Multi-block ADMM Heuristics for Mixed-Binary Optimization on Classical and Quantum Computers

Solving combinatorial optimization problems on noisy intermediate-scale quantum
(NISQ) devices is currently being advocated for (and restricted to) binary polynomial optimization with equality constraints. This is achieved using, e.g., the variational quantum eigensolver (VQE), a hybrid quantum/classical heuristic approach. We present a decomposition-based approach to extend the applicability of current approaches to mixed binary optimization (MBO) problems, so as to solve a broad class of real-world optimization problems. In the MBO framework, we show that the alternating direction method of multipliers (ADMM) can split the MBO into a binary unconstrained problem (that can be solved via variational quantum approaches), and continuous constrained convex subproblems (that can be solved cheaply with classical optimization solvers). The validity of the approach is then showcased by numerical results obtained on several optimization problems via simulations with VQE on the quantum circuits implemented in Qiskit, an open-source quantum computing software development framework.