Gradient-descent methods for fast quantum state tomography

Quantum state tomography (QST) is a standard tool used in quantum technology for characterizing the state of a system. However, it is cursed with two fundamental difficulties: both computational and experimental complexity increase exponentially with the number of qubits, making implementation and data post-processing difficult even for moderately sized systems. Here, we propose and benchmark gradient descent (GD) algorithms for QST, employing various density-matrix parameterizations using Cholesky decomposition, Stiefel manifold, and projective normalization to ensure valid reconstruction. Furthermore, wherever possible, we compare the performance of GD-QST techniques against several existing methods, including constrained convex optimization (CCO), conditional generative adversarial networks (CGANs), nonconvex-Riemannian GD (RGD), and accelerated projected-gradient based maximum likelihood estimation (APG-MLE). Our comparison mainly focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. Numerical analysis for up to 7 qubits demonstrates that GD-QST is computationally faster and outperforms other techniques in most scenarios, offering a promising path for characterization of noisy intermediate-scale quantum (NISQ) devices.