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 universal 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. The experiments successfully found the Baumert-Hall type Hadamard matrices up to 132. These results motivate further efforts to discover previously unknown Hadamard matrices and a prospect to ultimately demonstrate practical quantum advantages