Construction and the ergodicity properties of dual unitary quantum circuits

TL;DR

This paper explores the properties of dual unitary quantum circuits, including their construction, connections to various mathematical and physical theories, and their ergodicity and scrambling behavior, revealing surprising non-ergodic phenomena.

Contribution

It introduces new constructions of dual unitary gates, analyzes their ergodic properties, and uncovers unexpected non-ergodic behavior in certain models, including perfect tensor circuits.

Findings

Dual unitary circuits are exactly solvable for correlation functions.

Non-ergodic behavior can occur in multi-site correlations despite chaotic one-site correlations.

Certain perfect tensor circuits exhibit quantum revivals and non-ergodic features.

Abstract

We consider one dimensional quantum circuits of the brickwork type, where the fundamental quantum gate is dual unitary. Such models are solvable: the dynamical correlation functions of the infinite temperature ensemble can be computed exactly. We review various existing constructions for dual unitary gates and we supplement them with new ideas in a number of cases. We discuss connections with various topics in physics and mathematics, including quantum information theory, tensor networks for the AdS/CFT correspondence (holographic error correcting codes), classical combinatorial designs (orthogonal Latin squares), planar algebras, and Yang-Baxter maps. Afterwards we consider the ergodicity properties of a special class of dual unitary models, where the local gate is a permutation matrix. We find an unexpected phenomenon: non-ergodic behaviour can manifest itself in multi-site correlations, even in those cases when the one-site correlation functions are fully chaotic (completely thermalizing). We also discuss the circuits built out of perfect tensors. They appear locally as the most chaotic and most scrambling circuits, nevertheless they can show global signs of non-ergodicity: if the perfect tensor is constructed from a linear map over finite fields, then the resulting circuit can show exact quantum revivals at unexpectedly short times. A brief mathematical treatment of the recurrence time in such models is presented in the Appendix by Roland Bacher and Denis Serre.