Quantum measurement tomography via mini-batch stochastic gradient descent optimization

Drawing inspiration from earlier gradient-descent methods developed for quantum state tomography [Quantum Sci. Technol. 10, 045055 (2025)] and quantum process tomography [Phys. Rev. Lett. 130, 150402 (2023)], we complete the tomography trio by introducing stochastic gradient descent (SGD) algorithms for fast quantum measurement tomography (QMT), applicable to both discrete- and continuous-variable quantum systems. The goal of QMT is to estimate, from experimental data, the positive operator-valued measure (POVM) elements that characterize a measurement device or detector in a quantum experiment. To ensure physically valid POVM reconstructions, we propose two distinct parameterization schemes within the SGD framework: one based on Stiefel manifolds and one based based on a Hermitian operator normalization via eigenvalue scaling. Both parameterizations intrinsically enforce the positivity and completeness constraints required for valid POVMs. Within the SGD-QMT framework, we investigate two loss functions: mean squared error (MSE) and average negative log-likelihood. We benchmark performance against state-of-the-art constrained convex optimization methods, finding through numerical simulations that our SGD-QMT algorithms offer significantly lower computational cost with superior reconstruction fidelity and enhanced robustness to noise compared to standard methods.