Translating Constraints into QUBOs for the Quadratic Knapsack Problem.

One of the first fields where quantum computing will likely show its use is optimisation. Many optimisation problems naturally arise in a quadratic manner, such as the quadratic knapsack problem. The current state of quantum computers requires these problems to be formulated as a quadratic unconstrained binary optimisation problem, or QUBO. Constrained quadratic binary optimisation can be translated into QUBOs by translating the constraint. However, this translation can be made in several ways, which can have a large impact on the performance when solving the QUBO. We show six different formulations for the quadratic knapsack problem and compare their performance using simulated annealing. The best performance is obtained by a formulation that uses no auxiliary variables for modelling the inequality constraint.