Weyl's Relations, Integrable Matrix Models and Quantum Computation

• URL: http://arxiv.org/abs/2506.16841v1
• Date: Fri, 20 Jun 2025 08:45:44 GMT
• Title: Weyl's Relations, Integrable Matrix Models and Quantum Computation
• Abstract summary: We show that the Heisenberg commutation relations can be satisfied in a specific $N-1$ dimensional subspace.<n>This setup is used to construct a hierarchy of parameter-dependent commuting matrices in $N$ dimensions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Starting from a generalization of Weyl's relations in finite dimension $N$, we show that the Heisenberg commutation relations can be satisfied in a specific $N-1$ dimensional subspace, and display a linear map for projecting operators to this subspace. This setup is used to construct a hierarchy of parameter-dependent commuting matrices in $N$ dimensions. This family of commuting matrices is then related to Type-1 matrices representing quantum integrable models. The commuting matrices find an interesting application in quantum computation, specifically in Grover's database search problem. Each member of the hierarchy serves as a candidate Hamiltonian for quantum adiabatic evolution and, in some cases, achieves higher fidelity than standard choices -- thus offering improved performance.

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