# Quantum simulation of scattering in the quantum Ising model

## TL;DR

This paper explores quantum simulation of the real-time dynamics in the 1D quantum Ising model, demonstrating accurate particle evolution predictions with small interactions and analyzing implementation errors on near-term quantum computers.

## Contribution

It introduces a method to simulate quantum Ising model dynamics using Trotterization on quantum computers, with analytical solutions for small interactions and error analysis for near-term devices.

## Key findings

- Discrete Bessel functions accurately model particle evolution for small J.
- Boundary conditions significantly affect long-time evolution.
- Error analysis for Trotter steps and quantum hardware noise.

## Abstract

We discuss real time evolution for the quantum Ising model in one spatial dimension with $N_s$ sites. In the limit where the nearest neighbor interactions $J$ in the spatial directions are small, there is a simple physical picture where qubit states can be interpreted as approximate particle occupations. Using exact diagonalization, for initial states with one or two particles, we show that for small $J$, discrete Bessel functions provide very accurate expressions for the evolution of the occupancies corresponding to initial states with one and two particles. Boundary conditions play an important role when the evolution time is long enough. We discuss a Trotter procedure to implement the evolution on existing quantum computers and discuss the error associated with the Trotter step size. We discuss the effects of gate and measurement errors on the evolution of one and two particle states using 4 and 8 qubits circuits approximately corresponding to existing or near term quantum computers.