Quantum state tomography estimates an unknown density operator from measurement data. Dense reconstruction however, can be impractical for many-qubit systems because the Hilbert-space dimension grows exponentially. This contribution develops a rank-adaptive matrix-free approach to low-rank quantum state tomography based on rank-one atomic coordinates. The density operator is represented as a convex combination of pure-state atoms, which preserves positivity and unit trace while avoiding dense density, measurement, and gradient matrices. The resultant algorithm combines atom updates, simplex-constrained coefficient reweighting, and periodic spectral refactorization using only predicted probability vectors, atom vectors, and descriptor-level measurement actions. These computations decompose across measurement outcomes, and admit a master--worker implementation with the measurements partitioned across workers. A rank penalty adapts the representation size during optimization. Provable feasibility, prediction consistency, monotone descent of the penalized objective, and an exact spectral proximal characterization of the refactorization step are established. Simulations with Pauli measurements show favorable accuracy--runtime--memory tradeoffs relative to competing alternatives.
Quantum state tomography estimates an unknown density operator from measurement data. Dense reconstruction however, can be impractical for many-qubit systems because the Hilbert-space dimension grows exponentially. This contribution develops a rank-adaptive matrix-free approach to low-rank quantum state tomography based on rank-one atomic coordinates. The density operator is represented as a convex combination of pure-state atoms, which preserves positivity and unit trace while avoiding dense density, measurement, and gradient matrices. The resultant algorithm combines atom updates, simplex-constrained coefficient reweighting, and periodic spectral refactorization using only predicted probability vectors, atom vectors, and descriptor-level measurement actions. These computations decompose across measurement outcomes, and admit a master--worker implementation with the measurements partitioned across workers. A rank penalty adapts the representation size during optimization. Provable feasibility, prediction consistency, monotone descent of the penalized objective, and an exact spectral proximal characterization of the refactorization step are established. Simulations with Pauli measurements show favorable accuracy--runtime--memory tradeoffs relative to competing alternatives.
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.
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.
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.
Reconstruction of the quantum density operator ρ from finite projective measurement statistics is a foundational requirement for quantum characterization, verification, and benchmarking (QCVB) in modern quantum processors. However, standard linear inversion systematically fails non-negativity (λmin < 0) in the presence of statistical shot noise, yielding unphysical indefinite operators that violate the Born rule and predict negative probabilities. Conversely, standard iterative Maximum Likelihood Estimation (MLE) via Hradil's RρR map or Semidefinite Programming (SDP) incurs high computational overhead and exhibits sensitivity to rank deficiencies, creating severe latency bottlenecks in automated quantum processing unit (QPU) calibration sweeps. Here, we present IdemQST, a deterministic quantum state tomography framework governed by idempotent projection operator theory (Π2 = Π). In an informationally complete orthogonal Pauli basis, sequential projection across measurement hyperplanes collapses in exactly one pass to the unconstrained linear inversion estimate ρlin, and subsequent Euclidean simplex spectral projection ΠΔ yields the exact global Frobenius-closest physical state on the quantum state cone Sd, establishing mathematical equivalence to Projected Linear Inversion (PLI; Smolin, Gambetta, and Smith, 2012). Rather than asserting novel algorithmic optimality for the simplex step, IdemQST re-articulates this foundation within metric idempotent operator algebra, integrates zero-heap static workspace caching (IdemKV), provides empirical dynamic drift cancellation via Dynamic Permutation Networks (DPN), and extends to incomplete or shadow tomography (M < 4N - 1) via cyclic Projections Onto Convex Sets (POCS). Across simulated projective Born measurements on one-, two-, and three-qubit systems under realistic Poisson shot noise (Nshots ∈ [102, 2 × 104]), IdemQST unconditionally eliminates negative eigenvalues (λmin ≥ 0), conserves unit trace to machine precision (10-15), and achieves state fidelities F > 0.994 under pure prior projection or large shot budgets. Crucially, across N=30 repeated timing benchmarks, IdemQST executes the core simplex spectral projection in 96.7 ± 44.3 μs for two-qubit Bell states (39.3× speedup over Hradil MLE and 41.1× over Diluted MLE) and 0.145 ± 0.035 ms (145.0 μs) for three-qubit GHZ states (d=8), achieving a 111.0× speedup over iterative Hradil MLE (16.08 ± 2.17 ms) and 113.6× over accelerated Diluted MLE (16.46 ± 1.58 ms; Šeháček et al., 2007). Furthermore, we incorporate State Preparation and Measurement (SPAM) readout mitigation via simplex-constrained assignment inversion, providing a deterministic, hardware-ready platform for real-time automated QPU calibration.
We introduce a method to reconstruct the density matrix $\ensuremath{\rho}$ of a system of n qubits and estimate its rank d from data obtained by quantum-state-tomography measurements repeated m times. The procedure consists of minimizing the risk of a linear estimator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ of $\ensuremath{\rho}$ penalized by a given rank (from 1 to 2n), where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of the penalized estimator, evaluated with the Frobenius norm, which is of order dn(4/3)n/m and consistent for the estimator of the rank. The proposed methodology is computationally efficient and is illustrated with some example states and real experimental data sets.
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 .