Design rules for fault-tolerant multi-gate teleportation

Multi-gate teleportation (MGT) packages $n$ remote gates into a single ebit via a 1-ebit fan-out quantum circuit, saving $n{-}1$ entangled pairs relative to sequential gate teleportation. The cost is a correlated failure mode: a single network fault propagates through the fan-out tree, injecting a weight-$n$ Pauli error. We derive a design rule for fault-tolerant packet sizes, $\nmax^{\text{corr}}(d) = \lceil d/2 \rceil$ for rotated surface codes of distance~$d$ with a correlation-aware decoder ($\nmax^{\text{naive}} = \lfloor d/2 \rfloor$ without), bounding how many gates can be packaged whilst preserving fault tolerance. Simulation with PyMatching shows that the standard MWPM decoder built from the packet circuit's noise model naturally corrects the correlated error: at network-to-local noise ratios $\gamma = \pnet/\pgate$ up to $100$, the packet matches or surpasses the per-link sequential LER at moderate-to-high $\gamma$, with the advantage growing with both $\gamma$ and $d$, whilst reducing the entanglement cost from $n$ ebits to~$1$. Packetisation wins when the network is the bottleneck ($\gamma \gg 1$); at $\gamma \approx 1$ the $n{-}1$ extra local fan-out gates offset the network savings. No custom decoder is required: the circuit-level noise model already encodes the correlation. These results enable noise-aware distributed circuit compilers to favour fan-out packetisation without sacrificing fault tolerance.