GHZ-preserving Gates and Optimized Purification Circuits

Greenberger-Horne-Zeilinger (GHZ) states are fundamental resources for entanglement-based protocols. However, the presence of noise, natually, is a big chanllenge when working with GHZ states. In this work, we introduce a framework for faster simulation and optimization of circuits that purify the entanglement structure of GHZ states. This approach significantly reduces computational complexity from O(n) in the stabilizer formalism to a constant O(1). By decomposing the distillation operations into two subgroups, our framework offers a more scalable solution for simulating large GHZ-purifying circuits. We further apply this method to optimize entanglement purification protocols, demonstrating improved fidelity with fewer resources compared to existing methods. Additionally, we extend our approach to circular and linear graph states, showcasing the versatility of this framework across different quantum systems. These findings have broad implications for quantum networks, where efficient entanglement manipulation is critical for tasks such as distributed quantum computation and quantum teleportation.