Variational Quantum Computation For Electronic Structure Calculations

Quantum computation technology provides an efficient pathway for simulating quantum systems.Quantum chemistry is regarded as one of the most promising fields to achieve quantum computational advantage.On a classical computer,chemical systems accurately simulated are limited to 20 to 30 orbitals due to the high order or even exponentially scaling computational cost of wave function based methods,while such simulations are expected to be performed using only 40 to 60 qubits on a quantum computer.The variational quantum eigensolver(VQE)is a typical quantum algorithm for chemical simulations on noisy intermedia-scale quantum(NISQ)devices.VQE implements an integrated procedure of quantum circuit evolution with classical numerical optimization and is capable of performing accurate simulations with a lower order computational complexity.Nevertheless,there is still a gap between the practical application of VQE in quantum chemical simulations,which is currently limited to molecular systems at small scale and demanding elegant software platforms for further developments.In this thesis,I present my contribution to narrowing this gap by developing quantum algorithms for band structure calculations of periodic systems and for efficient simulations of large molecules,and by developing a novel software platform assisting the studies of these algorithms.VQE encodes the electronic wave function into the quantum state of qubits using a parametric quantum circuit.The central ingredient of VQE is the quantum circuit ansatz,which is also recognized as ’wave function ansatz’.A good ansatz should be capable of approaching exact wave function at high accuracy,meanwhile easy to perform numerical optimizations.The most widely used wave function ansatz in molecular simulations on a quantum computer is the unitary coupled-cluster(UCC)ansatz.Combining UCC with iterative algorithms,accurate ground states can be obtained for small molecules,and excited-state properties can be calculated using post-VQE methods such as quantum subspace expansion(QSE).However,these methods cannot be directly implemented in periodic systems due to the so-called ’residual error’ which arises from the non-zero imaginary part of anti-Hermitian contracted Schrodinger equation(ACSE).Such errors become more significant if denser k point grids(more qubits)are involved.In this thesis,I present a new wave function ansatz which is able to systematically eliminate the residual error.On the top of the new ansatz,I present a projected quantum equation-of-motion(EOM)algorithm which is inspired from classical EOM coupledcluster(EOM-CC)method for calculating quasi-particle band structures.Demonstrative calculations for silicon and diamond are performed,showing good agreement with classical EOM-CC results.The simulations for periodic systems at high accuracy usually suffer from significant shortage in qubit resources due to the orbital-to-qubit mapping in common VQE ansatzes.In classical computational chemistry algorithms,orbitals(basis functions)corresponds to the dimension of the wave function represented using a tensor.The memory and computational overhead of such a high dimensional tensor can be greatly alleviated using the tensor network algorithm.Inspired from classical matrix product state ansatz,I present a quantum circuit matrix product(QCMPS)ansatz for VQE simulation of molecules,which maps orbitals to blocks of a quantum circuit instead of qubits.QCMPS successfully simulated the ground state of a 50-orbital molecule using only 6 qubits up to chemical accuracy.The results presented here indicate that QCMPS can provide a solution for simulating large chemical systems with limited qubit resources.Since the scale of and access to current NISQ devices are both limited,a software platform which performs simulations of quantum algorithms on a classical computer plays a crucial role in designing quantum algorithms for quantum chemistry applications.A good platform should be easy to use which provides user-friendly translations between classical chemistry and quantum simulation data.Also,it should be flexible to design custom quantum algorithm routines based on existing algorithms or user-defined methods.In addition,the simulation backend should be sufficiently efficient to perform quantum circuit simulations on classical computers.Currently,various quantum simulation packages are accessible.However,disadvantages in performance and functionalities are significant.In order to address these points and to explore the formalism of domestic quantum computation softwares,I present a newly-developed software package Q2 Chemistry for designing quantum algorithms for chemical applications.Q2 Chemistry implements a modular design,which is built on the top of three central modules:classical computational chemistry interface,quantum algorithm composer and quantum circuit executor.In addition to quantul circuit simulators,Q2Chemistry also reserves interfaces for the upcoming quantum hardware.The benchmark presented in this thesis indicates that Q2 Chemistry is efficient and suitable as a versatile platform for studying quantum computational chemical algorithms.