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arxiv: 2506.18802 · v2 · submitted 2025-06-23 · 🪐 quant-ph

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

Abigail N. Poteshman , Jiwon Yun , Tim H. Taminiau , Giulia Galli This is my paper

Pith reviewed 2026-05-19 07:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Bayesian inferencereversible-jump MCMCparallel temperingmodel selectionparameter estimationquantum sensingspin defectsill-posed inverse problems
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The pith

Hybridized MCMC recovers nuclear spin parameters and model dimension from an order of magnitude less data than existing methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Bayesian framework that combines several MCMC techniques to estimate both continuous parameters and discrete model dimensions from sparse, noisy experimental data. This addresses ill-posed inverse problems where traditional deterministic or machine-learning approaches often fail due to limited data quality and quantity. The method integrates sampling over mixed parameter spaces, reversible-jump moves to change model dimension, and parallel tempering to explore complex posteriors. It is applied to recovering the locations and hyperfine couplings of nuclear spins around a spin defect in a semiconductor from coherence data. The approach produces meaningful posteriors with far less data than prior techniques and is validated on real experimental measurements.

Core claim

By hybridizing MCMC sampling for mixed continuous-discrete spaces, reversible-jump MCMC to select model dimension, and parallel tempering to accelerate mixing, the framework recovers informative posterior distributions over physical parameters and model dimension even when data are sparse and noisy, as demonstrated on experimental coherence measurements of nuclear spins surrounding a semiconductor spin defect.

What carries the argument

The hybridized MCMC sampler that uses reversible-jump proposals to move between models of different dimensions while employing parallel tempering to improve exploration of the joint posterior over parameters and model index.

If this is right

  • Model dimension and continuous parameters can be inferred jointly rather than by fixing the number of spins in advance.
  • Posterior distributions remain informative with roughly ten times fewer measurements than required by existing approaches.
  • The same sampler structure applies directly to other nonlinear inverse problems that mix continuous and discrete unknowns under realistic noise.
  • Experimental validation confirms that recovered posteriors are consistent with independent measurements on actual quantum devices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-throughput quantum sensing experiments could characterize many more defects per unit time by reducing the number of coherence measurements needed per device.
  • The framework may extend naturally to other data-limited domains such as single-molecule spectroscopy or sparse astrophysical signals where model order is uncertain.
  • If the assumed noise model deviates substantially from experiment, the recovered posteriors will broaden or shift even when the true Hamiltonian lies inside the model family.

Load-bearing premise

The true underlying Hamiltonian belongs to the discrete family of models the reversible-jump sampler is allowed to visit, and the noise statistics are known well enough that model mismatch does not dominate the posterior.

What would settle it

Synthetic data generated from a spin configuration outside the allowed model family in which the sampler nevertheless reports high posterior probability on an incorrect model dimension and parameters would show the method fails to produce reliable results.

Figures

Figures reproduced from arXiv: 2506.18802 by Abigail N. Poteshman, Giulia Galli, Jiwon Yun, Tim H. Taminiau.

Figure 1
Figure 1. Figure 1: Schematic of workflow. The input (left) to the hybrid MCMC algorithm is a fixed set of data dd ∈ R d , a family of candidate Hamiltonians {Hk1 , Hk2 , . . . , Hkj } which are parameterized by aki where the number of parameters for different Hamiltonians Hki and Hkj can be different. We also take in a forward model fk(Hk(ak), b, c) that generates a set of data from candidate Hamiltonian Hk, and a likelihood… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of algorithms (a) random walk Metropolis Hastings in over a continuous domain (Alg. 1), (b) random walk Metropolis Hastings over a discrete domain (Alg. 1), (c) reverse jump Markov chain Monte Carlo (Alg. 2), and (d) parallel tempering (Alg. 3). We address the challenge of recovering parameters that may be drawn from different domains (i.e., the set or distribution from which parameters are sampl… view at source ↗
Figure 3
Figure 3. Figure 3: Recovery in sparse data limit. (a) We simulate a system of ten 13C surrounding a nitrogen vacancy (NV) center in diamond, and plot the resulting coherence signal for a 16-pulse dynamical decoupling experiment in an external magnetic field of 311 G. We sample a varying number of data points with noise ϵ ∼ N (0, 0.001), and we plot the coherence signal from the nuclear spin parameters recovered from the spec… view at source ↗
Figure 4
Figure 4. Figure 4: Recovery in noisy data limit. (a) We simulate a system of seventeen 13C surrounding a nitrogen vacancy (NV) center in diamond, and plot the resulting coherence signal sampled uniformly for 250 τj for a 16-pulse dynamical decoupling experiment in an external magnetic field of 311 G. We add varying amounts of noise to each data point, and we plot the coherence signal from the nuclear spin parameters recovere… view at source ↗
Figure 5
Figure 5. Figure 5: Validation of hybrid MCMC approach on experimental data: (a) Average error per data point as a function of the number of MCMC steps, with five random ensembles initialized for 25,000 steps each; the first 10,000 steps are treated as burn-in, and the remaining 15,000 steps are the posterior spin configurations. (b) Experimental coherence data (250 data points from 6 µs to 8 µs of a 32-pulse CPMG sequence) t… view at source ↗
Figure 6
Figure 6. Figure 6: Error trajectories of 5 ensembles as a function of walker steps for varying σ 2 . The recovery algorithm was applied to the same coherence signal generated using N = 16 CP pulses, ε = 0.001 shot noise, hyperfine perturbation 0.001, τmax = 0.008 ms, and 250 interpulse times sampled from 15 simulated spins with Rspin = 5 Å. There are a variety of hyperparameters in the hybrid algorithm, including the walker … view at source ↗
Figure 7
Figure 7. Figure 7: Error trajectories of 5 ensembles as a function of walker steps for varying Rspin. The recovery algorithm was applied to the same coherence signal generated using N = 16 CP pulses, ε = 0.001 shot noise, hyperfine perturbation 0.001, τmax = 0.008 ms, and 250 interpulse times sampled from 15 simulated spins with σ 2 = 0.1. B Performance of hybrid algorithm in dense, low noise limit We demonstrate that the pr… view at source ↗
Figure 8
Figure 8. Figure 8: Recovery method on data with experimental parameters in machine learning regime. (a) Detection rate for hyperfine couplings of the posterior distribution, (b) detection rate for the number of spins in the modal configurations of the posterior distribution, and (c) false positive rate for hyperfine couplings for spins in the modal spin configuration over the posterior distribution for data generated with 15… view at source ↗
Figure 9
Figure 9. Figure 9: Recovery method accuracy versus hyperfine coupling perturbations. (a) Detection rate for hyperfine couplings of the posterior distribution, (b) discrepancy between the number of spins recovered and the number of spins simulated, and (c) false positive rate for hyperfine coupling per￾turbations of different magnitudes. The recovery algorithm was applied to the same coherence signal generated using N = 16 CP… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a Bayesian framework that hybridizes MCMC sampling techniques—including methods for mixed continuous-discrete spaces, reversible-jump MCMC for trans-dimensional model selection, and parallel tempering—to estimate both physical parameters and model dimension from sparse, noisy data. The method is applied to recovering nuclear-spin locations and hyperfine couplings around a spin defect in a semiconductor from coherence measurements, with the central claim that meaningful posteriors can be recovered using an order of magnitude less data than existing approaches and that results are validated on experimental measurements.

Significance. If the central claims hold, the work provides a general strategy for principled inference in ill-posed inverse problems common to quantum sensing and characterization, where data acquisition is resource-intensive. The hybridization of sampling methods directly targets challenges in high-dimensional, nonlinear spaces with discrete model choices.

major comments (1)
  1. [Abstract] Abstract: The claim that the hybridized MCMC recovers accurate posteriors with ~10× fewer coherence measurements rests on the unverified assumption that the true nuclear-spin Hamiltonian lies exactly within the discrete family of models over which the reversible-jump sampler can move. No controlled synthetic-data test is described in which the ground-truth Hamiltonian is generated outside this family (or an ablation removing the reversible-jump component), leaving open the possibility that reported performance gains are dominated by model mismatch rather than by the data.
minor comments (1)
  1. The abstract would benefit from a single sentence clarifying the specific hybridization (e.g., how reversible-jump proposals are combined with parallel tempering) to aid readers unfamiliar with the subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the scope of our validation experiments. We address the comment in detail below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the hybridized MCMC recovers accurate posteriors with ~10× fewer coherence measurements rests on the unverified assumption that the true nuclear-spin Hamiltonian lies exactly within the discrete family of models over which the reversible-jump sampler can move. No controlled synthetic-data test is described in which the ground-truth Hamiltonian is generated outside this family (or an ablation removing the reversible-jump component), leaving open the possibility that reported performance gains are dominated by model mismatch rather than by the data.

    Authors: We agree that the synthetic-data experiments generate ground-truth Hamiltonians from within the discrete family explored by the reversible-jump sampler. This choice is deliberate: the method is intended for settings in which the true physical model belongs to the considered family (as is the case for the nuclear-spin Hamiltonian around a spin defect), and the trans-dimensional sampler is used to infer the unknown dimension within that family. Existing approaches to which we compare also operate under comparable model assumptions, so the reported reduction in required data is measured under consistent conditions. We acknowledge, however, that an explicit out-of-family mismatch test and an ablation isolating the reversible-jump component would strengthen the claims. We will add a dedicated paragraph in the revised manuscript (likely in Section 4 or a new subsection of the methods) that (i) states the modeling assumption explicitly, (ii) discusses the implications for performance claims, and (iii) presents a limited ablation removing the reversible-jump moves to quantify their contribution. We will also revise the abstract to avoid any implication that the method has been validated under arbitrary model misspecification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; MCMC framework derives posteriors directly from data without self-referential reduction

full rationale

The paper introduces a hybridized MCMC sampler combining reversible-jump, parallel tempering, and mixed continuous-discrete sampling to compute posteriors over Hamiltonian parameters and model dimension. The reported performance (order-of-magnitude data efficiency) is presented as an empirical outcome of applying this sampler to coherence measurements, with validation on experimental data. No quoted derivation step equates a claimed prediction or first-principles result to a fitted input or self-citation by construction. The method conditions outputs on observed data under stated assumptions about the model family and noise; these assumptions are external to the sampling procedure itself and do not create a closed loop where the result is forced by redefinition or renaming of inputs. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5777 in / 1069 out tokens · 32284 ms · 2026-05-19T07:55:30.796189+00:00 · methodology

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Reference graph

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