  We extend the recently introduced notion of tensor enumerator to the circuit
enumerator. We provide a mathematical framework that offers a novel method for
analyzing circuits and error models without resorting to Monte Carlo
techniques. We introduce an analogue of the Poisson summation formula for
stabilizer codes, facilitating a method for the exact computation of the number
of error paths within the syndrome extraction circuit of the code that does not
require direct enumeration. We demonstrate the efficacy of our approach by
explicitly providing the number of error paths in a distance five surface code
under various error models, a task previously deemed infeasible via simulation.
We also show our circuit enumerator is related to the process matrix of a
channel through a type of MacWilliams identity.
