Variational principle for optimal quantum controls in quantum metrology

TL;DR

This paper introduces a variational principle to optimize quantum controls and initial states for enhanced quantum metrology, demonstrating near-Heisenberg scaling even with limited controls in many-body systems.

Contribution

It develops a variational framework for optimal quantum control in metrology, including methods to approximate solutions under control restrictions using Floquet engineering.

Findings

Optimal controls depend on probe time with restrictions

Heisenberg scaling achievable with limited controls in spin chains

Floquet engineering approximates unconstrained control solutions

Abstract

We develop a variational principle to determine the quantum controls and initial state which optimizes the quantum Fisher information, the quantity characterizing the precision in quantum metrology. When the set of available controls is limited, the exact optimal initial state and the optimal controls are in general dependent on the probe time, a feature missing in the unrestricted case. Yet, for time-independent Hamiltonians with restricted controls, the problem can be approximately reduced to the unconstrained case via the Floquet engineering. In particular, we find for magnetometry with a time-independent spin chain containing three-body interactions, even when the controls are restricted to one and two-body interaction, that the Heisenberg scaling can still be approximately achieved. Our results open the door to investigate quantum metrology under a limited set of available controls, of relevance to many-body quantum metrology in realistic scenarios.