Complexity phase diagram for interacting and long-range bosonic Hamiltonians

TL;DR

This paper introduces a complexity phase diagram for bosonic lattice models, showing how their classical simulability changes over time and with interactions, revealing two types of phase transitions related to quantum correlation spread.

Contribution

It extends previous complexity classifications to include on-site interactions and long-range hopping, establishing a detailed phase diagram with bounds on the transition.

Findings

Classical simulability transitions from easy to hard over time.

On-site interactions do not shift the transition point but change its nature.

Two types of phase transitions, sharp and coarse, are identified.

Abstract

We classify phases of a bosonic lattice model based on the computational complexity of classically simulating the system. We show that the system transitions from being classically simulable to classically hard to simulate as it evolves in time, extending previous results to include on-site number-conserving interactions and long-range hopping. Specifically, we construct a "complexity phase diagram" with "easy" and "hard" phases, and derive analytic bounds on the location of the phase boundary with respect to the evolution time and the degree of locality. We find that the location of the phase transition is intimately related to upper bounds on the spread of quantum correlations and protocols to transfer quantum information. Remarkably, although the location of the transition point is unchanged by on-site interactions, the nature of the transition point changes dramatically. Specifically, we find that there are two kinds of transitions, sharp and coarse, broadly corresponding to interacting and noninteracting bosons, respectively. Our work motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena.