Discrete Optimization: A Quantum Revolution?

Abstract

We develop several quantum procedures and investigate their potential to solve discrete optimization problems. First, we introduce a binary search procedure and illustrate how it can be used to effectively solve the binary knapsack problem. Next, we introduce two other procedures: a hybrid branch-and-bound procedure that allows to exploit the structure of the problem and a random-ascent procedure that can be used to solve problems that have no clear structure and/or are difficult to solve using traditional methods. We explain how to assess the performance of these procedures and perform a computational experiment. Our results show that we can match the performance of the best classical algorithms when solving the binary knapsack problem. After improving and generalizing our procedures, we show that they can solve any discrete optimization problem using at most O(µ2^(0.5nb)) operations, where µ is the number of operations required to evaluate the feasibility of a solution, n is the number of decision variables, and 2^b is the number of discrete values that can be assigned to each decision variable. In addition, we demonstrate that our procedures can also be used as heuristics to find (near-) optimal solutions using far less than O(µ2^(0.5nb)) operations. Not only does our work provide the tools required to explore a myriad of future research directions, it also has the potential to revolutionize the field of discrete optimization.

Keywords: quantum, computing, algorithm, knapsack, Grover