Locality and digital quantum simulation of power-law interactions

TL;DR

This paper derives a new Lieb-Robinson bound for quantum systems with power-law interactions, providing a tighter effective light cone and analyzing the implications for digital quantum simulation efficiency.

Contribution

It introduces a novel Lieb-Robinson bound for power-law interactions and connects it to quantum simulation error bounds, improving understanding of information propagation.

Findings

New Lieb-Robinson bound for $1/r^eta$ interactions

Tighter effective light cone than previous bounds

Quantum simulation gate count scales better for $eta>3D$

Abstract

The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance $r$ as a power law, $1/r^\alpha$. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when $\alpha>3D$ (where $D$ is the number of dimensions).