Finding a Hadamard Matrix by Simulated Quantum Annealing

Area of Science:

• Computational physics
• Quantum computing
• Combinatorial optimization

Background:

• Hard problems are a significant challenge in modern computing.
• Heuristic approaches inspired by physical phenomena are actively researched.
• Hadamard matrices are crucial in various fields but constructing them is computationally intensive.

Purpose of the Study:

• To propose and evaluate simulated quantum annealing (SQA) for generating Hadamard matrices.
• To reformulate the Hadamard matrix problem as an energy minimization task for spin vectors.
• To compare the efficacy of SQA against traditional simulated annealing (SA).

Main Methods:

• Reformulating Hadamard matrix construction as an energy minimization problem.
• Employing path-integral Monte-Carlo (PIMC) based SQA on a spin vector system.
• Applying a transverse magnetic field with time-decreasing strength.

Main Results:

• SQA successfully generated low-order Hadamard matrices, including those not constructible by the Sylvester method.
• Numerical experiments indicated favorable scaling properties for the SQA method.
• Residual energy measurements showed SQA's superiority over SA for this problem.

Conclusions:

• SQA is an effective method for tackling the computationally hard problem of finding Hadamard matrices.
• The proposed SQA approach outperforms traditional SA in solving this specific combinatorial optimization problem.
• This research opens avenues for applying quantum-inspired methods to complex mathematical constructions.