Quantum measurement tomography with mini-batch stochastic gradient descent

TL;DR

Stochastic gradient descent algorithms for fast quantum measurement tomography (QMT) are introduced, applicable to both discrete- and continuous-variable quantum systems -- thus completing the tomography trio.

Abstract

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}.

Figures (8)

• Figure 1: Quantum circuit illustrating the three fundamental components of a quantum experiment: the quantum state, described by the density matrix $\rho$; the quantum process, represented by the process matrix $\chi$; and the measurement apparatus, characterized by a set of POVM elements $\{ \Pi_i \}$.
• Figure 2: Time complexity and performance metrics of different QMT data-processing methods for random full-rank POVM sets. The upper and lower lines for each marker are error bars denoting one standard deviation. The first row shows runtime (in seconds, on a $\log_{10}$ scale), while the second and third rows report the Frobenius distance and the Wasserstein distance, respectively, for the reconstructed POVMs (both also on a $\log_{10}$ scale). Each metric on the $y$ axes is averaged over multiple randomly generated POVM sets [20 for ($N \leq 4$ & $K \leq 16$), 5--10 for ($N=5,6$ & $K=32$)] of the indicated size. The first column corresponds to QMT with $k=8$ POVM elements, while second and third columns correspond to $k=16$ and $k=32$, respectively. In all cases, the $x$ axis denotes the number of qubits, with the maximum number of iterations fixed at 3000.
• Figure 3: SGD-QMT performance for rank-1 Pauli projective measurements. (a) Runtime (seconds), (b) Frobenius distance, and (c) Wasserstein distance, all as functions of qubit number $N$. The bar plots show HONEST-MSE (teal), HONEST-MLE (orange), SM-MSE (red), SM-MLE (green) and CCO-CVX (purple, up to four qubits). All three performance metrics are averaged over 15 randomly generated Pauli projective measurements. The error bars represent standard deviation across these instances. The maximum number of iterations is fixed at 1000 in all cases.
• Figure 4: Performance of SGD-QMT data-processing algorithms for four-qubit projective measurements in the computational basis $\Pi = \{ |0\rangle\langle 0|, |1\rangle\langle 1| \}^{\otimes 4}$, evaluated with respect to loss (first row), Frobenius norm (second row) and Wasserstein distance (third row), as a function of the number of iterations. The columns correspond to results for HONEST-MSE, HONEST-MLE, SM-MSE, and SM-MLE, in that order. In the second row, each POVM element's Frobenius distance is plotted in a distinct color, with the black dashed line indicating the average Frobenius norm over all 16 POVM elements and gray shading indicating one standard deviation for that average. In the third row, the solid orange line denotes the average Wasserstein distance computed over 16 input states, and the shaded region indicates the standard deviation. In the second and third rows, the results obtained with CCO-CVX are shown as dashed red lines. The maximum number of iterations is set to 1500, with the average total runtime for all SGD-QMT methods below 4 seconds. Note that the $y$ axis scales differ across both rows and columns.
• Figure 5: Performance of SGD-QMT for photon detection. (a) Loss and (b) average Frobenius norm, as functions of the iteration number for HONEST-MSE (teal), HONEST-MLE (orange), SM-MSE (red), and SM-MLE (green). (c) The reconstructed POVM elements, $\Pi_{1}$ and $\Pi_{2}$, obtained using HONEST-MLE. Note that the POVM heatmaps are displayed with nonlinear scaling, enhancing the visibility of small entries while compressing larger ones, to highlight fine details in the low-value regions.