Weyl's Relations, Integrable Matrix Models and Quantum Computation

TL;DR

This paper generalizes Weyl's relations to construct commuting matrices linked to quantum integrable models, demonstrating their potential to enhance quantum algorithms like Grover's search through improved adiabatic Hamiltonians.

Contribution

It introduces a new hierarchy of parameter-dependent commuting matrices based on generalized Weyl relations, connecting them to quantum integrable models and quantum computation.

Findings

Hierarchy of commuting matrices constructed from generalized Weyl relations.

Application of these matrices as Hamiltonians in quantum adiabatic algorithms.

Potential for improved fidelity in Grover's search using new Hamiltonians.

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.