Generation of Frequency-Bin-Encoded Dual-Rail Cluster States via Time-Frequency Multiplexing of Microwave Photonic Qubits

Cluster states are a class of multiqubit 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 copropagating microwave photons, where each time bin contains a dual-rail qubit encoded in two distinct frequency modes, and adjacent time bins are entangled to form the cluster state. 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 localizable entanglement. Our implementation achieves a state fidelity exceeding <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>50</a:mn> <a:mi mathvariant="normal">%</a:mi> </a:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mn>50</d:mn> <d:mi mathvariant="normal">%</d:mi> </d:math> for states with up to eight logical qubits, and the entanglement persists across chains of up to 11 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.