Fig. 7: Encoded fusion (blue connection) between qubits (green circles) of two six-ring graph states.
From: Boosted Bell-state measurements for photonic quantum computation
(Figure adapted from ref.1). Using the (2, 2)-Shor encoding, each logical qubit is encoded in four entangled physical qubits, denoted with index A to D. All physical fusions measure the parity of the stabilizer generators XX and ZZ on the physical qubits 1i and \({4}_{{\rm{i}}}^{{\prime} }\). Upon fusion failure, we lose the parity of ZZ in fusions A and C and the parity of XX in fusions B and D, guaranteeing a fusion failure probability of the encoded fusion of \({p}_{{\rm{fail}},{\rm{enc}}}={p}_{{\rm{fail}},{\rm{phys}}}^{2}\), where pfail,phys denotes the failure probability of the physical fusion. If there are no other sources of error (apart from fusion failure), we always lose the ZZ measurement outcome of the encoded fusion if it fails.
