Estimating gate complexities for the site-by-site preparation of fermionic vacua

TL;DR

This paper analyzes the overlap of ground states in fermionic Hamiltonians to evaluate the efficiency of site-by-site quantum state preparation, proposing a recursive alternative to improve the process.

Contribution

It provides analytical and numerical insights into ground state overlaps for fermionic systems, and introduces a recursive method for state preparation.

Findings

Overlap remains large for most models up to large lattice sizes.

Near phase transitions or gapless modes, the overlap decreases significantly.

A recursive alternative to site-by-site preparation is proposed.

Abstract

An important aspect of quantum simulation is the preparation of physically interesting states on a quantum computer, and this task can often be costly or challenging to implement. A digital, ``site-by-site'' scheme of state preparation was introduced in arXiv:1911.03505 as a way to prepare the vacuum state of certain fermionic field theory Hamiltonians with a mass gap. More generally, this algorithm may be used to prepare ground states of Hamiltonians by adding one site at a time as long as successive intermediate ground states share a non-zero overlap and the Hamiltonian has a non-vanishing spectral gap at finite lattice size. In this paper, we study the ground state overlap as a function of the number of sites for a range of quadratic fermionic Hamiltonians. Using analytical formulas known for free fermions, we are able to explore the large-$N$ behavior and draw conclusions about the state overlap. For all models studied, we find that the overlap remains large (e.g. $> 0.1$) up to large lattice sizes ($N=64,72$) except near quantum phase transitions or in the presence of gapless edge modes. For one-dimensional systems, we further find that two $N/2$-site ground states also share a large overlap with the $N$-site ground state everywhere except a region near the phase boundary. Based on these numerical results, we additionally propose a recursive alternative to the site-by-site state preparation algorithm.