Trans-dimensional Hamiltonian model selection and parameter estimation from sparse, noisy data

Abigail N. Poteshman; Jiwon Yun; Tim H. Taminiau; Giulia Galli

doi:10.22331/q-2026-04-08-2055

Trans-dimensional Hamiltonian model selection and parameter estimation from sparse, noisy data

Abigail N. Poteshman^1,2, Jiwon Yun³, Tim H. Taminiau^3,4, and Giulia Galli^2,5,6

¹Committee on Computational and Applied Mathematics, University of Chicago, Chicago, IL 60637, USA
²Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA
³QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
⁴Kavli Institute of Nanoscience, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
⁵Department of Chemistry, University of Chicago, Chicago, IL 60637, USA
⁶Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

Published:	2026-04-08, volume 10, page 2055
Editor:	Marco Cerezo
Eprint:	arXiv:2506.18802v2
Doi:	https://doi.org/10.22331/q-2026-04-08-2055
Citation:	Quantum 10, 2055 (2026).

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Abstract

High-throughput characterization often requires estimating parameters and model dimension from experimental data of limited quantity and quality. Such data may result in an ill-posed inverse problem, where multiple sets of parameters and model dimensions are consistent with available data. This ill-posed regime may render traditional machine learning and deterministic methods unreliable or intractable, particularly in high-dimensional, nonlinear, and mixed continuous and discrete parameter spaces. To address these challenges, we present a Bayesian framework that hybridizes several Markov chain Monte Carlo (MCMC) sampling techniques to estimate both parameters and model dimension from sparse, noisy data. By integrating sampling for mixed continuous and discrete parameter spaces, reversible-jump MCMC to estimate model dimension, and parallel tempering to accelerate exploration of complex posteriors, our approach enables principled parameter estimation and model selection in data-limited regimes. We apply our framework to a specific ill-posed problem in quantum information science: recovering the locations and hyperfine couplings of nuclear spins surrounding a spin-defect in a semiconductor from sparse, noisy coherence data. We show that a hybridized MCMC method can recover meaningful posterior distributions over physical parameters using an order of magnitude less data than existing approaches, and we validate our results on experimental measurements. More generally, our work provides a flexible, extensible strategy for solving a broad class of ill-posed inverse problems under realistic experimental constraints.

Featured image: Schematic of inputs and outputs to a hybrid MCMC algorithm

Popular summary

Modern experiments—especially in areas like quantum materials and nanotechnology—often face a frustrating challenge: trying to extract detailed physical information from limited and noisy data. In many cases, there isn’t a single “correct” answer. Instead, multiple different physical models can explain the same measurements, making the problem fundamentally ambiguous, or ill-posed.

This work introduces a computational approach to tackle that ambiguity head-on. Rather than trying to find one best solution, the method uses Bayesian statistics to map out all plausible explanations consistent with the data—and how likely each one is. The key innovation is combining several advanced sampling techniques into a single framework that can handle both continuous parameters (like interaction strengths) and discrete parameters (like how many particles are present).

We demonstrate the utility of this approach on a problem from quantum information science: identifying the positions and interactions of nuclear spins surrounding a quantum defect in a solid. These nuclear spins' interactions subtly influence the defect’s behavior, but the available measurements are sparse and noisy, making the reconstruction especially difficult. The new method is able to recover meaningful information about the spin environment using far less data than previous techniques, and its predictions agree with experimental results.

Beyond this specific application, the framework offers a powerful and flexible way to solve a wide range of inverse problems in physics—particularly in situations where data are scarce, models are complex, and uncertainty cannot be ignored.

► BibTeX data

@article{Poteshman2026transdimensional,
  doi = {10.22331/q-2026-04-08-2055},
  url = {https://doi.org/10.22331/q-2026-04-08-2055},
  title = {Trans-dimensional {H}amiltonian model selection and parameter estimation from sparse, noisy data},
  author = {Poteshman, Abigail N. and Yun, Jiwon and Taminiau, Tim H. and Galli, Giulia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2055},
  month = apr,
  year = {2026}
}

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