Discrete optimization: A quantum revolution (Part II)

Abstract

In part I of this paper, we introduce the basics of quantum computing, and use Grover's algorithm as a subroutine in a binary search procedure that can solve any discrete optimization problem. In part II, we improve the performance of this binary search procedure. For this purpose, we propose two new procedures, and use them to solve 66000 instances of the binary knapsack problem. Our results show that both procedures can match the performance of the best classical algorithms. Whereas the first procedure (a hybrid branch-and-bound algorithm) can exploit the structure of the problem, the second procedure (a random-ascent algorithm) can be used to solve discrete optimization problems that have no clear structure and/or are difficult to solve using traditional methods. After improving and generalizing these 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 show that both procedures can also be used as a heuristic to find (near-) optimal solutions using far less than O(µ2^(0.5nb)) operations.

Keywords: Quantum, computing, algorithm, knapsack