Quantum Circuit Tensors and Enumerators With Applications to Quantum Fault Tolerance

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 exact number of error paths in a distance five surface code under various error models, a task infeasible via direct enumeration. We also show our circuit enumerator is related to the process matrix of a channel through a type of MacWilliams identity.