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arXiv:2508.06431 · Physical Review Letters

Nonparametric Learning Non-Gaussian Quantum States of Continuous Variable Systems

Liubov A. Markovich, Xiaoyu Liu, Jordi Tura

Published Jun 9, 2026• 0 citations
arXivHTMLPDF

Abstract

Continuous-variable quantum systems are foundational to quantum computation, communication, and sensing. While traditional representations using wave functions or density matrices are often impractical, the tomographic picture of quantum mechanics provides an accessible alternative by associating quantum states with classical probability distribution functions called tomograms. Despite its advantages, including compatibility with classical statistical methods, the tomographic method remains underutilized due to a lack of robust estimation techniques. This Letter addresses this gap by introducing a nonparametric kernel quantum state estimation (KQSE) framework for reconstructing quantum states and their trace characteristics from noisy data, without prior knowledge of the state. In contrast to existing methods, KQSE yields estimates of the density matrix in various bases, as well as trace quantities such as purity, higher moments, overlap, and trace distance, with a near-optimal convergence rate of O[over ˜](T^{-1}), where T is the total number of measurements. KQSE is robust for multimodal, non-Gaussian states, making it particularly well suited for characterizing states essential for quantum science.

DOI
10.1103/xdcg-6df5

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Continuous-variable quantum systems are foundational to quantum computation, communication, and sensing. While traditional representations using wave functions or density matrices are often impractical, the tomographic picture of quantum mechanics provides an accessible alternative by associating quantum states with classical probability distribution functions called tomograms. Despite its advantages, including compatibility with classical statistical methods, the tomographic method remains underutilized due to a lack of robust estimation techniques. This Letter addresses this gap by introducing a nonparametric kernel quantum state estimation (KQSE) framework for reconstructing quantum states and their trace characteristics from noisy data, without prior knowledge of the state. In contrast to existing methods, KQSE yields estimates of the density matrix in various bases, as well as trace quantities such as purity, higher moments, overlap, and trace distance, with a near-optimal convergence rate of Oover ˜, where T is the total number of measurements. KQSE is robust for multimodal, non-Gaussian states, making it particularly well suited for characterizing states essential for quantum science.
We further develop Kernel Quantum State Estimation (KQSE), a fully data-driven nonparametric method for continuous variable quantum state reconstruction and characterization, introduced in our recent work and rooted in nonparametric kernel density estimation (KDE). Unlike approaches relying on finite-dimensional basis truncations, parametric ansätze, or prior models, KQSE combines a new kernel estimator of the characteristic function of the tomographic quadrature distribution with suitable kernel integral transformations. The characteristic function is estimated directly from experimental homodyne or heterodyne data and subsequently transformed to estimate quantum state representations and characteristics, including the Wigner function and the density matrix kernel. Observing that several other physically relevant representations and characteristics admit transformations of a closely related form, we derive convergence rates for the corresponding broad class of KQSE-based estimators. The resulting framework covers all phase-space representations, the photon-number tomogram, trace products of quantum states, and purity. We derive mean squared error convergence rates for these kernel transformed estimators and show that the corresponding KQSE applications inherit the near optimal rate O(T^{-1}), where TTT is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate the advantages of KQSE over state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.
Abstract Quantum state estimation is a central task in quantum information theory, aiming to reconstruct an unknown density matrix from a finite number of measurement outcomes. Conventional approaches such as kernel smoothing or maximum likelihood estimation (MLE) often rely on restrictive parametric assumptions and become computationally prohibitive for high-dimensional quantum systems. In this work, we introduce a nonparametric quantum state estimation framework based on the k-Nearest Neighbor (k-NN) approach. The method leverages the local geometric structure of measurement data within the quantum feature space to infer the underlying quantum state without any predefined model. Using intrinsic quantum distance metrics—such as the trace distance and quantum fidelity—the proposed estimator identifies neighboring states and performs a weighted aggregation, followed by a positive-semidefinite projection to ensure physical admissibility. We establish the consistency and asymptotic convergence of the NN-based estimator under mild regularity conditions. Numerical simulations on single- and two-qubit systems confirm that the proposed method achieves high reconstruction accuracy, robustness to measurement noise, and superior computational efficiency compared to kernel- and likelihood-based estimators. Overall, this study highlights the potential of data-driven, geometry-aware, nonparametric learning for scalable quantum state reconstruction, providing a promising alternative for modern quantum information processing.
This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".
Quantum state tomography, aimed at deriving a classical description of an unknown state from measurement data, is a fundamental task in quantum physics. In this work, we analyse the ultimate achievable performance of tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems: not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but it also exhibits a dramatic scaling with the trace-distance error, even for low-energy states, in stark contrast with the finite-dimensional case. On a more positive note, we prove that tomography of Gaussian states is efficient. To accomplish this, we answer a fundamental question for the field of continuous-variable quantum information: if we know with a certain error the first and second moments of an unknown Gaussian state, what is the resulting trace-distance error that we make on the state? Lastly, we demonstrate that tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.
Continuous-variable quantum state tomography is limited by the cost of resolving non-Gaussian structure in high-dimensional phase space. We introduce QST-Flow, a quantum state tomography framework via flow-based generative modeling that represents experimentally accessible phase-space quasiprobability distributions with normalized, samplable neural densities rather than a truncated density matrix. The framework has two variants: QST-QFlow models the positive Husimi-QQQ function with a single normalizing flow, while QST-WFlow models sign-changing Wigner functions as a trainable difference of two normalized flows. This construction preserves quasiprobability normalization and enables exact density evaluation, direct sampling, and importance-sampled learning from finite phase-space measurements without a fixed grid. Benchmarks on non-Gaussian cat, binomial, Gottesman-Kitaev-Preskill, number, and Fock states show accurate single-mode reconstructions, extension to multimode states, robustness on noisy Wigner data, and improved reconstruction error compared with prior machine-learning tomography methods. QST-Flow opens a promising route toward scalable, measurement-efficient phase-space tomography of nonclassical bosonic systems.
Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.