Simulation-Based Quantum System Inference with Neural Posterior Estimation

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

quant-phSep 28, 2026

Simulation-Based Quantum System Inference with Neural Posterior Estimation

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

Organizations: Department of Computer Science and Engineering, Chalmers University of Technology and University of Gothenburg, 41296 Gothenburg, Sweden · Department of Microtechnology and Nanoscience, Chalmers University of Technology, 41296 Gothenburg, Sweden · Department of Chemistry and Chemical Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden

Abstract

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.

Figures & tables

Figure 1: Simulation-based quantum system inference. Top, schematic of the framework. Training phase (solid black arrows): parameters $\bm{\theta}_{1:N}$ sampled from a prior $\pi(\bm{\theta})$ are passed through a forward simulator to produce synthetic observations $\mathbf{x}_{1:N}$ , on which a neural density estimator $q_{\bm{\phi}}(\bm{\theta}|\mathbf{x})$ learns to approximate the posterior. Inference phase (red arrows): given a target observation $\mathbf{x}_{0}$ from a quantum processing unit (QPU), the trained estimator efficiently evaluates the posterior $q_{\bm{\phi}}(\bm{\theta}|\mathbf{x}_{0})$ without further simulation. Sequential training (dashed black arrow): an optional sequential refinement round that reuses the posterior as a new proposal. Bottom, the representative applications demonstrated in this work, including Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning.

Figure 2: Brickwall circuit template used in this study, shown for

n=4

. Each qubit

q

is initialized by a single-qubit rotation

U_{q}

, and each gate

\text{CNOT}_{k,k+1}

is followed by a 2-local Pauli noise channel

\Lambda_{k}

Figure 3: Scalable Pauli noise learning with NPE. (a) Posterior distributions of Pauli error coefficients for four representative CNOT gates within a 50-qubit brickwall circuit, shown as box plots over 2,000 posterior samples. Gates are ordered by brickwall layer, with odd-control gates first, rather than by qubit index. The inference is performed using an amortized NPE model trained on 30,000 simulations. Red crosses denote ground-truth values. Broad posteriors on specific parameters reflect gauge ambiguity (see main text and Appendix D ). (b) Convergence of nMAE and

R^{2}

as a function of cumulative training data across SNPE rounds for the 50-qubit model, starting from 10,000 simulations and incrementing by 5,000 per round. Metrics are shown for both the full parameter space and the gauge-free subset. (c) Training-data budget required to reach high-accuracy thresholds (

R^{2}_{\text{gauge-free}}\geq 0.99

and

\geq 0.999

) as a function of system size, extracted from SNPE training analogous to (b) for each system size. (d) Tracking selected drifting Pauli noise coefficients for the CNOT 1→2 gate shown in panel (a). Prescribed sinusoidal ground-truth profiles (purple curves) are compared with posterior means (teal circles), with shaded bands indicating

\pm 1

posterior standard deviation.

Figure 4: ZNE and PEC based on the SBI noise posterior. All protocols are parameterized by the ground-truth noise (red), the SBI posterior median (green), and the posterior mean (blue). (a) Zero-noise extrapolation (ZNE) of the expectation value

\langle Z_{5}Z_{6}Z_{7}Z_{8}\rangle

for a 15-qubit probe state. Mitigated estimates at the zero-noise limit (

G=0

, star) are obtained via linear (left), quadratic (middle), and exponential (right) extrapolation from unmitigated (

G=1

) and probabilistic-error-amplified (

G\in\{1.6,2.2,2.8\}

) results, with each expectation value estimated using 2,000 random circuit samples per gain level. (b) Convergence of the PEC estimate for the same observable as a function of the total quasi-probability samples (in thousands). The inset shows the sampling overhead of PEC under each noise model. (c) Comparison of the mitigated results from panels (a) and (b): ZNE with different extrapolation fits, and PEC evaluated at a sample size of 2,000 (2k). The dashed black and dash-dotted red lines indicate the ideal and unmitigated values, respectively.

Figure 5: Digital-twin ML-QEM with the SBI noise posterior. (a) Workflow. Training phase: the noise posterior drives a classical simulator to produce paired data $\{(\mathbf{x}_{i}^{\text{dt}},\mathbf{x}_{i}^{\text{exact}})\}$ for training a denoising network (black arrows), with optional fine-tuning on real data (gray arrows). Deployment phase: the pre-trained network maps noisy outputs $\mathbf{x}^{\text{noisy}}$ to mitigated estimates $\mathbf{x}^{\text{mit}}$ (red arrows). (b, c) Application to Trotterized dynamics of 12-qubit Ising chains, comparing ideal, raw noisy, linear and quadratic ZNE (gate folding at noise gains $G\in\{1,3,5\}$ ; see Appendix F.2 ), and SBI-DT. (b) Time-evolved transverse magnetization $M_{x}$ (left) and nearest-neighbor correlations $C_{ZZ}$ (right), averaged over 50 test coupling strengths $j\sim\mathcal{U}[-\pi,0]$ . The vertical dotted line at $t=1.4$ separates the training and extrapolation regions. (c) Time-resolved $\mathrm{MAE}(t)=\frac{1}{|\mathcal{S}|}\sum_{O\in\mathcal{S}}|\langle O\rangle(t)-\langle O\rangle_{\text{exact}}(t)|$ across all observables $\mathcal{S}=\{X_{i}\}_{i}\cup\{Z_{i}Z_{i+1}\}_{i}$ , with shaded $\pm 1$ standard deviation over test $j$ (left), and MAE aggregated over all $j$ and $t$ , with improvement factors relative to the noisy baseline (right).

Figure 6: Quantum state tomography with NPE. (a) Workflow, illustrated using a PQC ansatz and an MPS simulator. Randomly sampled circuit parameters define states whose observables are evaluated by MPS contraction to train a neural posterior estimator. The trained estimator infers posterior distributions over circuit parameters from measurement data of new experimental instances, and posterior samples are mapped through the ansatz to reconstructed MPSs. (b) Reconstruction infidelity of simulated 12-qubit pure states versus simulation budget: $1-F_{\mathrm{median}}$ (blue) and $1-F_{\mathrm{max}}$ (red), computed at each budget from 500 posterior samples for each of 200 test states ( $10^{5}$ samples in total). (c) Probability density of the fidelity $F$ at the largest budget ( $9\times 10^{5}$ ), excluding outliers beyond $1.5\times$ the interquartile range. Dashed lines mark the median (blue) and maximum (red) over all samples.

Figure 7: Posterior localization of atomic displacements in simulated $9\times 9$ Rydberg arrays. (a) Posterior distribution of in-plane displacements, $\Delta x$ and $\Delta y$ (in nm), for individual atoms within a single array instance. Each square represents a single array site, with filled contours indicating confidence intervals (color bar). The ground-truth displacements (red stars), posterior means (green circles), and posterior medians (orange diamonds) are indicated. An enlarged $3\times 3$ view highlights a representative patch. (b) Amortized posterior precision evaluated across 100 independent displacements. Top, parameter-wise regression for individual displacement components; vertical bars denote the posterior standard deviation. Bottom, probability density of the parameter vector error norms, $||\bm{\theta}-\bm{\theta}_{\text{true}}||_{2}$ , contrasting the broad prior (grey) with the localized posterior (blue).

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Neural spline flow architecture. (a) A base sample $\mathbf{z}$ is transformed through $T$ autoregressive spline layers into a posterior parameter sample $\bm{\theta}^{(T)}$ , with each layer conditioned on the context $\mathbf{c}$ derived from the observation $\mathbf{x}$ . (b) Structure of an autoregressive spline layer, illustrated for three coordinates. A masked autoregressive MLP generates the spline parameters $\bm{\psi}_{i}$ from the preceding coordinates $\bm{\theta}_{<i}^{(t-1)}$ and the context $\mathbf{c}$ . The scalar transforms $g_{i}$ then map each input coordinate to $\theta_{i}^{(t)}$ . (c) A monotonic rational-quadratic spline illustrated with $K+1=9$ knots, specified by bin widths $w^{(k)}$ , heights $h^{(k)}$ , and knot derivatives $d^{(k)}$ , with identity tails outside $[-B,B]$ .

Figure 9: Benchmarking the simulator for Pauli noise learning. (a) Mean absolute error (MAE) between our method and Qiskit Aer density-matrix simulator over all

k=12n-9

Pauli observables (system size

n=2

–

14

). (b) Wall time comparison for both methods over the same setting as in panel (a). (c) Wall time of our method for a single observable

\langle Z_{1}Z_{2}\rangle

as a function of

n=10

–

1000

, with an

\mathcal{O}(n)

reference. (d) Total wall time for all

k

observables, with an

\mathcal{O}(n^{2})

reference. Reference curves are initialized at the leftmost data point. All quantities are averaged over five independent random seeds (Pauli rotation angles and Pauli noise coefficients).

Figure 10: Benchmarking the simulator for Rydberg Hamiltonian learning. (a) Absolute error between the weight-truncated Pauli propagation and the Qiskit Aer statevector simulator for the observable

\langle Z_{1}Z_{2}\rangle

, shown for truncation orders

w\leq 1

–

5

2\times 2

3\times 3

, and

4\times 4

lattices (

n=4,9,16

). (b) Wall time comparison between the statevector reference and Pauli propagation over the same setting as in panel (a). (c) Estimated

\langle Z_{1}Z_{2}\rangle

as a function of system size up to the

12\times 12

lattice (

n=144

), across the same truncation orders. (d) Wall time for the same scaling sweep as in panel (c). All quantities are averaged over twenty random displacement samples.

Figure 11: Posterior Pauli noise distributions of a five-qubit circuit with brickwall connectivity. Layer one consists of

\text{CNOT}_{1\to 2}

and

\text{CNOT}_{3\to 4}

; layer two consists of

\text{CNOT}_{2\to 3}

and

\text{CNOT}_{4\to 5}

Figure 12: Parameter-wise regression for 50-qubit Pauli noise learning. Posterior means versus ground-truth values for all 735 parameters (left) and the 447-parameter gauge-free subset (right). Vertical bars denote

\pm 1

posterior standard deviation among 2,000 samples.

Figure 13: Complete posterior distributions for 50-qubit Pauli noise learning. Posterior samples (box plots) and ground-truth values (red crosses) for all 49 CNOT gates in the brickwall circuit. For each box, the central line marks the median, the box spans the interquartile range (25th–75th percentiles), and the whiskers extend to

1.5\times

the interquartile range; outliers beyond the whiskers are omitted. Each box summarizes 2,000 posterior samples. Gates are grouped by sublayer: odd-control (top five rows) and even-control (bottom five rows).

Figure 14: Overall MAE of the NPE-based SBI-DT, shown in direct comparison with the results in Fig. 5 . The plot contrasts this variant against the noisy baseline, quadratic ZNE, and the regression-based SBI-DT. Improvement factors are relative to the noisy baseline.

Figure 15: Amortized quantum state tomography with the Cholesky parameterization. (a) Heatmap of the median reconstruction infidelity over 2,000 posterior samples, where each cell $(\theta_{x},\theta_{y})$ represents a specific $4$ -qubit pure state. (b) Posterior fidelity distribution for a representative state prepared at $(\theta_{x},\theta_{y})=(\pi/3,\pi/3)$ . (c) Real (top) and imaginary (bottom) components of the reconstructed density matrix at the median fidelity in panel (b).

	Pauli noise	QST	Rydberg	SBI-DT
Flow transforms	3
Hidden features	2048
Spline bins	16 per dimension
Learning rate	$10^{-4}$
Early stopping	100 epochs without improvement
Standardize $\mathbf{x}$	yes	yes	yes	no

Appendix

Table 1: Neural spline flow architecture and training hyperparameters. Upper block: settings shared across all models; lower block: model-specific.

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Abstract

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