Geometric Optimization of Quantum Control with Minimum Cost

TL;DR

This paper presents a geometric approach to quantum control optimization by framing the problem as finding geodesics on Riemannian manifolds, applicable to systems with SU(2) and SU(1,1) symmetries, enabling efficient control protocols.

Contribution

It introduces a novel geometric framework for quantum control optimization using Riemannian geometry, applicable to a broad class of quantum systems with specific symmetries.

Findings

Optimal control corresponds to geodesics on the manifold.

The method applies to systems with SU(2) and SU(1,1) symmetries.

Provides a geometric approach to shortcuts to adiabaticity.

Abstract

We investigate the optimization of quantum control from a differential geometric perspective. In our approach, optimal control minimizes the cost associated with evolving a quantum state, with the cost quantified by the length of the trajectory on a relevant Riemannian manifold. We demonstrate the optimization protocol in systems with SU(2) and SU(1,1) dynamical symmetries, which encompass a broad range of physical systems. In these systems, the time evolution can be represented by trajectories on a three-dimensional manifold. Given the initial and final states, the minimum-cost quantum control corresponds to a geodesic on the manifold. When the trajectory between the initial and final states is specified, the minimum-cost control corresponds to a geodesic within a submanifold embedded in the three-dimensional space. This framework provides a geometric method for optimizing shortcuts to adiabatic driving.