[Paper Review] Optimal control for Hamiltonian parameter estimation in non-commuting and bipartite quantum dynamics

Shushen Qin, M. Cramer+2 more|arXiv (Cornell University)|May 5, 2022

Quantum Information and Cryptography4 citations

TL;DR

This paper develops optimal control strategies using Krotov's method to enhance Hamiltonian parameter estimation in noisy, non-commuting, and bipartite quantum systems. It demonstrates that controlled dynamics restore the Heisenberg limit in single-qubit non-commuting field estimation and achieve a sevenfold QFI improvement in two-qubit transverse coupling estimation, with enhanced precision, time stability, and robustness to control imperfections.

ABSTRACT

The ability to characterise a Hamiltonian with high precision is crucial for the implementation of quantum technologies. In addition to the well-developed approaches utilising optimal probe states and optimal measurements, the method of optimal control can be used to identify time-dependent pulses applied to the system to achieve higher precision in the estimation of Hamiltonian parameters, especially in the presence of noise. Here, we extend optimally controlled estimation schemes for single qubits to non-commuting dynamics as well as two interacting qubits, demonstrating improvements in terms of maximal precision, time-stability, as well as robustness over uncontrolled protocols.

Motivation & Objective

• To extend optimal control methods beyond single-qubit, commuting systems to non-commuting and two-qubit interacting systems for improved parameter estimation.
• To address the challenge of achieving high-precision Hamiltonian parameter estimation under noise and non-commuting dynamics.
• To demonstrate that optimal control can restore the Heisenberg limit in non-commuting scenarios where standard protocols fail.
• To evaluate robustness of controlled schemes against control field imperfections and initial estimation errors.
• To explore practical feasibility through pulse shape simplicity, directional constraints, and tolerance to timing/amplitude errors.

Proposed method

• Uses Krotov’s optimal control algorithm to design time-dependent control pulses that maximize quantum Fisher information (QFI).
• Applies the method to single-qubit systems with non-commuting dynamics and two-qubit systems with ZZ and XX couplings under dephasing noise.
• Optimizes control fields to steer arbitrary initial states toward decoherence-free subspaces and enhance parameter sensitivity.
• Employs the quantum Cramér-Rao bound as a precision benchmark, with QFI as the key performance metric.
• Implements numerical simulations to evaluate QFI, time stability, and robustness under deviations from ideal controls.
• Compares controlled protocols against uncontrolled ones to quantify precision gains and assess practical viability.

Experimental results

Research questions

• RQ1 Can optimal control restore the Heisenberg limit in non-commuting single-qubit parameter estimation under dephasing noise?
• RQ2 How does optimal control improve estimation precision in two-qubit systems with transverse (XX) and longitudinal (ZZ) couplings?
• RQ3 To what extent do controlled protocols maintain high QFI over long time spans, ensuring time stability?
• RQ4 How robust are the controlled schemes to imperfections in control pulse amplitudes and timings?
• RQ5 Can simple, unidirectional pulse shapes achieve near-optimal performance, enabling practical implementation?

Key findings

• Optimal control restores the Heisenberg limit in non-commuting single-qubit field direction estimation, overcoming the breakdown of standard protocols under non-commuting dynamics.
• A sevenfold increase in quantum Fisher information (QFI) is achieved in two-qubit transverse (XX) coupling estimation compared to uncontrolled protocols.
• Controlled schemes maintain near-maximal QFI over extended time intervals, demonstrating superior time stability compared to uncontrolled dynamics.
• The method remains effective even under significant deviations from ideal control pulses, showing robustness to amplitude and timing errors.
• Simple, unidirectional pulse shapes achieve near-optimal performance, reducing experimental complexity and enhancing practical feasibility.
• Arbitrary initial probe states can be effectively steered into decoherence-free subspaces by optimal controls, making probe state preparation less critical.