A Research On Hybrid Classical-Quantum Algorithm And Application To Quantum Chemistry

Quantum computing is a new paradigm of computing that was first proposed by Feynman in 1982.It was firstly suggested to solve problems in quantum simulation,that is,simulate any quantum system with controllable quantum systems.Quantum com-puting as a solution to it is mainly motivated by the difficulties to simulate quantum systems with a classical computer.In the early stage of the quantum computing,algo-rithms focus on solving eigenstates and dynamical evolution have been proposed.Since the prototype of quantum computer has been built,we were and we will stay in the noisy intermediate-scale quantum era for a long period.Aiming to show quantum advantage with NISQ devices,a series of quantum algorithms for those devices have been de-veloped.In the NISQ era,hybrid classical-quantum algorithms,also called variational quantum algorithms have been proposed such as Variational Quantum Eigensolver and Quantum Approximate Optimization Algorithm.Those algorithms take advantages of the hybrid framework which utilize quantum and classical computing resource to tackle different tasks.For example,VQE has been used to calculate the ground state of Hamiltonians and applied in quantum chemistry while QAOA has been used in solving combinatorial optimizations.In this thesis,we will start with algorithms like VQE,QAOA,study how to solve problems in quantum chemistry by designing algorithms for NISQ devices.We also simulate our algorithms numerically with molecules such as H₂,LiH to show how those algorithms perform when calculating ground states and excited states as well as finding the equilibrium geometry of specific molecule.In the first chapter of this thesis,we will first make an introduction to the back-ground of noisy-intermediate-scale quantum era,then discuss the significance of study-ing variational quantum algorithms.Lastly we will introduce the innovative point of the content which we will study in thesis.In the second chapter of this thesis,we will first explain the basic concepts of quantum computation.Quantum algorithms proposed in the early days will also be in-troduced in this chapter.Then we will introduce variational quantum algorithms which is the base of the study in this thesis,including variational quantum eigensolver and quantum approximate optimization algorithm.In the third chapter of this thesis,we will propose an algorithm call mutual gra-dient descent,and the method we adopted is to utilize the derivative information of Hamiltonians under the framework of VQE.With this algorithm,the parameters of the parameterized wave function and Hamiltonians are optimized simultaneously.This al-gorithm is characterized by the ability to find the equilibrium geometry variationally with a quantum computer,and it can also control the consumption of quantum com-puting resource and classical computing resource.Besides,we proposed a differential equation method to calculate the potential energy surface of molecules.In the fourth chapter of this thesis,we will use energy variance as the object fucn-tion of the optimization algorithm.The variance is calculated with quantum covariance matrix and we will show the advantage of this method while calculating the excited states of the molecule.The last chapter is the summary and outlook of our work of this thesis.