Optimization of a Quantum Subset Sum Oracle

TL;DR

This paper develops optimized quantum oracles for the Subset Sum problem, reducing resource requirements and enabling approximate solutions using Grover's algorithm, with practical implementation and experimental validation.

Contribution

It introduces techniques to minimize qubits and gates in quantum Subset Sum oracles, including new comparison methods and approximate solution strategies.

Findings

Techniques effectively reduce qubit and gate counts.

Experimental results validate the efficiency of the proposed methods.

Approximate solutions are feasible with modified oracles.

Abstract

We investigate the implementation of an oracle for the Subset Sum problem for quantum search using Grover's algorithm. Our work concerns reducing the number of qubits, gates, and multi-controlled gates required by the oracle. We describe the compilation of a Subset Sum instance into a quantum oracle, using a Python library we developed for Qiskit and have published in GitHub. We then present techniques to conserve qubits and gates along with experiments showing their effectiveness on random instances of Subset Sum. These techniques include moving from fixed to varying-width arithmetic, using partial sums of a set's integers to determine specific integer widths, and sorting the set to obtain provably the most efficient partial sums. We present a new method for computing bit-string comparisons that avoids arbitrarily large multiple-control gates, and we introduce a simple modification to the oracle that allows for approximate solutions to the Subset Sum problem via Grover search.