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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• Quantum algorithms & computation
• Integrable systems