[Paper Review] Hamiltonian Learning with Online Bayesian Experiment Design in Practice

Ian Hincks, Thomas Alexander+3 more|arXiv (Cornell University)|Jun 6, 2018

Machine Learning in Materials Science4 citations

TL;DR

This paper proposes an online Bayesian experiment design framework for Hamiltonian learning in quantum systems, applying it experimentally to a nitrogen-vacancy (NV) center in diamond. By adaptively selecting experiments based on real-time Bayesian inference, the method achieves median posterior variances 10–100× lower than offline pre-determined sweeps, significantly improving parameter estimation efficiency for quantum sensing and device characterization.

ABSTRACT

Estimating parameters of quantum systems is usually done by performing a sequence of predetermined experiments and post-processing the resulting data. It is known that online design, where the choice of the next experiment is based on the most up-to-date knowledge about the system, can offer speedups to parameter estimation. We apply online Bayesian experiment design to a Nitrogen Vacancy (NV) in diamond to learn the values of a five-parameter model describing its Hamiltonian and decoherence process. Comparing this to standard pre-determined experiment sweeps, we find that we can achieve median posterior variances on some parameters that are between 10 and 100 times better given the same amount of data. This has applications to NV magnetometry where one of the Hamiltonian coefficients is the parameter of interest. Furthermore, the methods that we use are generic and can be adapted to any quantum device.

Motivation & Objective

• To improve the efficiency of quantum Hamiltonian parameter estimation by replacing fixed experiment sequences with adaptive, online experiment design.
• To evaluate whether online Bayesian experiment design can outperform standard offline experiment sweeps in real experimental settings with noisy, non-trivial quantum systems.
• To demonstrate the practical feasibility and performance gains of a fully Bayesian, risk-minimizing approach to experiment selection in a real-world quantum device—specifically, an NV center in diamond.
• To provide a generic, adaptable framework for Hamiltonian learning applicable to a wide range of quantum devices beyond NV centers.
• To quantify the improvement in estimation accuracy and consistency by comparing posterior variance reduction and trial-to-trial variability between online and offline heuristics.

Proposed method

• Employ sequential Bayesian inference to update the posterior distribution over Hamiltonian parameters after each experiment, maintaining a full probability distribution over unknowns.
• Use a weighted Bayes risk minimization criterion to select the next experiment in real time, where the risk is defined over a set of possible future experiments and their expected information gain.
• Implement a variety of experiment types (e.g., Ramsey, Rabi) with tunable control parameters (e.g., pulse durations, delays) to explore the parameter space adaptively.
• Integrate the inference engine with a real experimental setup controlling the NV center’s qutrit manifold, including optical initialization, microwave control, and fluorescence readout.
• Allow dynamic switching of control fields during experiments to increase experimental flexibility and information content.
• Compare multiple heuristics: offline (e.g., Alternating Linear, Ramsey Sweeps) and online (e.g., Magnetometry Weighted Risk), all evaluated under identical data collection conditions.

Experimental results

Research questions

• RQ1 Can online Bayesian experiment design significantly reduce posterior variance in Hamiltonian parameter estimation compared to standard offline experiment sweeps in a real experimental system?
• RQ2 How does the performance of online heuristics compare to offline heuristics in terms of convergence speed and consistency across independent trials?
• RQ3 To what extent do online heuristics prioritize experiments that are most informative for the parameter of interest (e.g., magnetic field sensitivity via ωₑ)?
• RQ4 Does the use of a full Bayesian risk minimization framework lead to more predictable and robust estimation outcomes than heuristic-based or fixed-sweep approaches?
• RQ5 What is the scaling behavior of posterior variance reduction in online learning, and does it approach the standard quantum limit?

Key findings

• The online heuristic called Magnetometry Weighted Risk reduced median posterior variance for the magnetic field parameter ωₑ by over two orders of magnitude after 200 experiments compared to the offline Alternating Linear sweep.
• In scenarios with a tight prior on all parameters except ωₑ, online heuristics achieved approximately one order of magnitude better posterior variance reduction than the best offline heuristic.
• Online heuristics exhibited significantly more consistent performance across independent trials, with posterior variance variation less than one order of magnitude, compared to up to four orders of magnitude variation in the Ramsey Sweeps offline heuristic.
• The online experiment design strategy achieved exponential-to-standard-quantum-limit (SQL) scaling of posterior variance, consistent with theoretical expectations and prior Hamiltonian estimation work.
• When prioritizing ωₑ estimation, online heuristics predominantly selected Ramsey experiments, confirming their ability to focus on the most informative experiment types for the target parameter.
• The transient phase of learning showed exponential decay in variance with respect to effective sample size (ESM), transitioning into SQL scaling once system coherence and experimental duration were optimized.