  We construct a gate and time-independent noise model that results in the
output of a logical randomized benchmarking protocol oscillating rather than
decaying exponentially. To illustrate our idea, we first construct an example
in standard randomized benchmarking where we assume the existence of ``hidden''
qubits, permitting a choice of representation of the Clifford group that
contains multiplicities. We use the multiplicities to, with each gate
application, update a hidden memory of the gate history that we use to
circumvent theorems which guarantee the output decays exponentially. In our
focal setting of logical randomized benchmarking, we show that the presence of
machinery associated with the implementation of quantum error correction can
facilitate non-exponential decay. Since, in logical randomized benchmarking,
the role of the hidden qubits is assigned to the syndrome qubits used in error
correction and these are strongly coupled to the logical qubits via a decoder.
