A Quantum Approximate Optimization Method For Finding Hadamard Matrices

TL;DR

This paper introduces a qubit-efficient quantum algorithm using QAOA to find Hadamard matrices, reducing qubit requirements and demonstrating its feasibility on simulators and real quantum hardware.

Contribution

It presents a novel, resource-efficient quantum algorithm for Hadamard matrix search that overcomes previous limitations by eliminating the need for ancillary qubits.

Findings

Successful implementation on quantum simulator

Experimental validation on real quantum hardware

Reduced qubit requirements from O(M^2) to O(M)

Abstract

Finding a Hadamard matrix of a specific order using a quantum computer can lead to a demonstration of practical quantum advantage. Earlier efforts using a quantum annealer were impeded by the limitations of the present quantum resource and its capability to implement high order interaction terms, which for an $M$-order matrix will grow by $O(M^2)$. In this paper, we propose a novel qubit-efficient method by implementing the Hadamard matrix searching algorithm on a gate-based quantum computer. We achieve this by employing the Quantum Approximate Optimization Algorithm (QAOA). Since high order interaction terms that are implemented on a gate-based quantum computer do not need ancillary qubits, the proposed method reduces the required number of qubits into $O(M)$. We present the formulation of the method, construction of corresponding quantum circuits, and experiment results in both a quantum simulator and a real gate-based quantum computer.