Finding Small and Large k-Clique Instances on a Quantum Computer

TL;DR

This paper develops and evaluates practical quantum algorithms for finding small and large k-cliques, addressing implementation challenges on current and near-future quantum hardware.

Contribution

It introduces a gate-based approach for k-clique problems, compares theoretical and practical implementations, and estimates when these methods will be feasible on real quantum devices.

Findings

Quantum algorithms for triangle and k-clique problems are feasible on NISQ devices with optimized circuits.

Simulations show error impacts on amplitude damping and solution accuracy.

Estimated timeline for practical implementation on IBM quantum hardware based on error growth and quantum volume.

Abstract

Algorithms for triangle-finding, the smallest nontrivial instance of the k-clique problem, have been proposed for quantum computers. Still, those algorithms assume the use of fixed access time quantum RAM (QRAM). We present a practical gate-based approach to both the triangle-finding problem and its NP-hard k-clique generalization. We examine both constant factors for near-term implementation on a Noisy Intermediate Scale Quantum computer (NISQ) device, and the scaling of the problem to evaluate long-term use of quantum computers. We compare the time complexity and circuit practicality of the theoretical approach and actual implementation. We propose and apply two different strategies to the k-clique problem, examining the circuit size of Qiskit implementations. We analyze our implementations by simulating triangle finding with various error models, observing the effect on damping the amplitude of the correct answer, and compare to execution on six real IBMQ machines. Finally, we estimate the date when the methods proposed can run effectively on an actual device based on IBM's quantum volume exponential growth forecast and the results of our error analysis.