Quantum two-player SDP solver with stochastic trajectory-level analysis

Develop a quantum two-player semidefinite-programming solver that retains the \(\widetilde O(\sqrt n+\sqrt m)\) dimension dependence of existing quantum SDP algorithms while improving their dependence on the effective inverse-accuracy parameter \(\gamma=Rr/\varepsilon\) through a trajectory-level stochastic analysis.

Background

The paper classically combines stochastic optimization, sampled rank-one matrix responses, constraint sampling, and martingale concentration to obtain an additive dependence on the matrix dimension and number of constraints while improving accuracy dependence. Existing quantum SDP solvers retain the $\widetilde O(\sqrt n+\sqrt m)$ dimensional scaling but rely primarily on oracle-based matrix-multiplicative-weights frameworks, in which each iteration requires sufficiently accurate constraint information.

The unresolved direction is to transfer the paper’s trajectory-level stochastic perspective to the quantum setting. The authors identify dynamic quantum Gibbs sampling and cumulative-feedback control, previously developed for linear programs, as possible ingredients, but note that matrix-valued analogues and suitable control of matrix-response and constraint-feedback errors remain to be developed.