Quantum filter algorithms for solving ground-state problem of local Hamiltonians

Abstract

Simulating the quantum many-body systems is an important task in many fields, such as quantum chemistry, condensed-matter physics, and high-energy physics. However, it is difficult to solve this kind of task for conventional computers, especially when the system scale is large. Quantum computation provides an efficient way to simulate the quantum system of interest, may overcoming the large-size problem. One of the key problems in this field is to solve the ground state problem of local Hamiltonian. This thesis presents two approaches for this crucial task.

The first part of this study proposes an Inverse Iteration Quantum Eigensolver, which enhances the classical inverse power iteration method with the capabilities of quantum computing. A pivotal element of this method involves the construction of an inverse Hamiltonian as a linear combination of coherent Hamiltonian evolution. This task is achieved using a continuous-variable quantum mode, which enables the encoding of a linear combination as an integral into a quantum mode resource state. The algorithm’s effectiveness is demonstrated through numerical simulations for a range of physical systems, including molecules and quantum many-body models. A hybrid quantum-classical algorithm, leveraging continuous-variable resources for reducing coherent evolution time of Hamiltonians, is also introduced and compared with the purely quantum approach.

The latter part of the thesis introduces the Quantum Gaussian Filter (QGF), an algorithm that efficiently projects a superposed quantum state onto a target state, provided sufficient overlap exists between the two states. The QGF algorithm employs a Gaussian function of the system Hamiltonian as the filter operator. A hybrid quantum-classical implementation is presented, feasible on near-term quantum computers, which realizes the Quantum Gaussian Filter as a linear combination of Hamiltonian evolution at various time points. Significantly, the linear combination coefficients are determined classically and can be optimized post-processing. This algorithm is demonstrated using numerical simulations for the quantum Ising model under noise. A full-quantum realization of the Quantum Gaussian Filter with an ancillary continuous- variable is also provided for comparison, emphasizing the hybrid approach’s flexibility due to post-processing capabilities on classical computers.