  As quantum devices make steady progress towards intermediate scale and
fault-tolerant quantum computing, it is essential to develop rigorous and
efficient measurement protocols that account for known sources of noise. Most
existing quantum characterization protocols such as gate set tomography and
randomized benchmarking assume the noise acting on the qubits is Markovian.
However, this assumption is often not valid, as for the case of 1/f charge
noise or hyperfine nuclear spin noise. Here, we present a general framework for
quantum process tomography (QPT) in the presence of time-correlated noise. We
further introduce fidelity benchmarks that quantify the relative strength of
different sources of Markovian and non-Markovian noise. As an application of
our method, we perform a comparative theoretical and experimental analysis of
silicon spin qubits. We first develop a detailed noise model that accounts for
the dominant sources of noise and validate the model against experimental data.
Applying our framework for time-correlated QPT, we find that the number of
independent parameters needed to characterize one and two-qubit gates can be
compressed by 10x and 100x, respectively, when compared to the fully generic
case. These compressions reduce the amount of tomographic measurements needed
in experiment, while also significantly speeding up numerical simulations of
noisy quantum circuit dynamics compared to time-dependent Hamiltonian
simulation. Using this compressed noise model, we find good agreement between
our theoretically predicted process fidelities and two qubit interleaved
randomized benchmarking fidelities of 99.8% measured in recent experiments on
silicon spin qubits. More broadly, our formalism can be directly extended to
develop efficient and scalable tuning protocols for high-fidelity control of
large-arrays of quantum devices with non-Markovian noise.
