Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography

In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a O \(sqrt{ ( 1 / t\) d log t } ) rate, where d and t denote the dimension and number of iterations, respectively. Its per-iteration time complexity is O \(d³\), independent of the sample size. To the best of our knowledge, this is currently the computationally fastest stochastic first-order method for maximum-likelihood quantum state tomography.