From Symmetric Toeplitz Hamiltonians to Quantum Circuits

TL;DR

This paper presents a quantum circuit synthesis method for simulating symmetric Toeplitz Hamiltonians, simplifying their decomposition into diagonal matrices and constructing explicit circuits for the 1D Poisson equation.

Contribution

It introduces a novel framework for decomposing symmetric Toeplitz Hamiltonians into classified diagonal matrices for efficient quantum simulation.

Findings

Decomposition of symmetric Toeplitz Hamiltonians into specific diagonal matrices.

Simplification of matrices when indices are powers of two or belong to congruence classes.

Explicit quantum circuit construction for the 1D discrete Poisson equation.

Abstract

This work introduces a quantum circuit synthesis framework for simulating the unitary time evolution under a subclass of symmetric Toeplitz Hamiltonians by decomposing them into specific diagonal matrices $M_k$. These matrices are then classified, to achieve significant simplification, into, $M_k$ when $k$ is a power of two, and congruence classes with constant coefficients. Finally, we construct the explicit quantum circuit for the one-dimensional discrete Poisson equation. This research was conducted under the supervision of Benoit Valiron during a Master's internship.