  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.
