Noisy intermediate-scale quantum algorithm for semidefinite programming

Semidefinite programs (SDPs) are convex optimization with vast applications in control theory, quantum information, combinatorial optimization, and operational research. Noisy intermediate-scale (NISQ) algorithms aim to make an efficient use of the current generation hardware. However, optimizing variational is a challenge as it nondeterministic polynomial time-hard problem that general requires exponential time solve can contain many far from optimal local minima. Here, we present term NISQ algorithm for solving SDPs. The classical program our solver another SDP over lower dimensional ansatz space. We harness SDP-based formulation Hamiltonian ground-state design eigensolver. Unlike eigensolvers, eigensolver be solved number parameters, every minimum global minimum. find numeric evidence improve estimation energies scalable manner. Further, efficiently constrained problems calculate excited states Hamiltonians, lowest energy symmetry determine measurements state discrimination. demonstrate potential approach by finding largest eigenvalue up ${2}^{1000}$ matrices graph related contextuality. also discuss rank-constrained Our work extends application computers onto one most successful algorithmic frameworks past few decades.