Journal of High Energy Physics

, 2019:63 | Cite as

Cayley graphs and complexity geometry

  • Henry W. LinEmail author
Open Access
Regular Article - Theoretical Physics


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 identity, 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 δ-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.


AdS-CFT Correspondence Random Systems 


Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Jadwin HallPrinceton UniversityPrincetonU.S.A.
  2. 2.Facebook AI Research, FacebookNew YorkU.S.A.

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