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 $FF^\dagger$, where the factor $F$ 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.
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†, where the factor F 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.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
Quantum state tomography (QST) has attracted considerable attention due to its fundamental role in quantum information processing. In this paper, we develop a unified theoretical framework for analyzing the sample complexity of structured QST under noisy observations arising from state preparation noise, measurement noise, and finite-shot statistical noise. The proposed framework applies to a broad family of structured quantum-state classes, including general mixed states, sparse states, low-rank states, matrix product states (MPSs), matrix product operators (MPOs), projected entangled-pair states (PEPSs), and projected entangled-pair operators (PEPOs), while further introducing physically consistent structured models---including low-rank and sparse states, low-rank MPOs (LR-MPOs), and low-rank PEPOs (LR-PEPOs)---that simultaneously exploit low-dimensional structures and preserve the physical constraint. Within this framework, we derive unified non-asymptotic sample complexity guarantees for two constrained least-squares estimators under noisy observations: a noise-aware estimator that incorporates the calibrated noise model and a noise-unaware estimator based on the ideal Born measurement model. For the noise-aware estimator, we derive unified trace-norm recovery guarantees that explicitly characterize the dependence of the sample complexity on three fundamental quantities: the complexity of the underlying structured state class, the complexity of the measurement ensemble, and the state preparation and measurement noise levels. For the noise-unaware estimator, we establish a unified non-asymptotic recovery guarantee consisting of a statistical error term and an additional deterministic bias term arising from the mismatch between the assumed reconstruction model and the noisy observation process.
Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.
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 .
Matrix product operators (MPOs) provide a scalable approach for quantum state tomography (QST) by offering a compact representation of many-body mixed states with limited entanglement, using only a number of parameters that scales polynomially with the system size. In this paper, we study QST for quantum density matrices that can be represented by MPOs. We first derive an equivalent characterization of Hermiticity in terms of the MPO core tensors and show that the coefficient tensor of an MPO under the Pauli or generalized Gell-Mann basis admits a real-valued low tensor-train (TT) rank structure. This establishes an explicit connection between MPO-based QST and noisy low-rank tensor completion. Motivated by this formulation, we develop an online Riemannian gradient descent (oRGD) algorithm that sequentially incorporates measurement data during the reconstruction process. With a proper initialization, we prove that oRGD converges linearly to the target MPO and succeeds with a number of distinct measurement settings that scales quadratically with the system size. As a byproduct, our analysis also yields a significantly improved sample complexity bound for the low TT rank tensor completion task. Furthermore, we propose a tailored spectral initialization method and establish its theoretical guarantee. Numerical experiments on several classes of quantum states validate the effectiveness and scalability of the proposed method.
Existing quantum state tomography methods are limited in scalability due to their high computation and memory demands, making them impractical for the recovery of large quantum states. In this work, we address these limitations by reformulating the maximum likelihood estimation problem using the Burer--Monteiro factorization, resulting in a nonconvex but low-rank parametrization of the density matrix. We derive a fully unconstrained formulation by analytically eliminating the trace-one and positive-semidefinite constraints, thereby avoiding the need for projection steps during optimization. Furthermore, we determine the Lagrange multiplier associated with the unit-trace constraint a priori, reducing computational overhead. The resulting formulation is amenable to scalable first-order optimization, and we demonstrate its tractability using the limited-memory Broyden-Fletcher-Goldfarb-Shanno method. Importantly, we also propose a low-memory version of the above algorithm to fully recover certain large quantum states with Pauli-based positive-operator-value measurements. Our low-memory algorithm avoids explicitly forming any density matrix, and does not require the density matrix to have a matrix product state or other tensor structure. For a fixed number of measurements and fixed rank, our algorithm requires just O(dlog2d) complexity per iteration to recover a $d\ifmmode\times\else\texttimes\fi{}d$ density matrix. Additionally, we derive a useful error bound that can be used to give a rigorous termination criterion. We numerically demonstrate that our method is competitive with state-of-the-art algorithms for moderately sized problems, and then demonstrate that our method can solve a 20-qubit problem on a laptop in under 5 hours.