Mixed-Integer Programming by Continuous Variable Quantum Computation*

We propose a continuous-variable quantum computation (CVQC) approach to solving mixed-integer programming problems using a circuit-model quantum optical device. Our strategy is to use the photon number operator associated with each optical mode in the device to encode the integer variables of an optimization problem, and the corresponding quadrature operators to encode the continuous variables of it. Following an adiabatic ground-state preparation scheme, optimal integer and continuous variable solutions are obtained by respectively performing photon number resolving detection and homodyne measurements on the corresponding modes. We demonstrate our method by studying its application to a wide range of important optimization problems, including integer linear and nonlinear programming, continuous nonlinear programming, and mixed-integer programming problems, using real-world instances of these optimization problems  such as the integer knapsack problem, the maximum clique problem, and various instances of sparse optimization problems.