Complexity growth for one-dimensional free-fermionic lattice models

TL;DR

This paper analytically and numerically investigates the growth of complexity in one-dimensional free-fermionic lattice models, revealing linear growth and saturation behaviors, with differences between short-range and long-range hopping models.

Contribution

It provides the first analytical lower bound on complexity growth in non-interacting fermionic lattice models and compares it with the upper bound, highlighting different growth regimes.

Findings

Lower bound grows linearly and saturates for short-range models.

For long-range models, the bound grows sub-linearly before saturation.

Upper bound also grows linearly and saturates, exceeding the lower bound.

Abstract

Complexity plays a very important part in quantum computing and simulation where it acts as a measure of the minimal number of gates that are required to implement a unitary circuit. We study the lower bound of the complexity [Eisert, Phys. Rev. Lett. 127, 020501 (2021)] for the unitary dynamics of the one-dimensional lattice models of non-interacting fermions. We find analytically using quasiparticle formalism, the bound grows linearly in time and followed by a saturation for short-ranged tight-binding Hamiltonians. We show numerical evidence that for an initial Neel state the bound is maximum for tight-binding Hamiltonians as well as for the long-range hopping models. However, the increase of the bound is sub-linear in time for the later, in contrast to the linear growth observed for short-range models. The upper bound of the complexity in non-interacting fermionic lattice models is calculated, which grows linearly in time even beyond the saturation time of the lower bound, and finally, it also saturates.