Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

TL;DR

This paper analyzes the spectral form factor of dual-unitary quantum circuits with disorder, showing it matches circular unitary ensemble predictions under certain conditions, revealing insights into quantum chaos and spectral statistics.

Contribution

It provides an explicit construction of the commutant for dual-unitary circuits and proves the spectral form factor matches CUE predictions for specific cases.

Findings

Spectral form factor equals CUE prediction for dual-unitary circuits with non-interacting gates.

Explicit construction of the commutant for dual-unitary circuits.

Results extend to circuits with time-reversal symmetry and higher moments.

Abstract

We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over ${\rm SU}(d)$, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time $t$ in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of $t$ sites. For general dual unitary circuits of qubits $(d=2)$ and a family of their extensions to higher $d>2$, we provide explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all $t$, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to a weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and to computing higher moments of the spectral form factor.