Feasibility of Logical Bell State Generation in Memory Assisted Quantum Networks
This study explores the feasibility of utilizing quantum error correction (QEC) to generate and store logical Bell states in heralded quantum entanglement protocols, crucial for quantum repeater networks. Two lattice surgery-based protocols (local and non-local) are introduced to establish logical Bell states between distant nodes using an intermediary node. We simulate the protocols using realistic experimental parameters, including ion trap memories, noisy optical channels, frequency conversion, and non-destructive detection of photonic qubits. The study evaluates rotated and planar surface codes alongside Bacon-Shor codes for small code distances ($d = 3, 5$) under depolarizing and physical noise models. Pseudo-thresholds are identified, with physical error rates above $p_{\text{err}} \sim 10^{-3}$ offering no advantage over unencoded Bell states under depolarizing noise. Pseudo-thresholds are also reevaluated in terms of gate error rates $p_{\text{err}_H}$, $p_{\text{err}_{CX}}$, and $p_{\text{err}_M}$. For a distance of 1 km between the end node and the intermediary, an advantage over unencoded Bell-state heralded protocols requires reducing gate error rates by an order of magnitude ($0.1p_{\text{err}_H}$, $0.1p_{\text{err}_{CX}}$, and $0.1p_{\text{err}_M}$). These results highlight the need for significant hardware improvements to implement logical Bell state protocols with quantum memories. Additionally, the non-local protocol rate was analyzed, achieving rates up to $(32.53 \pm 1.53) \, \mathrm{Hz}$ over distances of $1$ to $80 \, \mathrm{km}$ between the end node and the intermediary node.