# Bounds on success probability of an algorithm under depolarizing noise

The problem I have is that, I have a quantum circuit performing a certain task (e.g. Shor's algorithm) and the circuit is subject to depolarizing noise, Pauli errors, after each gate. And I want to investigate the correlation between the success probability of the final output and physical error rate, given no error correction is applied. I have trouble with such analysis ... since for each gate applied there is a new conditional probability, it gets quite messy ... and I don't think it's a Markov chain? I think this question arises very naturally but I'm unable to find resources on such theoretical analysis. Could anyone give suggestions or what mathematical tools might be useful? Thanks.

For example, find a way to convert from depolarization strength to probability-of-gate-failing $$p$$. If each gate has a probability $$p$$ of failing and $$\overline{p}$$ of succeeding, and there are $$g$$ gates, then to first order the algorithm as a whole has a probability $$\overline{p}^g$$ of success. This ignores the fact that errors can cancel each other out, but that's only important if $$p$$ is large (like, larger than 2%) and if $$p$$ is large you're in nothing-is-going-to-work-this-estimate-doesn't-matter territory.
Actually, since $$p$$ must be small for anything to work, you can use the approximation $$\overline{\overline{p}^g} \approx p \cdot g$$ for the algorithm's probability of failure. Technically this approximation completely breaks when the gate count becomes large (it can return probabilities above 100%) but, similar to the large-p-doesn't-matter thing, if $$g$$ is large enough that this approximation starts breaking then you are probably in nothing-is-going-to-work-this-estimate-doesn't-matter territory.
Another benefit of the $$p \cdot g$$ approximation is that it only overestimates the amount of error. So you can use it to prove upper bounds on the error. You can use it to show the algorithm will succeed a certain amount, and then the actual algorithm will succeed even more than that. It's good for showing feasibility.