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arXiv:2607.21584

Flow-based Phase-space Tomography of Continuous-variable Quantum States

Owen Dugan, Rumen Dangovski, Peter Y. Lu, Di Luo

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

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-$Q$ 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.

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Quantum state tomography (QST) is a fundamental technique in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from experimental measurements. However, traditional QST methods often suffer from significant computational demands and are highly susceptible to measurement noise, hindering their practical application, particularly with near-term quantum computers. This work proposes a novel approach to QST utilizing variational quantum algorithms (VQAs). VQAs iteratively optimize a parameterized quantum circuit to minimize a cost function related to the fidelity between the desired state and the state predicted by the circuit. This approach dramatically reduces the number of required measurements and offers the potential for achieving high-fidelity state reconstruction with lower quantum resource requirements. We detail the theoretical framework, outlining the core algorithms and mathematical formulations. The key advantage lies in the ability to learn the state directly, mitigating the limitations of traditional methods. We explore the optimization landscape and discuss strategies for improving convergence and robustness. The results demonstrate a promising path toward scalable and robust QST on quantum devices.
Quantum state tomography (QST) is a crucial technique for characterizing quantum states, vital for verifying quantum device performance and implementing quantum information processing. However, traditional QST methods are notoriously computationally demanding, requiring extensive experimental data acquisition and subsequent processing, often leading to high resource consumption. This work presents a novel framework for QST utilizing probabilistic programming. The core idea is to model the quantum state as a probability distribution and employ Bayesian inference, coupled with variational methods, to efficiently infer the density matrix from limited measurement data. We demonstrate the efficacy of this approach, showcasing its potential to significantly reduce experimental costs and improve the accuracy of QST. The framework's automation and optimization capabilities represent a substantial advancement in the field, paving the way for more accessible and efficient quantum state characterization.
Abstract Quantum state tomography (QST) is a widely employed technique for characterizing the state of a quantum system. However, it is plagued by two fundamental challenges: computational and experimental complexity grows exponentially with the number of qubits, rendering experimental implementation and data post-processing arduous even for moderately sized systems. Here, we introduce gradient-descent (GD) algorithms for the post-processing step of QST in discrete- and continuous-variable systems. To ensure physically valid state reconstruction at each iteration step of the algorithm, we use various density-matrix parameterizations: Cholesky decomposition, Stiefel manifold, and projective normalization. These parameterizations have the added benefit of enabling a rank-controlled ansatz, which simplifies reconstruction when there is prior information about the system. We benchmark the performance of our GD-QST techniques against state-of-the-art methods, including constrained convex optimization, conditional generative adversarial networks, and iterative maximum likelihood estimation. Our comparison focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. We find that rank-controlled ansatzes in our stochastic mini-batch GD-QST algorithms effectively handle noisy and incomplete data sets, yielding significantly higher reconstruction fidelity than other methods. Simulations achieving full-rank seven-qubit QST in under three minutes on a standard laptop, with 18 GB of RAM and no dedicated GPU, highlight that GD-QST is computationally more efficient and outperforms other techniques in most scenarios, offering a promising avenue for characterizing noisy intermediate-scale quantum devices. Our Python code for GD-QST algorithms is publicly available at github.com/mstorresh/GD-QST .
Since the complexity of quantum state tomography (QST) scales exponentially with system size, exploiting priors such as low-rankness, tensor-network structures, and neural-network representations is essential for scalable QST in terms of sample complexity and parameter complexity. In this paper, we introduce a unified framework, termed structured factorization, that builds on BurerMonteiro-type factorization by parametrizing the density matrix as FF†FF^\daggerFF†, where the factor FFF is constrained to belong to a structured model class. This factorization guarantees physical validity by construction while allowing a broad range of structural priors to be incorporated directly through the choice of the factor space, ranging from the generic Cholesky decomposition to low-rank matrices, matrix product operators, and neural density operators based on multilayer perceptron and transformer architectures. Building on this structured factorization framework, we formulate QST as an optimization problem over the factor space from measurement data. We first develop a unified statistical analysis of the sample complexity of least-squares estimation for a broad class of structured quantum states. We then propose a projected gradient descent method that operates directly on the factor space and accommodates a wide range of structural parametrizations and reconstruction objectives. To further exploit the geometry of the maximum-likelihood estimation formulation and the constraints on the factors, we derive a power method that yields a step-size-free algorithm with fast convergence, recovering Covers method as a special case when the factor is unconstrained.