Rank-1 Factored Gradient Descent without Restricted Isometry: Application to Quantum Pure State Tomography on Multi-Qubits

Factored gradient descent (FGD) is a scalable nonconvex method for low-rank positive semidefinite matrix recovery, in which the rank and positivity constraints are satisfied automatically by the factorization. Most of the existing theory that certifies benignity of its loss landscape, however, relies on a restricted isometry property (RIP) of the measurement map. In quantum state tomography, one of its applications, the measurements that can be acquired with stable statistics are limited by detection efficiency and drift to Pauli observables of low weight; we first show that under such partial-weight Pauli measurements RIP fails in the worst possible way. We then bundle the mutually commuting observables setting by setting, pushing the state forward to the probability distribution of each setting, and introduce a max-sliced total-variation loss. On the stratum where the setting attaining the maximum is unique there is no spurious local minimum, and provided only that the measurement map is injective on generic pure states, the global minimum is the truth up to a phase. Finally we verify these claims numerically on four qubits and compare against FGD with the ordinary loss.