Non-Exponential Behaviour in Logical Randomized Benchmarking

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.