On Complexity and Duality

TL;DR

This paper investigates how complexity and duality influence the evolution of local and non-local operators in quantum systems, revealing nuanced behaviors in different models and highlighting the intricate relationship between non-locality, complexity, and duality.

Contribution

It uncovers the dynamics of operator complexity under duality mappings in quantum models, especially contrasting open and periodic chains, and elucidates their implications for understanding quantum complexity growth.

Findings

Non-local operators mimic local operator growth under duality.

Boundary term mappings reveal multiple complex operator branches.

Complexity saturation values differ significantly in parity-mixing operators.

Abstract

We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behavior that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examing periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.