Statistical modeling of quantum error propagation

TL;DR

This paper introduces a new statistical model for quantum error propagation, providing algorithms and measures to analyze error patterns, and demonstrates its effectiveness through simulations and applications to quantum circuits.

Contribution

It presents a novel statistical framework for quantum error propagation, including graph constructions, complexity analysis, and practical bounds for specific quantum circuits.

Findings

Error pattern finding is in P, error distribution calculation is NP-complete.

Error propagation in transversal CNOT circuits is bounded by n/27.

Error threshold decreases significantly in surface code with global connections.

Abstract

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.