Fig. 1. Resource state and fusion operation. (a) Two steps of FBQC^[28], the black lines connecting photons indicate that a control-Z operation was applied to the two photonic qubits each initially prepared in the + state; (b) schematic of a fusion measurement implemented via a probabilistic linear-optical Bell state analyzer^[30]; (c) projective states and parity outcomes for the fusion success and failure cases, with ψ±=(01±10)/2, ϕ±=(00±11)/2

Fig. 2. Fault-tolerant fusion network^[22]. (a) An example of a fusion network; (b) the symptom graph corresponding to Fig. 2(a), vertices represent parity checks (symptoms) and the edges represent the measurements of the corresponding Pauli operators; (c) the 6-ring fault-tolerant fusion network; (d) the symptom graph corresponding to Fig. 2(c), each parity check has 12 incident edges, the vertical edges correspond to the XX type fusion outcome and diagonal edges correspond to the ZZ outcome; (e) an example of a 2D fusion network; (f) an example architecture which could create the fusion network shown in Fig. 2(e)

Fig. 3. Different FBQC schemes. (a) 4-star fusion network^[22]; (b) foliated Floquet color code^[23]; (c) fault-tolerant regions for different fusion-based constructions,two situations are considered: the fusion result is erased due to fusion failure or photon loss, and the fusion result is flipped due to Pauli error; (d) the 6-ring resource state with every qubit encoded in a (2,2)-Shor code^[22]; (e) encoded-fusion scheme^[25]; (f) photon loss thresholds for different fusion success probabilities^[25]

Fig. 4. Schemes for generating high-dimensional multiphoton entangled states. (a) The schematic diagram of preparing four-photon three-dimensional GHZ state is shown on the left, and the post-selection process of any dimension using auxiliary entanglement is shown on the right^[65]; (b) entanglement generated by path identity^[72]; (c) time-bin entangled photons generated by quantum frequency comb^[82]

Fig. 5. Deterministic generation of resource state with quantum emitters. (a) A possible schematic diagram of the energy level structure of a quantum emitter^[26]; (b) quantum circuits to generate linear and encoded linear chain states; (c) using a deterministic source of caterpillar graph states followed by fusion gate to produce a 24-photon 6-ring resource state encoded with (2,2)-Shor code^[96]