Ground State Preparation via Qubitization

TL;DR

This paper introduces a quantum algorithm leveraging qubitization to efficiently prepare the ground state of a Hamiltonian by approximating imaginary time evolution, demonstrated on specific models.

Contribution

It presents a novel quantum protocol combining qubitization and Chebyshev polynomial expansion for ground state preparation.

Findings

Successfully prepares ground states of models tested

Uses qubitization to implement imaginary time evolution

Achieves projection onto ground state with initial overlap

Abstract

We describe a protocol for preparing the ground state of a Hamiltonian $H$ on a quantum computer. This is done by designing a quantum algorithm that implements the imaginary time evolution operator: $e^{-\tau H}$. The method relies on the so-called ``qubitization'' procedure of Low and Chuang which, assuming the existence of a unitary encoding of the Hamiltonian $H = \langle G| U_H |G\rangle$, produces a new operator $W_H$ whose moments are the Chebyshev polynomials of $H$ when projected on $|G\rangle$. Using this result and the expansion of $e^{-\tau H}$ in terms of Chebyshev polynomials we construct a circuit that implements an approximation of the imaginary time evolution operator which, at large time, projects any state on the ground state, provided a non-trivial initial overlap between the two. We illustrate our method on two models: the transverse field Ising model and a single qubit toy model.