Cayley graphs and complexity geometry
Abstract
The basic idea of quantum complexity geometry is to endow the space of unitary matrices with a metric, engineered to make complex operators far from the origin, and simple operators near. By restricting our attention to a finite subgroup of the unitary group, we observe that this idea can be made rigorous: the complexity geometry becomes what is known as a Cayley graph. This connection allows us to translate results from the geometrical group theory literature into statements about complexity. For example, the notion of -hyperbolicity makes precise the idea that complexity geometry is negatively curved. We report an exact (in the large N limit) computation of the average complexity as a function of time in a random circuit model.
Cite
@article{arxiv.1808.06620,
title = {Cayley graphs and complexity geometry},
author = {Henry W. Lin},
journal= {arXiv preprint arXiv:1808.06620},
year = {2019}
}
Comments
16 pages, 3 figures