efficiency of quantum control optimization, enabling the handling of larger systems by reducing computational effort per parameter compared to traditional GRAPE.

Contribution

The paper presents a novel Krylov-based modification of the GRAPE algorithm that significantly improves scalability for quantum many-body control tasks.

Findings

K-GRAPE achieves constant elementary computational effort per parameter.

The method scales linearly with system dimension, unlike traditional super-quadratic scaling.

Benchmark results demonstrate superior performance on the XXZ spin-chain model.

Abstract

The Gradient Ascent Pulse Engineering (GRAPE) is a celebrated control algorithm with excellent converging rates, owing to a piece-wise-constant ansatz for the control function that allows for cheap objective gradients. However, the computational effort involved in the exact simulation of quantum dynamics quickly becomes a bottleneck limiting the control of large systems. In this paper, we propose a modified version of GRAPE that uses Krylov approximations to deal efficiently with high-dimensional state spaces. Even though the number of parameters required by an arbitrary control task scales linearly with the dimension of the system, we find a constant elementary computational effort (the effort per parameter). Since the elementary effort of GRAPE is super-quadratic, this speed up allows us to reach dimensions far beyond. The performance of the K-GRAPE algorithm is benchmarked in the paradigmatic XXZ spin-chain model.