Explicit mathematical reconstructions of quantum combs play a significant role in developing quantum information science. However, tremendous parameter requirements and physical constraint implementations have become computationally nonignorable encumbrances. In this Letter, we propose an efficient method for quantum comb tomography by learning isometries on the Stiefel manifold via solving a series of unconstrained optimization problems with significantly fewer parameters. The stepwise isometry estimation shows the capability for providing information of the truncated quantum comb while processing the tomography. Remarkably, this method enables the dimension-reduced quantum comb tomography by reducing the ancillary dimensions of isometries with bounded error. As a result, our proposed method exhibits high accuracy and efficiency.
Explicit mathematical reconstructions of quantum combs play a significant role in developing quantum information science. However, tremendous parameter requirements and physical constraint implementations have become computationally nonignorable encumbrances. In this Letter, we propose an efficient method for quantum comb tomography by learning isometries on the Stiefel manifold via solving a series of unconstrained optimization problems with significantly fewer parameters. The stepwise isometry estimation shows the capability for providing information of the truncated quantum comb while processing the tomography. Remarkably, this method enables the dimension-reduced quantum comb tomography by reducing the ancillary dimensions of isometries with bounded error. As a result, our proposed method exhibits high accuracy and efficiency.
Explicit mathematical reconstructions of quantum networks play a significant role in developing quantum information science. However, tremendous parameter requirements and physical constraint implementations have become computationally non-ignorable encumbrances. In this work, we propose an efficient method for quantum network tomography by learning isometries on the Stiefel manifold. Tasks of reconstructing quantum networks are tackled by solving a series of unconstrained optimization problems with significantly fewer parameters. The stepwise isometry estimation shows the capability for providing information of the truncated quantum network while processing the tomography. Remarkably, this method enables the dimension-reduced quantum network tomography by reducing the ancillary dimensions of isometries with bounded error. As a result, our proposed method exhibits high accuracy and efficiency.
Characterizing non-Markovian quantum dynamics is currently hindered by the self-inconsistency and high computational complexity of existing quantum comb tomography (QCT) methods. In this work, we propose a self-consistent framework that unifies the quantum comb, instrument set, and initial states into a single geometric entity, termed as the Comb-Instrument-State (CIS) set. We demonstrate that the CIS set naturally resides on a product Stiefel manifold, allowing the tomography problem to be solved via efficient unconstrained Riemannian optimization while automatically preserving physical constraints. Numerical simulations confirm that our approach is computationally scalable and robust against gate definition errors, significantly outperforming conventional isometry-based QCT methods. Our work indicates the potential to efficiently learn quantum comb with fewer computational resources.
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.
Flexible characterization techniques that provide a detailed picture of the experimental imperfections under realistic assumptions are crucial to gain actionable advice in the development of quantum computers.Gate set tomography self-consistently extracts a complete tomographic description of the implementation of an entire set of quantum gates, as well as the initial state and measurement, from experimental data.It has become a standard tool for this task but comes with high requirements on the number of sequences and their design, making it already experimentally challenging for only two qubits.In this work, we show that low-rank approximations of gate sets can be obtained from significantly fewer gate sequences and that it is sufficient to draw them at random.This coherent noise characterization however still contains the crucial information for improving the implementation.To this end, we formulate the data processing problem of gate set tomography as a rank-constrained tensor completion problem.We provide an algorithm to solve this problem while respecting the usual positivity and normalization constraints of quantum mechanics.For this purpose, we combine methods from Riemannian optimization and machine learning and develop a saddle-free second-order geometrical optimization method on the complex Stiefel manifold.Besides the reduction in sequences, we numerically demonstrate that the algorithm does not rely on structured gate sets or an elaborate circuit design to robustly perform gate set tomography.Therefore, it is more flexible than traditional approaches.We also demonstrate how coherent errors in shadow estimation protocols can be mitigated using estimates from gate set tomography.
Drawing inspiration from gradient-descent methods developed for data processing in quantum state tomography [\href{https://iopscience.iop.org/article/10.1088/2058-9565/ae0baa}{Quantum Sci.~Technol.~\textbf{10} 045055 (2025)}] and quantum process tomography [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.150402}{Phys.~Rev.~Lett.~\textbf{130}, 150402 (2023)}], we introduce stochastic gradient descent (SGD) algorithms for fast quantum measurement tomography (QMT), applicable to both discrete- and continuous-variable quantum systems -- thus completing the tomography trio. A measurement device or detector in a quantum experiment is characterized by a set of positive operator-valued measure (POVM) elements; the goal of QMT is to estimate these operators from experimental data. To ensure physically valid (positive and complete) POVM reconstructions, we propose two distinct parameterization schemes within the SGD framework: one leveraging optimization on a Stiefel manifold and one based on Hermitian operator normalization via eigenvalue scaling. Within the SGD-QMT framework, we further investigate two loss functions: mean squared error, equivalent to L2 or Euclidean norm, and average negative log-likelihood, inspired by maximum likelihood estimation. We benchmark performance against state-of-the-art constrained convex optimization methods. Numerical simulations demonstrate that, compared to standard methods, our SGD-QMT algorithms offer significantly lower computational cost, superior reconstruction fidelity, and enhanced robustness to noise. We make a Python implementation of the SGD-QMT algorithms publicly available at \href{https://github.com/agtomo/SGD-QMT}{github.com/agtomo/SGD-QMT}.
Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process tomography, in which the goal is to retrieve the underlying quantum process based on a given set of measurement data. In this paper, we introduce a modified version of stochastic gradient descent on a Riemannian manifold that integrates recent advancements in numerical methods for Riemannian optimization. This approach inherently supports the physically driven constraints of a quantum process, takes advantage of state-of-the-art large-scale stochastic objective optimization, and has superior performance to traditional approaches such as maximum likelihood estimation and projected least squares. The data-driven approach enables accurate, order-of-magnitude faster results, and works with incomplete data. We demonstrate our approach on simulations of quantum processes and in hardware by characterizing an engineered process on quantum computers.