Finding Hadamard Matrices by a Quantum Annealing Machine

Area of Science:

• Quantum Computing
• Computational Mathematics
• Combinatorial Optimization

Background:

• Finding Hadamard matrices (H-matrices) is a computationally challenging problem.
• Quantum computing offers a potential avenue for solving this problem.
• Existing quantum annealing machines (QAMs) have limitations in handling higher-order terms naturally present in H-matrix formulations.

Purpose of the Study:

• To propose a method for formulating the Hamiltonian of the H-matrix problem.
• To address the implementation limitations of QAMs for H-matrix problems.
• To explore solutions for related problems, including orthogonal binary vectors and deleted vectors of H-matrices.

Main Methods:

• Formulating the Hamiltonian for the H-matrix problem.
• Employing symbolic computing techniques to manage the increased variable complexity due to QAM limitations.
• Investigating three related cases: N

Main Results:

• The study details how QAM limitations necessitate a significant increase in variables for H-matrix formulation.
• Symbolic computing is demonstrated as a viable method to manage this complexity.
• The research discusses potential solutions for the three related cases using both simulated and actual quantum annealing hardware.

Conclusions:

• The formulation of H-matrix problems for QAMs is feasible but requires advanced techniques like symbolic computing to overcome hardware limitations.
• The study provides a framework for tackling H-matrix and related combinatorial problems on quantum hardware.
• Further research can explore optimized implementations and larger-scale problem instances.