# Polynomial Scaling of Numerical Diagonalization of the 1D Transverse Field Ising Model into a Commuting Basis using the Pauli Product Representation

## TL;DR

This paper demonstrates that the diagonalization of the 1D transverse field Ising model can be achieved with polynomial complexity using the Pauli product representation, facilitating efficient quantum simulation of many-particle Hamiltonians.

## Contribution

It introduces a polynomial-scaling diagonalization method for the 1D transverse field Ising model using the Pauli product basis, advancing quantum simulation techniques.

## Key findings

- Diagonalization scales polynomially with the number of spins.
- Number of Jacobi transformations grows polynomially.
- Results suggest feasible quantum circuit construction for simulation.

## Abstract

We report numerical results on the diagonalization of 1D transverse field Ising model. Numerical simulations using the Pauli product representation yield diagonalization from 3 spins to 22 spins in the transverse field Ising model with the number of global Jacobi unitary transformations and number of final terms in diagonalized spin z representation both grew polynomial with the number of spins. These results computed on a classical computer show promise in constructing a quantum circuit to simulate diagonalized generic many-particle Hamiltonians using polynomial number of gates.