A quantum algorithm for solving 0-1 Knapsack problems

TL;DR

This paper introduces a quantum algorithm called the Quantum Tree Generator that efficiently solves 0-1 knapsack problems, demonstrating potential quantum advantage in large-scale combinatorial optimization.

Contribution

The paper presents the Quantum Tree Generator and a new runtime prediction method, enabling practical quantum advantage for large 0-1 knapsack instances.

Findings

Quantum Tree Generator generates all feasible solutions in superposition.

The method shows competitive runtimes for instances with 100 variables.

Runtime predictions extend to problems with up to 600 variables.

Abstract

Here we present two novel contributions for achieving quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the "Quantum Tree Generator", an approach to generate in superposition all feasible solutions of a given instance, yielding together with amplitude amplification the optimal solutions for 0-1 knapsack problems. The QTG offers massive memory savings and enables competitive runtimes compared to the classical state-of-the-art knapsack solvers (such as COMBO, Gurobi, CP-SAT, Greedy) already for instances involving as few as 100 variables. (2) By introducing a new runtime calculation technique that exploits logging data from the classical solver COMBO, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for various benchmark instances with up to 600 variables. Combining both of these innovations, we demonstrate the QTG's potential practical quantum advantage for large-scale problems, indicating an effective approach for combinatorial optimisation problems.