Variational quantum algorithms for local Hamiltonian problems

TL;DR

This paper explores variational quantum algorithms, especially VQE, for solving local Hamiltonian problems, analyzing their performance, limitations like barren plateaus, and applications in quantum machine learning and state fidelity estimation.

Contribution

It provides new numerical insights into VQE performance on spin and Hubbard models, derives bounds on derivative variance related to circuit structure, and explores quantum classification and fidelity bounds.

Findings

VQE effectively approximates ground states in tested models.

Barren plateau phenomenon is linked to circuit and Hamiltonian structure.

Derived lower bounds on derivative variance depend on ansatz and Hamiltonian decomposition.

Abstract

Variational quantum algorithms (VQAs) are a modern family of quantum algorithms designed to solve optimization problems using a quantum computer. Typically VQAs rely on a feedback loop between the quantum device and a classical optimization algorithm. The appeal of VQAs lies in their versatility, resistance to noise, and ability to demonstrate some results even with circuits of small depth. We primarily focus on the algorithm called variational quantum eigensolver (VQE), which takes a qubit Hamiltonian and returns its approximate ground state. We first present our numerical findings regarding VQE applied to two spin models and a variant of the Hubbard model. Next, we briefly touch the topic of quantum machine learning by developing a quantum classifier to partition quantum data. We further study the phenomenon of vanishing derivatives in VQAs, also known as barren plateaus phenomenon. We derive a new lower bound on variance of the derivatives, which depends on the causal structure of the ansatz circuit and the individual terms entering the Pauli decomposition of the problem Hamiltonian. In the final chapter of the thesis, we present our results on bounding the fidelity of experimentally prepared Clifford states using their parent Hamiltonians.