Local Stochastic Factored Gradient Descent for Distributed Quantum State Tomography
This addresses the problem of characterizing quantum system states and noise in quantum computation for researchers in quantum information, but it appears incremental as it builds on existing gradient descent methods for distributed settings.
The paper tackles distributed quantum state tomography by proposing Local Stochastic Factored Gradient Descent (Local SFGD) to estimate low-rank density matrices, proving local convergence to a small neighborhood of the global optimum at a linear rate with constant step sizes and validating this with numerical simulations on the GHZ state.
We propose a distributed Quantum State Tomography (QST) protocol, named Local Stochastic Factored Gradient Descent (Local SFGD), to learn the low-rank factor of a density matrix over a set of local machines. QST is the canonical procedure to characterize the state of a quantum system, which we formulate as a stochastic nonconvex smooth optimization problem. Physically, the estimation of a low-rank density matrix helps characterizing the amount of noise introduced by quantum computation. Theoretically, we prove the local convergence of Local SFGD for a general class of restricted strongly convex/smooth loss functions, i.e., Local SFGD converges locally to a small neighborhood of the global optimum at a linear rate with a constant step size, while it locally converges exactly at a sub-linear rate with diminishing step sizes. With a proper initialization, local convergence results imply global convergence. We validate our theoretical findings with numerical simulations of QST on the Greenberger-Horne-Zeilinger (GHZ) state.