  Mid-circuit measurement (MCM) is a crucial ingredient in the development of
fault-tolerant quantum computation. While there have been rapid experimental
progresses in realizing MCM, a systematic method for characterizing noisy MCM
is still under exploration. In this work we develop an algorithm to
characterize noisy MCM, via a generalization of cycle benchmarking -- a
standard approach for characterizing the Pauli noise channel of Clifford gates.
The key idea is to use a joint Fourier transform on the classical and quantum
registers and then estimate parameters in the Fourier space, analogous to Pauli
fidelities used in cycle benchmarking. Furthermore, we develop a theory of the
noise learnability of MCM, which determines what information can be learned
about the noise model (in the presence of state preparation and measurement
noise) and what cannot, which shows that all learnable information can be
learned using our algorithm. As an application, we show how to use the learned
information to test the independence between measurement noise and state
preparation noise in a MCM. Finally, we conduct numerical simulations to
illustrate the practical applicability of the algorithm. Similar to cycle
benchmarking, we expect the algorithm to provide a useful toolkit that is of
experimental interest.
