A Quantum Approximate Optimization Method For Finding Hadamard Matrices

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.