Abstract
Solving combinatorial optimization problems on current noisy quantum devices is currently being advocated for (and restricted to) binary polynomial optimization with equality constraints via quantum heuristic approaches. This is achieved using, e.g., the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). We present a decomposition-based approach to extend the applicability of current approaches to "quadratic plus convex" 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 with quantum algorithms), 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 and QAOA on the quantum circuits implemented in Qiskit, an open-source quantum computing software development framework.
Highlights
Mixed-binary optimization (MBO) has been studied for decades in mathematical programming, because of the widespread range of applications in several domains [1]–[3], and the inherent difficulties posed by integer variables
Our method instead does not involve a projection step and makes use of the alternating direction method of multipliers (ADMM) operatorsplitting procedure to devise a decomposition for certain classes of MBOs into the following: 1) a quadratic unconstrained binary optimization (QUBO) subproblem to be solved by a QUBO solver, e.g., on noisy quantum devices via quantum variational algorithms, such as variational quantum eigensolver (VQE) [11], quantum approximate optimization algorithm (QAOA) [8], or with Grover-search-based algorithms [43], or quantum-based semidefinite programming (SDP) relaxations [44]
The algorithm has been implemented in Python on a machine with 2.2 GHz, Intel Core i7 processor, and a RAM of 16 GB; the simulations on quantum devices to solve the QUBOs have been conducted by using the Qiskit framework [79], while IBM ILOG CPLEX 12.8 has been chosen as classical optimization solver
Summary
Mixed-binary optimization (MBO) has been studied for decades in mathematical programming, because of the widespread range of applications in several domains [1]–[3], and the inherent difficulties posed by integer variables. Our method instead does not involve a projection step and makes use of the ADMM operatorsplitting procedure to devise a decomposition for certain classes of MBOs into the following: 1) a QUBO subproblem to be solved by a QUBO (approximate) solver, e.g., on noisy quantum devices via quantum variational algorithms, such as VQE [11], QAOA [8], or with Grover-search-based algorithms [43], or quantum-based semidefinite programming (SDP) relaxations [44]. The mathematical contribution of this article is a multiblock ADMM heuristic (M-ADMM-H) algorithm for MBO, for which we present the following points: 1) a decomposition approach suitable for computation on current quantum devices; 2) conditions for convergence, feasibility, and optimality. The indicator function of any closed set is lower semicontinuous
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