Fig. 1: Measurement-free logical state teleportation with the [[4, 1, 2]]-code.

a Stabilizers S_Z, S_X and logical operators of the [[4, 1, 2]]-code. b Experimental logical quantum state tomography for FT logical state initialization. The black dashed boxes correspond to ideal values in a fault-free case. Any deviation from the black dashed boxes, e.g., unboxed gray, blue or brown areas, indicates that noise reduces the overall fidelity. c High-level circuit for measurement-based modular logical teleportation. The source (S) and target (T) code blocks are merged by measuring the joint logical \({X}_{{{{\rm{L}}}}}^{{{{\rm{S}}}}}{X}_{{{{\rm{L}}}}}^{{{{\rm{T}}}}}\)-operator via an auxiliary register (Aux.) and applying a Z-type feedback operation based on the measurement outcome (first green box). The two blocks are then split again by measuring \({Z}_{{{{\rm{L}}}}}^{{{{\rm{S}}}}}\) and applying an X-type operation to the target register conditioned on the measurement outcome (second green box). d Schematic illustration of modular state teleportation, where the source and target registers are never directly coupled to one another, but only interact via an auxiliary quantum register (Aux). e High-level circuits for measurement-free logical teleportation and experimental logical quantum state tomography. We replace the measurements and feed-forward operations with coherent feedback operations to teleport a state without mid-circuit measurements (blue). An additional H_L is applied to the target state using the circuit shown in orange. The reset operation can either be carried out explicitly by physically resetting the auxiliary qubits and reusing them afterwards, or implemented by replacing them with fresh qubits.