Quantum superoperator learning is a pivotal task in quantum information science, enabling accurate reconstruction of unknown quantum operations from measurement data. We propose a robust approach based on the matrix sensing techniques for quantum superoperator learning that extends beyond the positive semidefinite case, encompassing both quantum channels and Lindbladians. We first introduce a randomized measurement design using a near-optimal number of measurements. By leveraging the restricted isometry property (RIP), we provide theoretical guarantees for the identifiability and recovery of low-rank superoperators in the presence of noise. Additionally, we propose a blockwise measurement design that restricts the tomography to the sub-blocks, significantly enhancing performance while maintaining a comparable scale of measurements. We also provide a performance guarantee for this setup. Our approach employs alternating least squares (ALS) with acceleration for optimization in matrix sensing. Numerical experiments validate the efficiency and scalability of the proposed methods.
Quantum superoperator learning is a pivotal task in quantum information science, enabling accurate reconstruction of unknown quantum operations from measurement data. We propose a robust approach based on the matrix sensing techniques for quantum superoperator learning that extends beyond the positive semidefinite case, encompassing both quantum channels and Lindbladians. We first introduce a randomized measurement design using a near-optimal number of measurements. By leveraging the restricted isometry property (RIP), we provide theoretical guarantees for the identifiability and recovery of low-rank superoperators in the presence of noise. Additionally, we propose a blockwise measurement design that restricts the tomography to the sub-blocks, significantly enhancing performance while maintaining a comparable scale of measurements. We also provide a performance guarantee for this setup. Our approach employs alternating least squares (ALS) with acceleration for optimization in matrix sensing. Numerical experiments validate the efficiency and scalability of the proposed methods.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2 -norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
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.
Compressed sensing (CS) is a technique to faithfully estimate an unknown signal from relatively few data points when the measurement samples satisfy a restricted isometry property (RIP). Recently, this technique has been ported to quantum information science to perform tomography with a substantially reduced number of measurement settings. In this work we show that the constraint that a physical density matrix is positive semidefinite provides a rigorous connection between the RIP and the informational completeness of a POVM used for state tomography. This enables us to construct informationally complete measurements that are robust to noise using tools provided by the CS methodology. The exact recovery no longer hinges on a particular convex optimization program; solving any optimization, constrained to the cone of positive semidefinite matrices, effectively results in a CS estimation of the state. From a practical point of view, we can therefore employ fast algorithms developed to handle large dimensional matrices for efficient tomography of quantum states of a large dimensional Hilbert space.
Scalable characterization of quantum processors is crucial for mitigating noise and imperfections. While randomized measurement protocols enable efficient access to local observables, inferring a globally consistent description of multi-qubit processes remains challenging. Here we introduce a local-to-global reconstruction framework for one-dimensional multi-qubit states and channels. The method is efficient provided that correlations, as quantified by the conditional mutual information, decay exponentially. In particular, we prove that under this assumption, the required number of samples scales polynomially with the system size and the desired global reconstruction error. Our approach is based on combining local shadow tomography with locally optimal recovery maps obtained by convex optimization. We supplement these rigorous guarantees by studying the performance of the protocol numerically for a system evolving under a local Lindbladian and a noisy, shallow circuit. By employing a tensor networ representation, we reconstruct channels acting on up to 50 qubits and accurately recover global diagnostics such as the process fidelity, the Choi state purity, and Pauli-weight-resolved process matrix elements. Our work thus extends the powerful toolbox local shadow tomography to scalable channel characterization with access to global properties.
Factored gradient descent (FGD) is a scalable nonconvex method for low-rank positive semidefinite matrix recovery, in which the rank and positivity constraints are satisfied automatically by the factorization. Most of the existing theory that certifies benignity of its loss landscape, however, relies on a restricted isometry property (RIP) of the measurement map. In quantum state tomography, one of its applications, the measurements that can be acquired with stable statistics are limited by detection efficiency and drift to Pauli observables of low weight; we first show that under such partial-weight Pauli measurements RIP fails in the worst possible way. We then bundle the mutually commuting observables setting by setting, pushing the state forward to the probability distribution of each setting, and introduce a max-sliced total-variation loss. On the stratum where the setting attaining the maximum is unique there is no spurious local minimum, and provided only that the measurement map is injective on generic pure states, the global minimum is the truth up to a phase. Finally we verify these claims numerically on four qubits and compare against FGD with the ordinary loss.
Abstract Accurate reconstruction of quantum channels from experimental data is a fundamental challenge in quantum information science. The standard approach via the Choi–Jamiołkowski isomorphism relies on semidefinite programming to enforce physical completely positive and trace-preserving constraints, which becomes computationally prohibitive for high-dimensional systems. To circumvent this dimensionality curse this paper introduces a hybrid alternating least squares framework for quantum channel reconstruction utilizing a mixed unitary parameterization. The proposed approach directly targets the constituent operation elements by alternating between updating the unitary matrices and the associated probability distribution. The probability vector is efficiently resolved via standard constrained least squares. For the unitary elements the framework employs an exact O(n3) singular value decomposition Procrustes update in the single-pair setting, and transitions to a natural gradient flow to manage the simultaneous constraints of multi-pair process tomography. Because the underlying optimization landscape features continuous gauge symmetries and resides on a nonconvex product isospectral manifold establishing theoretical guarantees is highly nontrivial. By embedding the optimization dynamics within the Kurdyka–Łojasiewicz framework and establishing strict sufficient decrease and relative error bounds this work provides a rigorous proof of global convergence to a critical point. Numerical experiments validate these theoretical findings, demonstrating that the framework offers a highly scalable, robust and mathematically principled alternative for recovering physical quantum channels.