Learning to Reconstruct Wigner Functions in Phase Space

Abstract: Wigner function learning is a central tool for characterizing continuous variable quantum systems. A fundamental challenge in this setting is to infer a continuous phase-space function from sparse pointwise measurement data, a task that becomes increasingly demanding as the effective dimension enlarges. Here, we develop a general machine learning framework to reconstruct Wigner functions directly as continuous functions from sparse phase-space data. For states with sparse Fock-space or coherent-state representations, such as binomial code states and cat states, we devise provably efficient regression models whose measurement complexity scales only logarithmically with the effective Hilbert-space dimension. For more general states, such as the Gottesman-Kitaev-Preskill (GKP) states, we design a deep learning model that reconstructs the Wigner function from sparse measurements and generalizes to arbitrary phase-space resolution. We demonstrate the broad applicability of our framework on both simulated data and experimental data from a circuit quantum electrodynamic (circuit-QED) system. Interestingly, on experimental data, we find that our model reconstructs Wigner functions of GKP code states across multiple rounds of quantum error correction and identifies the dominant error process using significantly fewer measurements than conventional estimation techniques.

• The paper presents efficient ML methods to reconstruct Wigner functions, significantly reducing measurement complexity for continuous-variable quantum states.
• It details two primary approaches: structure-aware sparse regression and deep neural network surrogates, each tailored to different state characteristics.
• Empirical and experimental validations demonstrate high fidelity (≈0.99) reconstructions with dramatically fewer measurements, advancing quantum error correction and state diagnosis.

Efficient Data-Driven Wigner Function Reconstruction: Theory, Models, and Experimental Validation

Introduction

Accurate characterization of continuous-variable (CV) quantum states is a central requirement for quantum information processing in bosonic systems. The Wigner function, as a quasi-probability distribution in phase space, is a fundamental object for such characterization but suffers from extreme sampling complexity as the effective Hilbert space dimension increases, particularly for nonclassical states or in high-fidelity error correction settings. Conventional Wigner tomography often requires $d^2$ pointwise measurements for a $d$-dimensional truncation, rapidly becoming prohibitive. The paper "Learning to Reconstruct Wigner Functions in Phase Space" (2607.06232) introduces a machine learning framework for efficient Wigner function reconstruction from sparse measurement data, targeting both theoretically structured states and experimentally relevant settings.

Formalization and Learning Protocol

The authors formalize Wigner function reconstruction as a supervised learning problem: given sparse samples $\{(\alpha_i, y_i)\}$, where $y_i$ is the measured Wigner function at phase-space coordinate $\alpha_i$, train a surrogate model $\widehat{W}_{\bm\eta}$ to predict $W_\rho(\alpha)$ for arbitrary $\alpha$. The approach divides into two primary regimes: structure-aware regression with provable sample complexity for states with sparse basis support, and deep neural networks (DNNs) for more general or experimentally realized states.

Figure 1: Architecture overview—the regression model exploits explicit feature mappings, while the DNN improves the continuous prediction by an encoder-decoder pipeline over sparse input data.

Structure-Aware Sparse Regression for Fock- and Coherent-State Supported States

For states with a small support in either the Fock basis (e.g., binomial code states, Fock superpositions) or sparse coherent-state expansions (e.g., cat states), the authors leverage the linearity properties of Wigner functions and construct regression models whose sample and computational complexity depend on the "intrinsic" sparsity, not the truncation dimension.

Fock-Sparse Regression

Given a state $\rho = \sum_{mn} \rho_{mn} \ket{m}\bra{n}$ with at most $s^2$ nonzero coefficients, the model uses a Laguerre (Fock-state) feature map $d$0 and solves a Lasso-regularized regression problem. The authors prove that for fixed sparsity $d$1, the number of training samples to achieve prediction error $d$2 scales as $d$3, exponentially improving over the $d$4 grid-based tomography requirement.

Coherent-Sparse Regression (Gabor Frame Model)

For sparse superpositions of coherent states, the approach is to construct a finite Gabor frame as the feature set—overcomplete Gaussian envelopes modulated in phase space—which provides exponential tail decay for the Wigner function features. The sample complexity in this regime is bounded by $d$5 for support radius $d$6 and minimum feature separation $d$7, giving logarithmic scaling in phase-space region size.

Figure 2: Examples of CV state families for which efficient Wigner-function learning is provably tractable with the presented regression models.

Deep Neural Implicit Modeling Beyond Sparsity: DNN Surrogates for Arbitrary State Classes

For general CV states with smooth but non-sparse Wigner functions (e.g., practical GKP states, SNAP-circuit outputs, experimentally measured data), the authors propose a coordinate-conditioned DNN architecture inspired by super-resolution in computer vision. The model uses a convolutional encoder to synthesize a latent phase-space feature map from sparse measurement data and a small MLP decoder to regress the continuous Wigner function value at arbitrary queries, interpolating the latent representation for any coordinate.

This DNN-based surrogate is resolution-independent, handles arbitrary sampling patterns, and is robust to significant experimental noise. Empirical results demonstrate that the network reconstructs highly non-trivial Wigner patterns (GKP grids, SNAP-generated states) with less than one-ninth the measurement cost of conventional tomography.

Figure 3: The DNN learning pipeline leverages self-supervision to bridge low-resolution input and high-resolution Wigner function recovery.

Numerical and Experimental Results

Sample Efficiency and Scaling

Systematic simulation demonstrates the strong dependence of sample complexity on state structure:

• Binomial code states: The number of phase-space measurements required to achieve fidelity $d$8 remains nearly constant as the Fock space truncation $d$9 grows for fixed Fock-support size, confirming the predicted $\{(\alpha_i, y_i)\}$0 scaling.

  Figure 4: Data-efficiency evaluation for various binomial code parameterizations, showing orders-of-magnitude fewer measurements than required for informational completeness.

• Cat states: The regression model on Gabor-frame features reconstructs Wigner functions with high fidelity well below the $\{(\alpha_i, y_i)\}$1 threshold.

  Figure 5: Reconstruction examples for cat states—the model robustly captures both coherent peaks and quantum interference regions.

• GKP states & general non-sparse states: DNN models outperform regression-based surrogates as structure becomes less sparse and patterns smooth/compressible. For simulated and experimental GKP states, scaling of data requirements is substantially subquadratic in $\{(\alpha_i, y_i)\}$2. Classical runtime is also reduced compared to maximum-likelihood estimation (MLE) approaches, and the DNN remains robust under added photon-loss noise.

  Figure 6: DNN sample- and noise-robustness for GKP states versus conventional MLE, highlighting significant speed and reliability advantages.

• Random circuit outputs & Wigner negativity: As DNNs see states prepared by deeper SNAP/displacement circuits, reconstruction fidelity degrades smoothly as a function of Wigner negativity, providing a practical measure of the model's expressiveness for highly nonclassical scenes. For shallow circuits, fidelity is near unity; for deep/high-negativity states, it decays but remains practical.

  Figure 7: DNN fidelity versus circuit depth (left) and Wigner negativity (right); example reconstructions (bottom) show close agreement even for highly structured, random states.

Experimental Validation: GKP QEC and Error Diagnosis

The DNN model is applied to state-of-the-art circuit QED data (beyond-break-even GKP error-correction experiments), consuming only a sparse fraction of the real measurement grid (e.g., $\{(\alpha_i, y_i)\}$3 out of $\{(\alpha_i, y_i)\}$4 samples). The DNN:

• Recovers high-resolution Wigner functions with fidelity $\{(\alpha_i, y_i)\}$5 to dense experimental grids
• Outperforms interpolation and theoretical regression surrogates in noise
• Allows post-hoc extraction of logical and error subspaces (via density matrix estimation from DNN Wigner predictions), enabling physical diagnosis of error processes (single-photon addition as the principal error channel is robustly identified from sparse/noisy data)

  Figure 8: DNN-based Wigner function recovery from sparse experimental data; analysis of logical and error subspaces tracked through multiple rounds of QEC. Overlap and purity metrics confirm accurate codeword identification and resource tracking.

  

  Figure 9: Fidelity and purity of reconstructed GKP code states across QEC cycles, comparing DNN and full-data analyses; the DNN yields consistent optimal squeezing and code-space fidelity values.

Theoretical and Practical Implications

The results establish an explicit separation between dense-tomography and learning-based reconstruction. For practical CV quantum information experiments, especially as high-photon-number codes or error correction become routine, these approaches can reduce required measurements by orders of magnitude, notably for code diagnosis, benchmarking, and error tracking. The regression models constitute the first provably efficient surrogates for CV Wigner characterization under realistic measurement models, extending prior (qubit-focused) quantum learning theory into the CV regime in both Fock and coherent-state bases.

The neural DNN surrogates demonstrate that "physics-agnostic" AI can profitably recover complex quantum patterns where structure-aware regression is either inapplicable or brittle to noise, giving an experimentally robust, computationally scalable solution—crucial for ongoing developments in bosonic QEC, analog quantum simulation, and hybrid systems.

Outlook

Several lines of extension are suggested:

• Multi-mode generalization: Scaling these models to joint Wigner functions in higher-dimensional (multi-mode) phase spaces is a natural next target.
• Active/Adaptive sampling: Optimizing phase-space measurement grids (e.g., via information-theoretic or uncertainty reduction objectives) combined with learning surrogates may yield further efficiency improvements.
• Surrogates for additional quantum properties: Beyond Wigner functions, properties such as entropic quantities, nonclassicality indices, or process tomography objects can potentially be learned from sparse functional data with appropriate architectural modifications.
• Hybrid models and transfer learning: Combining explicit regression basis sets and DNN surrogates may further reduce measurement costs, particularly in state-preparation verification and hardware calibration settings.

Conclusion

This work delivers both theoretical and algorithmic advances for resource-efficient Wigner function reconstruction in CV quantum systems. Structure-aware regression models provide provable guarantees for sparse-support states, while DNN-based surrogates extend to general, experimentally relevant regimes, demonstrating significant reductions in quantum measurement and classical postprocessing requirements. The results have direct implications for the near-term scalability and fidelity of CV quantum computation, error correction, and quantum system benchmarking, and motivate future research in multi-mode and property-specific quantum learning surrogates.