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Simulation-Based Quantum System Inference with Neural Posterior Estimation

Hang Zou, Anton Frisk Kockum, Martin Rahm, Simon Olsson

arXiv:2609.34995v1quant-phcs.LG

TL;DR

Quantum-system characterization requires inferring parameters from measurements, but likelihood evaluation can require exponentially costly simulation. This paper pairs classical simulation with neural posterior estimation, demonstrating reusable inference across quantum applications and examples reaching 81 qubits and 735 parameters.

  • Problem

    Inferring quantum-system parameters from measurement data is hindered by intractable likelihoods that generally require exponentially costly full-dynamics simulation.

    Source: §I · p. 1

  • Method

    The framework pairs polynomial-cost classical simulators with neural posterior estimation to learn posteriors from simulations and reuse them on new experimental observations.

    Source: §I · p. 2

  • Results

    The framework is demonstrated on four representative inverse problems, with results spanning up to 81 qubits and 735 latent parameters.

    Source: §I · p. 2

  • Takeaways & Limitations

    Posterior uncertainty provides a diagnostic that may guide experimental design, while inference becomes a reusable capability that grows as simulators and generative models improve.

    Source: §VII · pp. 9–10

    In the n-qubit brickwall circuit, 6(n−2) error coefficients remain gauge-ambiguous and individually unidentifiable, even with tomographically complete inputs and measurements.

    Source: §III · p. 4

Abstract

from arXiv · show

Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.

I. INTRODUCTION

Quantum characterization requires inferring states, Hamiltonians, and noise from measurements, but likelihood evaluation can demand exponentially costly full-dynamics simulation. This work pairs classical simulators with neural posterior estimation to make inference reusable across experiments and demonstrates it in several quantum applications.

I. INTRODUCTION

  • Likelihood-based inference is impractical beyond small systems because evaluating the likelihood generally requires exponentially costly full quantum-dynamics simulation.Prior SBI approaches also face costly forward simulation, high-dimensional sample requirements, and per-observation reruns; this framework combines neural posterior estimation with classical simulators to amortize inference.Source: §I · pp. 1–2
  • Pauli-noise inference scales to 50-qubit shallow circuits, tracks parameter drift, and uses posterior uncertainty to diagnose identifiability.The training-data budget for high-accuracy inference grows approximately linearly with system size for the sparse Pauli noise model.Source: §I · p. 2
  • For Trotterized Ising dynamics, digital-twin ML-QEM reduces error by over twofold relative to ZNE baselines and extrapolates beyond trained circuit depths.PEC and ZNE using inferred noise posteriors achieve mitigation accuracy comparable to using ground-truth noise parameters.Source: §I · p. 2
  • For 12-qubit state tomography, a reusable estimator reconstructs randomly sampled states with fidelity typically exceeding 99%, without per-state optimization.The estimator uses structured parameterized quantum circuits.Source: §I · p. 2
  • In simulated 9 × 9 Rydberg arrays, inferred atomic displacements agree closely with prescribed values (R2 = 0.983), and posterior uncertainties flag sites for possible calibration.Inference uses local observables evaluated after short-time Hamiltonian evolution.Source: §I · p. 2
  • The demonstrations reach systems of up to 81 qubits and inference over as many as 735 latent parameters.The paper presents this parameter dimensionality as lying at the frontier of Bayesian quantum parameter estimation and SBI in other domains.Source: §I · p. 2

II. METHODS

The methods section introduces the inference framework from a Bayesian perspective and presents conditional normalizing flows as its posterior estimators.

II. METHODS

  • The section first presents the simulation-based inference framework from a Bayesian perspective.Source: §II · p. 2
  • It then introduces conditional normalizing flows.Source: §II · p. 2
  • The flows serve as the posterior density estimators used in this work.Source: §II · p. 2

A. Simulation-based quantum system inference

The framework casts quantum-system learning as Bayesian inference and uses simulator-generated data to train a neural posterior estimator. Once trained, the estimator returns posterior samples for new observations without further simulation; sequential refinement trades reuse across observations for lower simulation cost on one target.

A. Simulation-based quantum system inference

  • Each inference task is specified by latent parameters, their prior, an observation, and a scalable simulator approximating the quantum forward map.The exact quantum process defines the target likelihood, which is generally intractable.Source: A. Simulation-based quantum system… · p. 2
  • Prior-sampled parameters are propagated through a simulator to generate observations, implicitly encoding the likelihood for NPE training on parameter-observation pairs.Source: Figure 1 · p. 3, A. Simulation-based quantum system… · p. 3
  • After training, the estimator approximates the posterior and generates samples for a new observation without further simulation or iterative sampling.Source: A. Simulation-based quantum system… · p. 3
  • For a single target observation, sequential NPE reduces total simulation cost but produces a model that cannot be reliably reused across observations.At each round, the previous posterior serves as the proposal and new simulations are appended to the training set.Source: A. Simulation-based quantum system… · p. 3

B. Normalizing flows

Conditional normalizing flows represent the posterior as a tractable density obtained by transforming a simple latent distribution into parameter space, conditioned on the observation.

B. Normalizing flows

  • A conditional normalizing flow maps a latent variable from a simple base density through a smooth, invertible transformation into the parameter space.The transformation is conditioned on the observation.Source: B. Normalizing flows · p. 3
  • The change-of-variables formula makes the posterior density exactly evaluable under the flow.Source: B. Normalizing flows · p. 3
  • The paper uses neural spline flows, combining monotone rational-quadratic splines with autoregressive transformations for tractable Jacobian determinants.These design choices support fast training and inference.Source: B. Normalizing flows · p. 3

III. LEARNING SPARSE PAULI NOISE

The section infers gate-by-gate Pauli noise from brickwall-circuit measurements, then examines identifiability, training-data scaling, and tracking of drifting parameters.

Circuit and noise model

  • The characterization uses nearest-neighbor CNOT gates in odd- and even-control brickwall layers, each followed by its own two-qubit Pauli noise channel.The measurement ensemble contains weight-1 and nearest-neighbor weight-2 Pauli observables; SPAM and single-qubit gates are treated as ideal.Source: §III · p. 4

Inference setup

  • The latent model assigns 15 Pauli-error probabilities to each CNOT, while Pauli propagation computes the observations at O(n^2) cost.The prior is uniform over [0, 0.001], and the simulator takes approximately 0.1 s at n = 50 on a single CPU thread.Source: §III · p. 4

Identifiability

  • Gauge freedom leaves 6(n−2) coefficients unidentifiable in the brickwall model, even with tomographically complete inputs and measurements.The full-space metrics plateau, while the gauge-free subset converges with high accuracy.Source: §III · p. 4
  • Broad posteriors identify coefficients that composite circuit measurements cannot resolve, because adjacent-gate Pauli errors can enter gauge-equivalent combinations.The paper presents this numerical diagnostic as complementary to analytic analysis.Source: §III · p. 4

Scalable learning results

  • Training-data requirements grow approximately linearly from n = 10 to n = 50, while the 50-qubit, 735-parameter model concentrates most estimates near ground truth.The figure also compares convergence in the full parameter space with convergence in the gauge-free subset.Source: §III · p. 4, Figure 3 · p. 5

Drifting noise

  • Across prescribed sinusoidal drifts, pXX tracks accurately, pYY is systematically overestimated despite phase tracking, and pIX remains broad and out of phase.The examples span ten runs, representing temporal drift or cross-device variation, without per-instance calibration.Source: §III · pp. 4–5, Figure 3 · p. 5

IV. POSTERIOR-DRIVEN QUANTUM ERROR MITIGATION

The section considers error-mitigation protocols that use an explicit noise model and introduces posterior-driven model-based and classical-reference strategies.

Posterior-driven QEM

  • The inferred noise posterior is applied to probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE), which require an explicit noise model.These are presented as applications of the learned noise posterior.Source: §IV · p. 5

Sampling-based methods

  • ZNE amplifies noise and extrapolates expectation values to the zero-noise limit, whereas PEC reweights outcomes using quasiprobability representations of inverse noise channels.Both methods can require substantial sampling, with costs that may grow exponentially with circuit size.Source: §IV · pp. 5–6

Classical-reference methods

  • The section also proposes using classically tractable noisy reference circuits to reduce data-collection costs for methods that estimate corrections from reference circuits.Source: §IV · p. 6

A. Sampling-based QEM with inferred noise

The section tests posterior-parameterized ZNE and PEC against simulations using ground-truth noise. Both inferred-noise protocols perform comparably to ground-truth parameterizations, despite gauge non-identifiability.

Protocol setup

  • The posterior mean and median parameterize the noise models used to construct amplification channels for ZNE and quasiprobabilities for PEC.Noisy circuit simulations use ground-truth noise, while inferred models are used to sample random circuits for mitigation.Source: A. Sampling-based QEM with inferred… · p. 6

Mitigation results

  • Posterior-parameterized ZNE estimates improve over the unmitigated baseline and lie close to ground-truth-noise estimates across linear, quadratic, and exponential fits.The comparison concerns the 15-qubit probe-state observable ⟨Z5Z6Z7Z8⟩.Source: A. Sampling-based QEM with inferred… · p. 6
  • PEC estimates for ground-truth, posterior-median, and posterior-mean noise models converge closely toward the ideal value with nearly identical sampling overheads.Source: A. Sampling-based QEM with inferred… · p. 6

Robustness to non-identifiability

  • Across the tested parameterizations, QEM remains effective despite gauge non-identifiability when noise models represent the same effective circuit channel.Some posterior-based results are closer to ideal than ground-truth-parameterized results, but the paper attributes this possibly to finite-sampling fluctuations, not a systematic advantage.Source: A. Sampling-based QEM with inferred… · p. 6

B. Digital-twin ML-QEM

The digital-twin strategy uses SBI noise posteriors to generate simulated training pairs for a denoiser, which then mitigates new noisy outputs. In simulated 12-qubit Ising dynamics, SBI-DT improves observable estimates over noisy and ZNE baselines at most evolution times and generalizes beyond its training boundary.

B. Digital-twin ML-QEM

  • Noise parameters drawn from the SBI posterior parameterize a classical digital twin that generates noisy–ideal pairs for denoiser training.The denoiser learns a correction map from synthetic pairs, with optional fine-tuning on real data to absorb model mismatch.Source: B. Digital-twin ML-QEM · p. 6, Figure 5 · p. 7
  • For mitigation, the latent parameter is the observable correction, and deployment applies a trained regression network to noisy data in one forward pass.The network predicts a correction rather than a full posterior; the simulator is a Clifford simulator trained on Clifford variants of the target circuit.Source: B. Digital-twin ML-QEM · pp. 6–7
  • Beyond the training boundary at t = 1.4, SBI-DT generalizes while ZNE methods degrade significantly; ZNE is more accurate at short times.Across most evolution times, SBI-DT gives the most accurate observable estimates.Source: B. Digital-twin ML-QEM · p. 7, Figure 5 · p. 7
  • 5.3-fold improvement for SBI-DT, versus 2.1-fold for quadratic ZNE and 1.2-fold for linear ZNE, relative to the noisy baseline.SBI-DT is over twice as effective as quadratic ZNE and requires no additional quantum samples at deployment.Source: B. Digital-twin ML-QEM · pp. 7–8, Figure 5 · p. 7

V. QUANTUM STATE TOMOGRAPHY WITH PARAMETERIZED QUANTUM CIRCUITS

The paper presents amortized quantum state tomography using a parameterized quantum circuit (PQC) ansatz and simulated measurement data. On 12-qubit test states, reconstruction improves with simulation budget, and a complementary unstructured 4-qubit example also reports high median fidelity.

V. QUANTUM STATE TOMOGRAPHY WITH PARAMETERIZED QUANTUM CIRCUITS

  • One estimator learns from simulated observations of many random states and infers states across the ansatz family without retraining.Posterior parameter samples are mapped through the PQC ansatz to reconstructed states.Source: §V · p. 8
  • The PQC prior restricts inference to its ansatz manifold, while MPS simulation provides polynomial-cost evaluation for the chosen state family.For this ansatz, each cut is crossed by at most one CNOT gate, so the MPS simulation is exact at χ = 2.Source: §V · p. 8
  • At the largest training budget, median infidelity reaches approximately 2 × 10^-3, and 86% of posterior samples achieve fidelity above 0.99.The median infidelity falls by more than two orders of magnitude as the simulation budget grows; best-case infidelity remains below 1.5 × 10^-3 throughout.Source: §V · pp. 8–9
  • The complementary unstructured 4-qubit pure-state example achieves median reconstruction fidelities above 0.98 for most tested states.This uses a dense Cholesky parameterization under an unstructured prior.Source: §V · p. 9

VI. HAMILTONIAN LEARNING OF RYDBERG ATOM ARRAYS

The paper applies simulation-based inference to infer atomic displacements in simulated Rydberg arrays, which determine the interatomic couplings. Its estimator produces localized posteriors and accurate displacement estimates across 81 atoms.

VI. HAMILTONIAN LEARNING OF RYDBERG ATOM ARRAYS

  • Atomic displacements alter pairwise couplings because the Rydberg interaction strength depends on the sixth power of interatomic distance.The paper motivates rapid position inference as a potential input to feedback adjustments of the target interactions.Source: §VI · p. 9
  • The latent vector contains two in-plane displacement components for each atom, and the model retains nearest- and next-nearest-neighbor couplings.The short-time Trotterized observables are evaluated by truncated Pauli propagation at polynomial cost; all training and evaluation data use this simulator.Source: §VI · p. 9
  • Posterior means and medians cluster around the true displacements across the 81-atom array, with point estimates accurate at the nanometer scale.For most sites, the ground truth lies within high-confidence contours; the estimator was trained on 10^5 simulations.Source: §VI · p. 9
  • Across 100 independent configurations, parameter-wise displacement regression yields MAE = 12.2±9.8 nm.Each configuration is evaluated using 500 posterior samples.Source: §VI · p. 9
  • The full parameter-vector error norm is reduced by an order of magnitude relative to the broad prior, with R2 = 0.983.The authors note that experimental performance remains to be validated, despite the simulated accuracy being practically meaningful relative to reported tweezer-positioning uncertainties.Source: §VI · p. 9

VII. DISCUSSION AND CONCLUSION

The discussion presents simulation-based quantum system inference as a reusable framework for quantum characterization, with posterior uncertainty offering information beyond point estimates. It also identifies modeling and deployment limits and outlines possible extensions.

VII. DISCUSSION AND CONCLUSION

  • The framework turns per-experiment quantum parameter inference into a reusable capability that can grow as simulators and generative models improve.The authors’ central conclusion is that inference can reach regimes accessible to classical simulation.Source: §VII · p. 9
  • Its four stated features are generality, polynomial-cost simulator-based training, amortized posterior inference, and uncertainty that may guide experimental design.Training uses simulators such as Pauli propagation and tensor networks rather than density-matrix evolution.Source: §VII · pp. 9–10
  • Current Pauli-noise learning assumes Markovian dynamics and idealized SPAM, while simulator–device mismatch can bias posteriors on experimental data.The authors identify misspecification detection and posterior-uncertainty calibration as essential for reliable hardware deployment.Source: §VII · p. 10
  • Size-agnostic or gauge-aware architectures and flow matching are proposed as routes to address fixed-size restrictions, gauge redundancies, and high-dimensional training.The discussion also points to hybrid quantum–classical data generation and quantum generative models as future extensions.Source: §VII · p. 10
  • Unlike the cited variational Bayesian approach, this SBI method avoids online optimization across observations at the cost of upfront simulation and training.The authors describe the two inference frameworks as complementary.Source: §VII · pp. 10–11

DATA AVAILABILITY

The simulator benchmarks report numerical agreement with density-matrix simulation and polynomial-cost Pauli propagation, while spline flows provide a flexible posterior estimator. The appendix also describes Pauli propagation's exponential branching limitation and reports results for weight-truncated Hamiltonian simulation; data and code are listed as available at pending_publication.

DATA AVAILABILITY

  • Data and models, as well as source code and software details, are listed as available at pending_publication.The named reproduction packages include zuko, sbi, stim, Qiskit, and PyTorch.Source: DATA AVAILABILITY · p. 11

Appendix A: Neural spline flows

  • The posterior estimator composes autoregressive layers conditioned on an observation embedding, with masked networks generating coordinatewise monotone spline transforms from preceding coordinates.Alternating variable orderings couple parameters across layers, while rational-quadratic spline bins use widths, heights, and derivatives with identity tails outside [−B, B].Source: Appendix A · p. 11, Figure 8 · p. 12

Appendix B: Pauli propagation

  • Pauli propagation evolves observables backward in the Heisenberg picture and tracks sparse Pauli-string coefficients rather than constructing a dense 4^n-dimensional transfer matrix.Clifford gates and Pauli channels are 1-branching, whereas non-Clifford rotations can branch; increasing branching can grow the string count exponentially, motivating coefficient or weight truncation.Source: Appendix B · pp. 12–13

1. Simulator for Pauli noise learning

  • For Pauli noise learning, analytic factorization of terminal product-state expectations enables exact, non-truncated simulation with O(nk) cost for k target strings.The approach avoids the initial layer's branching overhead and is described as suitable for sparse Markovian Pauli noise in circuits exceeding 100 qubits.Source: §1 · p. 13
  • Numerical agreement within floating-point precision accompanies O(n) cost per observable and O(n^2) cost across all observables; n = 1000 takes under 4 ms per observable and roughly 40 s overall.The benchmark uses k = 12n −9 local Pauli observables; Qiskit Aer becomes prohibitive beyond n ∼14.Source: Figure 9 · p. 13, §1 · p. 14

2. Simulator for Rydberg Hamiltonian learning

  • For Rydberg Hamiltonian learning, truncation at weight 3 or higher yields negligible error for the tested local observables, with propagation remaining polynomial-cost up to n = 144.At n = 16, Pauli propagation is nearly an order of magnitude cheaper than statevector simulation; the main text uses weight-5 truncation.Source: §2 · p. 14

Appendix D: Local gauge ambiguity

These appendices explain how gauge freedom limits gate-level noise identification, report posterior accuracy on identifiable parameters, and detail alternative simulation and inference procedures for mitigation and tomography.

Appendix D: Local gauge ambiguity

  • Each bridge qubit creates three coupled Pauli-error pairs, yielding 18 unlearnable parameters in the five-qubit example and 6(n −2) in an n-qubit brickwall circuit.Errors can shift across adjacent CNOTs without changing the composite channel; independent gate calibrations or gauge-fixing elements are needed to resolve them.Source: Appendix D · pp. 14–15

Appendix E: Complete posterior distributions for 50-qubit Pauli noise

  • R2 = 0.996 and MAE = 2.3×10−5 on the 447 gauge-free parameters, versus R2 = 0.759 and MAE = 1.1 × 10−4 across all 735 parameters.The 50-qubit posteriors exhibit four gate categories: odd-layer top and bottom boundaries, odd-layer interior, and even-layer interior.Source: Appendix E · pp. 15–16, Figure 12 · p. 16

1. Probabilistic error cancellation

  • Probabilistic error cancellation samples Pauli corrections according to normalized absolute inverse-channel coefficients, with the product of per-gate overheads setting total sampling overhead.The inverse-channel coefficients can be negative because the inverse channel is non-physical.Source: §1 · p. 16

2. Zero-noise extrapolation

  • Zero-noise extrapolation uses posterior-informed probabilistic error amplification when a noise model is available, or gate folding when it is not.Probabilistic amplification permits non-integer gains; folding lengthens circuits and restricts gains to odd integers.Source: §2 · p. 16

Appendix G: NPE-based digital-twin ML-QEM

  • 2.4× aggregated MAE reduction relative to the noisy baseline, versus 2.1× for quadratic ZNE and 5.3× for direct regression.The NPE variant learns a full correction posterior; because Clifford-only corrections lack continuous support, training used random product-state augmentation.Source: Appendix G · pp. 16, 18, Figure 14 · p. 18

Appendix H: Matrix product states

  • The QST simulator contracts a bond-χ matrix product state without forming a dense statevector; evaluating all 12n −9 Pauli observables costs O(n^2χ^3).A left-to-right environment sweep evaluates each observable at O(nχ^3) cost.Source: Appendix H · p. 18

Appendix I: Quantum state tomography with Cholesky parameterization

  • Most (θx, θy) states achieve median infidelity below 0.04; at (π/3, π/3), Fmedian = 0.9901, Fmean = 0.9897, and Fmax = 0.9966.Infidelity rises near periodic boundaries, where equivalent rotation settings can make state reconstruction ambiguous.Source: Figure 15 · p. 19, Appendix I · p. 19

1. Neural posterior estimation

  • Neural posterior estimation uses shared flow settings across tasks, with task-specific batch size, training budget, and observation preprocessing; the Rydberg task uses a two-layer observation embedding.The Rydberg embedding has dimensions dimobs →512→256 with SiLU activations.Source: §1 · p. 19

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