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

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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.

Citation

Sewell, T., Bapat, A., & Jordan, S. (2022). “Estimating gate complexities for the site-by-site preparation of fermionic vacua”. arXiv preprint arXiv:2207.01692.

@article{sewell2022estimating,
  title={Estimating gate complexities for the site-by-site preparation of fermionic vacua},
  author={Sewell, Troy and Bapat, Aniruddha and Jordan, Stephen},
  journal={arXiv preprint arXiv:2207.01692},
  year={2022}
}