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Home > Journal of High Energy Physics (Springer/SISSA) > Cayley graphs and complexity geometry |

Lin, Henry (0000 0001 2097 5006, Jadwin Hall, Princeton University, Princeton, NJ, 08540, U.S.A.) (0000 0004 0615 529X, Facebook AI Research, Facebook, New York, NY, 10003, U.S.A.)

13 February 2019

**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 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.

**Published in: ****JHEP 1902 (2019) 063**
**Published by: **Springer/SISSA

**DOI: **10.1007/JHEP02(2019)063

**arXiv: **1808.06620

**License: **CC-BY-4.0