Fig. 5: Fault tolerance and scaling to higher code distances for measurement-free logical state teleportation.

a We prepare an auxiliary two-qubit GHZ-state to prevent single faults on auxiliary qubits from causing a logical failure. In addition, we map two stabilizer-equivalent representations of the joint logical operator with fully disjoint qubit support onto the auxiliary registers (1, 2). The same strategy is used in step 3, where two equivalent but fully disjoint logical Z-operators of the source register are mapped onto two physical qubits. b Numerically determined scaling of the logical infidelity for FT and non-FT logical teleportation and the H_L-gate, averaged over initial states \({| 0\rangle }_{{\mathrm{L}}}\) and \({|+\rangle }_{{\mathrm{L}}}\). We fix the error parameters \(\vec{p}=({p}_{{{{\rm{1}}}}},{p}_{{{{\rm{2}}}}},{p}_{{{{\rm{m}}}}},{p}_{{{{\rm{i}}}}},{p}_{{{{\rm{idle}}}}})\) to experimental error rates^7,17 and scale these with a common improvement factor λ. We identify a quadratic scaling of the infidelity for the FT protocols with λ, which indicates—as expected—that no single fault leads to a logical failure. The FT teleportation protocol outperforms its non-FT counterpart already for the current experimental noise parameters (λ = 1). The inset shows the logical infidelities at λ = 1 obtained from the experiment (darker color) and numerical simulations (lighter color). c Scaling measurement-free state teleportation to surface codes with higher distances d > 2. Each string of qubits connecting opposing boundaries supports a representation of a logical \({X}_{{{{\rm{L}}}}}^{{{{\rm{S}}}}}\) (upper lattice) and \({X}_{{{{\rm{L}}}}}^{{{{\rm{T}}}}}\) (lower lattice); one exemplary representation is shown in red. There are d equivalent representations that have fully disjoint support. Each one can be mapped onto an auxiliary d-qubit GHZ-state, and coherent feedback steps can be applied, which are controlled by the state on d physical auxiliary qubits.