Statistical modeling of quantum error propagation

In this paper, I design a new statistical abstract model for studying quantum error propagation. For each circuit, I give the algorithm to construct the Error propagation space-time graph(\textbf{EPSTG}) graph as well as the bipartite reverse spanning graph (\textbf{RSG}). Then I prove that the problem of finding an error pattern is $\mathcal{P}$ while calculate the error number distribution is $\textit{NP-complete}$. I invent the new measure for error propagation and show that for widely used transversal $CNOT$ circuit in parallel, the shift of distribution is bounded by $\frac{n}{27}$, where $n$ is the number of physical qubits. The consistency between the result of qiskit simulation and my algorithm justify the correctness of my model. Applying the framework to random circuit, I show that there is severe unbounded error propagation when circuit has global connection. We also apply my framework on parallel transversal logical $CNOT$ gate in surface code, and demonstrate that the error threshold will decrease from $0.231$ to $0.134$ per cycle.