More on complexity of operators in quantum field theory

TL;DR

This paper generalizes the complexity measure of SU(n) operators in quantum field theory using a bi-invariant Finsler geometry, connecting it with Schatten p-norms and analyzing its behavior over time.

Contribution

It relaxes axioms to extend the complexity formula to Schatten p-norms and clarifies its relation to other geometries and the time evolution pattern.

Findings

Complexity formula generalized to Schatten p-norms.

Results consistent with right-invariant geometries like k-local geometry.

Complexity exhibits linear growth up to a saturation time.

Abstract

Recently it has been shown that the complexity of SU($n$) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten $p$-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as $k$-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU($n$) groups.