Solving Quantum Many Body Problems with a NISQ-era Quantum Computer

借助NISQ時期的量子電腦解決量子多體問題

Student thesis: Doctoral Thesis

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Award date1 Aug 2023

Abstract

Calculating the ground-state properties of quantum many-body systems exactly is beyond the reach of polynomial-time algorithms, whether classical or quantum. However, for quantum chemistry applications, the main goal is to efficiently prepare a quantum state that approximates the true ground state well enough. To this end, various classical algorithms have been proposed, such as Hartree-Fock (HF), Coupled-Cluster (CC), Configuration Interaction (CI), and Density Functional Theory (DFT), each with different assumptions and limitations. On the other hand, quantum computers are expected to offer significant advantages in finding the eigenstates of challenging Hamiltonians in chemistry and physics. However, these advantages are hard to realize in the current noisy intermediate-scale quantum era. Therefore, it is crucial to identify and explore the boundary between feasible and infeasible improvements in quantum chemistry on quantum computers. In this thesis, we will discuss how to apply quantum chemistry on quantum computers and suggest some strategies to exploit the strengths and avoid the weaknesses of this approach.

First, we discuss how to use second quantization and mapping to encode the problem Hamiltonian on the quantum computer. We propose a classically efficient algorithm that can balance the trade-off between the number of qubits and the classical resource requirement, compared to the existing works. Then we compare it with the current mapping methods in terms of the number of qubits, circuit depth, and the number of Pauli terms(Measurement).

Second, we propose the imaginary-time control scheme for finding the ground state, which can work with both repeated and shallow quantum circuits. Through numerical experiments on a wide range of realistic models, such as molecular systems, 2D Heisenberg models, and Sherrington–Kirkpatrick models, we show that imaginary-time control can significantly speed up the imaginary time evolution for all systems, and even achieve orders of magnitude acceleration for challenging molecular Hamiltonians with small energy gaps as remarkable special cases. With a proper choice of the control Hamiltonian, the new variational quantum algorithm does not require additional measurements compared to the original variational quantum imaginary-time algorithm.

Third, we use a measurement-friendly parameterized quantum circuit, where the number of parameters scales linearly with the system size, to extract eigen spectrum information with the help of classical clustering methods like K-means. Through numerical examples and theoretical analysis, we show that the quantum enhanced classical clustering can obtain approximate eigen spectrum more efficiently than the existing excited state methods. Moreover, it can be used as a subroutine to accelerate existing classical or quantum methods to find approximate eigenstates. We also provide a theoretical analysis that gives the bound of the total convergence time and numerical results that show that the quantum enhanced classical procedure can polynomially speed up the pure quantum procedure

Finally, many studies suggest that classical algorithms can solve quantum many-body problems efficiently when there is no sign problem. By carefully investigating how to solve the sign problem with the quantum computer and combining it with a neural network-based eigen solver, we show the advantage of a quantum-neural hybrid algorithm that uses the quantum computer to mitigate the sign problem, compared to pure quantum or classical algorithms.

Overall, this thesis study has initiated a new exploration of solving quantum many-body problems in the era of quantum computing and the potential applications that can lead to quantum advantages in the future.