# Quantum Computation and Visualization of Hamiltonians using Discrete Quantum Mechanics and IBM QISKit

## TL;DR

This paper explores methods to translate various quantum systems into a form compatible with quantum computers, specifically using matrix formulations and visualization techniques, to enhance quantum simulation capabilities.

## Contribution

It introduces two approaches—position basis and Gaussian basis—for mapping systems onto quantum computers and visualizes wave functions for better understanding.

## Key findings

- Successful translation of quantum systems into matrix form for quantum algorithms
- Visualization of wave functions aligns with theoretical continuous operator results
- Demonstrates the applicability of discrete quantum mechanics in quantum computing

## Abstract

Quantum computers have the potential to transform the ways in which we tackle some important problems. The efforts by companies like Google, IBM and Microsoft to construct quantum computers have been making headlines for years. Equally important is the challenge of translating problems into a state that can be fed to these machines. Because quantum computers are in essence controllable quantum systems, the problems that most naturally map to them are those of quantum mechanics. Quantum chemistry has seen particular success in the form of the variational quantum eigensolver (VQE) algorithm, which is used to determine the ground state energy of molecular systems. The goal of our work has been to use the matrix formulation of quantum mechanics to translate other systems so that they can be run through this same algorithm. We describe two ways of accomplishing this using a position basis approach and a Gaussian basis approach. We also visualize the wave functions from the eigensolver and make comparisons to theoretical results obtained with continuous operators.