  Randomized benchmarking has emerged as a popular and easy-to-implement
experimental technique for gauging the quality of gate operations in quantum
computing devices. A typical randomized benchmarking procedure identifies the
exponential decay in the fidelity as the benchmarking sequence of gates
increases in length, and the decay rate is used to estimate the fidelity of the
gate. That the fidelity decays exponentially, however, relies on the assumption
of time-independent or static noise in the gates, with no correlations or
significant drift in the noise over the gate sequence, a well-satisfied
condition in many situations. Deviations from the standard exponential decay,
however, have been observed, usually attributed to some amount of time
correlations in the noise, though the precise mechanisms for deviation have yet
to be fully explored. In this work, we examine this question of randomized
benchmarking for time-correlated noise---specifically for time-correlated
dephasing noise for exact solvability---and elucidate the circumstances in
which a deviation from exponential decay can be expected.
