Simulation of Schrodinger's equation on Quantum Computers

Author

YuXuan Liu

Date of Award

2022

First Advisor

Eric Kramer

Second Advisor

Michael Bergman

Abstract

Quantum computation is a theoretical computation model that processes information in a quantum mechanical system. It is fundamentally different from the classical computation model, known as the Turing Machine because quantum computation takes advantage of quantum mechanical properties of superposition, entanglement, and nondeterminism. It has been proven that quantum computers are able to simulate a classical computer. Therefore quantum computation is at least as robust as classical computation. This thesis will explore the theory of how one can simulate quantum particles on a quantum computer. While the idea may sound intuitive, many challenges and limitations need to be overcome. We will start with a formal introduction to quantum gates, the fundamental building blocks of a quantum circuit that can simulate a quantum particle. Using this formulation, we then introduce the Schrodinger's equation simulation algorithm, which decomposes the hamiltonian into the diagonal matrix in a specific basis. This matrix will be represented/constructed using the quantum gates, utilizing theorems fully developed in the past by Suzuki-Trotter [3] and Shende [9]. I simulated three systems using the developed algorithm: • The Free space potential, with both stationary and gaussian waves • The Square well potential • The Harmonic potential, with both the eigenstates and coherent-states All of which have well-studied solutions, and we will compare the simulation result to the analytical results. All simulations were done on the IBM's QASM simulator using six qubits, and they produced results agreeing with the past analytical results. The free space stationary gaussian and square well potential circuit was also run on IBM's Manila quantum computer, a ve-qubit real quantum computer, to explore the behavior of this simulation scheme in a physical quantum computer subject to many errors inducing hardware limitations. The results are, euphemistically speaking, disastrous. This is because the developed uses many multi-qubit gate to set up correlation between the qubits, which under current hardware is still not a robust process, and is very much subject to error accumulation and environmental noise This results in a large amount of error build up throughout the simulation. From this, we conclude that simulation scheme of this form is hypothetically achievable, provided that a perfect quantum computer with perfect coherence exists. However, due to the massive number of multi-qubit gates required throughout the simulation, the present scheme cannot produce useful information from current widely available real quantum computers due to error rate in multi-qubit gates.

Recommended Citation

Liu, YuXuan, "Simulation of Schrodinger's equation on Quantum Computers" (2022). Senior Theses. 1586.
https://digitalcommons.bard.edu/sr-theses/1586