Design rules for fault-tolerant multi-gate teleportation

Abstract

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 on the target register (weight $n+1$ including the control). We establish a design rule for fault-tolerant packet sizes in rotated surface codes of odd distance~$d$: a rigorous decoder-independent floor $\lfloor d/2\rfloor$, extended to $\lceil d/2\rceil$ by a local weight argument confirmed by simulation when the decoder is correlation-aware. Simulation with PyMatching shows the standard MWPM decoder built from the packet circuit's detector error model (DEM) naturally corrects the correlated error: at $n=\lfloor d/2\rfloor$ the packet matches or surpasses the per-link sequential LER for moderate-to-high ${\hat{I}}^3$, with the crossover $γ^\star$ decreasing as $d$ grows (from $γ^\star{\approx}17$ at $d=5$ to $γ^\star{<}1$ at $d=9$, extrapolated), whilst reducing the entanglement cost from $n$ ebits to~$1$. Packetisation wins when the network is the bottleneck (${\hat{I}}^3\gg 1$); at ${\hat{I}}^3\approx 1$ the packet is marginally worse ($\le 1.13\times$ at $d\in \{5,7\}$, reversing by $d=9$) as the $n-1$ extra local fan-out gates offset the network savings. No custom decoding algorithm is required: the standard MWPM decoder, built from the packet circuit's DEM, already encodes the correlation.

4 pages, 2 figures, 1 table