  We study a classical model for the accumulation of errors in multi-qubit
quantum computations. By modeling the error process in a quantum computation
using two coupled Markov chains, we are able to capture a weak form of
time-dependency between errors in the past and future. By subsequently using
techniques from the field of discrete probability theory, we calculate the
probability that error quantities such as the fidelity and trace distance
exceed a threshold analytically. The formulae cover fairly generic error
distributions, cover multi-qubit scenarios, and are applicable to e.g. the
randomized benchmarking protocol. To combat the numerical challenge that may
occur when evaluating our expressions, we additionally provide an analytical
bound on the error probabilities that is of lower numerical complexity. Besides
this, we study a model describing continuous errors accumulating in a single
qubit. Finally, taking inspiration from the field of operations research, we
illustrate how our expressions can be used to e.g. decide how many gates one
can apply before too many errors accumulate with high probability, and how one
can lower the rate of error accumulation in existing circuits through simulated
annealing.
