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arXiv:2102.05102

Morse Potential on a Quantum Computer for Molecules and Supersymmetric Quantum Mechanics

Joshua Apanavicius, Feng, Yuan, Yasmin Flores, Mohammad Hassan, Michael McGuigan

Published Feb 9, 2021• 3 citations
arXivHTMLPDF

Abstract

In this paper we discuss the Morse potential on a quantum computer. The Morse potential is useful to describe diatomic molecules and has a finite number of bound states which can be measured through spectroscopy. It is also a example of an exactly soluble potential using supersymmetric quantum mechanics. Using the the supersymmetric quantum mechanics formalism one can derive a heirachy of Hamiltonians such that the ground state of the next rung on the heirarchy yeids the first excited state of the hamiltonian below it. Using this method one can determine all the states of the Morse potential by calculating all the ground states of the sequence of Hamiltonians in the heirarchy. We use the IBM QISKit software together with the Variational Quantum Eiegensolver (VQE) algorithm to calculate the ground state and first excited state energy of the Morse potential and find agreement with the exact expression for the bound state energies of the Morse Potential. We analyze different optimizers to study the numerical effect on the calculations. Finally we perform quantum computations for diatomic and triatomic molecules to illustrate the application of these techniques on near term quantum computers and find excellent agreement with experimental data.

DOI
10.48550/arxiv.2102.05102

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In this paper we discuss the Morse potential on a quantum computer. The Morse potential is useful to describe diatomic molecules and has a finite number of bound states which can be measured through spectroscopy. It is also a example of an exactly soluble potential using supersymmetric quantum mechanics. Using the the supersymmetric quantum mechanics formalism one can derive a heirachy of Hamiltonians such that the ground state of the next rung on the heirarchy yeids the first excited state of the hamiltonian below it. Using this method one can determine all the states of the Morse potential by calculating all the ground states of the sequence of Hamiltonians in the heirarchy. We use the IBM QISKit software together with the Variational Quantum Eiegensolver (VQE) algorithm to calculate the ground state and first excited state energy of the Morse potential and find agreement with the exact expression for the bound state energies of the Morse Potential. We analyze different optimizers to study the numerical effect on the calculations. Finally we perform quantum computations for diatomic and triatomic molecules to illustrate the application of these techniques on near term quantum computers and find excellent agreement with experimental data.
Quantum computing has emerged as a promising technology for solving problems that are intractable for classical computers. In this study, we introduce quantum computing and implement the Variational Quantum Eigensolver (VQE) algorithm using Qiskit on the IBM Quantum platform to calculate the ground state energy of a hydrogen molecule. We provide a theoretical framework of quantum mechanics, qubits, quantum gates, and the VQE algorithm. Our implementation process is described, and we simulate the results. Additionally, experiments are conducted on the IBM Quantum platform, and the results are analyzed. Our fi ndings demonstrate that VQE can effi ciently calculate molecular properties with high accuracy. However, limitations and challenges in scaling the algorithm for larger molecules are also identifi ed. This work contributes to the growing body of research on quantum computing and highlights the potential applications of VQE for real-world problem-solving.
We find theoretical results on energy eigenvalues and corresponding supersymmetric Hamiltonians reflect contradictory behavior for negative values of A. furthermore the resulting supersymmetric partners potentials can be model scattering states instead of bound states, However following the literature ( Flu 1979);J and A (J. Physics A 2020), we suggest a correct form of super potential , which remains valid for both positive and negative values of constants. A from this in complex space also the eigenvalues remain invariant without discussion of T-symmetry as the previous discussion appears incomplete.
Abstract Solving the electronic structure for multi-electronic systems is a hard problem. Even for small atoms and molecules, approximations have to be made in order to solve numerically the Schrödinger equation. Although different methods have been developed to take into account electron correlations, their computational cost reduces their feasibility. Quantum simulation provides an alternative to traditional computational methods for enquiring the electronic structure of molecules. Specifically, the Variational Quantum Eigensolver (VQE) algorithm provides a hybrid quantum-classical algorithm for the implementation on current near term quantum devices. In this work, we explore the implementation of VQE on Qiskit for calculating the ground-state energy of diatomic Hydrogen molecule.
Key properties of physical systems can be described by the eigenvalues of matrices that represent the system. Computational algorithms that determine the eigenvalues of these matrices exist, but they generally suffer from a loss of performance as the matrix grows in size. This process can be expanded to quantum computation to find the eigenvalues with better performance than the classical algorithms. One application of such an eigenvalue solver is to determine energy levels of a molecule given a matrix representation of its Hamiltonian using the variational principle. Using a variational quantum eigensolver, we determine the ground state energies of different molecules. We focus on the choice of optimization strategy for a Qiskit simulator on low-end hardware. The benefits of several different optimizers were weighed in terms of accuracy in comparison to an analytic classical solution as well as code efficiency.
Knowing the ground state energies of molecules and atoms gives scientists critical insight into these particles' behavior in reactions, thermodynamic and molecular properties, and excitation energies.However, due to the quantum mechanical nature of particles, finding the ground state of a system is intractable, both mathematically and experimentally.Computational methods are utilized to estimate the ground state energies.As we approach the limits of classical computers, quantum computers are viewed as promising alternatives that present a bright future for computational chemistry.In this paper, I review classical and quantum algorithms/methods for computational chemistry.Then, I discuss the current topics in using quantum computers for computational chemistry by analyzing the performance of the Variational Quantum Eigensolver (VQE) in different contexts on IBM's noisy quantum computer simulator, "QasmSimulator," and one of IBM's quantum computers, ibmq_belem.This review presents a theoretical and practical view of quantum computational chemistry.It allows the reader to learn the theory behind the algorithms and implement them with the guidance of the presented code and its detailed explanations.The purpose of this paper is to leave the reader with the necessary theory and adequate programming reference to implement the algorithms in various contexts.
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 use this translation to compute finite temperature quantities such as the average energy as a function of temperature. We do this by constructing a 0+1 dimensional thermal field theory, constructing a thermal double of the system and using tensor products to construct operators in the system coupled to a heat bath. We also discuss how to include a chemical potential in the quantum computations. We then connect the 0+1 formalism to gauge theory by using an effective matrix model description used in nuclear theory. We study effective potentials for components of the gauge field and use the VQE algorithm to calculate the ground state energies. We also visualize the wave functions from the eigensolver and make comparisons to theoretical results obtained with continuous operators.