Generation of frequency-bin-encoded dual-rail cluster states via time-frequency multiplexing of microwave photonic qubits

Cluster states are a class of multi-qubit entangled states with broad applications such as quantum metrology and one-way quantum computing. Here, we present a protocol to generate frequency-bin-encoded dual-rail cluster states using a superconducting circuit consisting of a fixed-frequency transmon qubit, a resonator and a Purcell filter. We implement time-frequency multiplexing by sequentially emitting co-propagating microwave photons of distinct frequencies. The frequency-bin dual-rail encoding enables erasure detection based on photon occupancy. We characterize the state fidelity using quantum tomography and quantify the multipartite entanglement using the metric of localizable entanglement. Our implementation achieves a state fidelity exceeding 50$\%$ for a cluster state consisting of up to four logical qubits. The localizable entanglement remains across chains of up to seven logical qubits. After discarding the erasure errors, the fidelity exceeds 50% for states with up to eight logical qubits, and the entanglement persists across chains of up to eleven qubits. These results highlight the improved robustness of frequency-bin dual-rail encoding against photon loss compared to conventional single-rail schemes. This work provides a scalable pathway toward high-dimensional entangled state generation and photonic quantum information processing in the microwave domain.