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arXiv:2312.01311 · IEEE International Conference on Rebooting Computing (ICRC)

Using non-convex optimization in quantum process tomography: Factored gradient descent is tough to beat

David Quiroga, Anastasios Kyrillidis

Published Dec 5, 2023• 2 citations
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Abstract

We propose a non-convex optimization algorithm, based on the Burer-Monteiro (BM) factorization, for the quantum process tomography problem, in order to estimate a low-rank process matrix χ for near-unitary quantum gates. In this work, we compare our approach against state of the art convex optimization approaches based on gradient descent. We use a reduced set of initial states and measurement operators that require 2 · 8ncircuit settings, as well as $\mathcal{O}(4^n)$ measurements for an underdetermined setting. We find our algorithm converges faster and achieves higher fidelities than state of the art, both in terms of measurement settings, as well as in terms of noise tolerance, in the cases of depolarizing and Gaussian noise models.

DOI
10.1109/icrc60800.2023.10386455

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We propose a non-convex optimization algorithm, based on the Burer-Monteiro (BM) factorization, for the quantum process tomography problem, in order to estimate a low-rank process matrix χ for near-unitary quantum gates. In this work, we compare our approach against state of the art convex optimization approaches based on gradient descent. We use a reduced set of initial states and measurement operators that require 2 · 8ncircuit settings, as well as O(4n)\mathcal{O}(4^n)O(4n) measurements for an underdetermined setting. We find our algorithm converges faster and achieves higher fidelities than state of the art, both in terms of measurement settings, as well as in terms of noise tolerance, in the cases of depolarizing and Gaussian noise models.
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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}.
Gradient descent algorithms on Riemannian manifolds have recently been proposed as a practical tool for optimizing quantum channels. We add rank-penalizing regularization terms to the cost functions of these methods (in the spirit of Lasso), driving the optimization towards channels with as few Kraus operators as possible. For quantum process tomography of rank-deficient channels, regularization both accelerates convergence and improves the fidelity of the solution compared to the unregularized case. For quantum classification tasks, it reduces the channel rank while maintaining classification accuracy, thereby revealing the minimum rank required by the given input data.
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)O(d{log}_{2}d)O(dlog2​d) 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.
We present a technique for performing quantum detector tomography (QDT) of phase insensitive quantum detectors, a category under which many detectors of interest fall under, using gradient descent-based optimization to learn the positive operator-valued measure (POVM) that best describes the data collected using the detector under study. We numerically benchmark our method against constrained convex optimization (CCO) and show that it reaches higher or comparable reconstruction fidelity in much less time even in the presence of noise and limited probe state resources. We also present a possible extension of our approach to the phase sensitive case via a parametrization of POVMs on the complex Stiefel manifold which enables gradient based optimization restricted to this manifold.
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 .