A quantum approximate optimization method for finding Hadamard matrices

Area of Science:

• Quantum Computing
• Quantum Algorithms
• Computational Mathematics

Background:

• Demonstrating practical quantum advantage is a key goal in quantum computing.
• Previous attempts to find Hadamard matrices on quantum annealers were limited by resource constraints and high-order interaction terms.
• The complexity of high-order interactions grows significantly with matrix order (M).

Purpose of the Study:

• To propose a novel, qubit-efficient method for searching Hadamard matrices.
• To leverage gate-based quantum computers for this task.
• To reduce the number of qubits required compared to previous methods.

Main Methods:

• Implementation of the Hadamard matrix searching algorithm on a gate-based quantum computer.
• Utilizing the Quantum Approximate Optimization Algorithm (QAOA).
• Formulation of the method and construction of corresponding quantum circuits.

Main Results:

• The proposed method reduces the required number of qubits to O(M).
• High-order interaction terms do not necessitate ancillary qubits on gate-based systems.
• Successful experiment results were obtained using both a quantum simulator and a real quantum computer.

Conclusions:

• The novel QAOA-based approach offers a more qubit-efficient solution for finding Hadamard matrices.
• This method paves the way for demonstrating practical quantum advantage.
• The findings are validated on both simulated and real quantum hardware.