Randomized Benchmarking, Correlated Noise, and Ising Models

We compute the expected randomized benchmarking sequence fidelity for a system subject to Gaussian time-correlated noise. For single qubit benchmarking we show that the expected sequence fidelity is given by the partition function of a long-range coupled spin-one Ising model, with each site in the Ising model corresponding to a free evolution interval. For d-state systems, the expected sequence fidelity is given by an Ising-like model partition function whose site variables are given by the weights of the adjoint representation of SU(d). A high effective temperature expansion for the partition function in the single qubit case shows decay of sequence fidelity varying from exponential for uncorrelated noise to a power law for quasistatic noise. Fitting an exponential to the sequence fidelity decay under correlated noise gives unreliable estimates of the average gate error rate.