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

Abstract

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.

Figures (18)

• Figure 1: Protocol for generating frequency-bin photons and dual-rail cluster states. (a) Schematic diagram of the system. The resonator--filter system has been simplified as an effective resonator. $\Omega_{f0g1}$ and $\Omega_{h0e1}$ represent the external drives for the $\ket{f0}$--$\ket{g1}$ and $\ket{h0}$--$\ket{e1}$ transitions, respectively. (b) Energy-level diagram of the system. The two transitions we drive, $\ket{f0}$--$\ket{g1}$ and $\ket{h0}$--$\ket{e1}$ transitions, are shown as solid arrows. (c) Frequency-bin dual-rail encoding. A logical qubit is defined by regarding the exclusive single-photon occupation of the frequency mode $\ket{\omega_1}$ ($\ket{\omega_2}$) as logical $\ket{0}_\mathrm{L}$ ($\ket{1}_\mathrm{L}$). (d) Quantum circuit used to generate a dual-rail frequency-bin cluster state. The part of the circuit enclosed by the black dashed box generates one frequency-bin photon pair. Here, H represents the Hadamard gate, and X represents the X gate (the bit-flip gate). Finally, at the end of the sequence, the qubit is projected on the $X$-axis by applying a Hadamard gate and $Z$-axis measurement (enclosed by a green dashed box). (e) Pulse sequence which is equivalent to the black dashed box in (c): (i) a $\pi_{ef}$ pulse, (ii) a $\pi_{ge}$ pulse, (iii) a $\pi_{fh}$ pulse, (iv) a $\pi_{ef}$ pulse, (v) a $\pi_{fh}/2$ pulse, and (vi) two simultaneous pulses for driving the $\ket{f0}$--$\ket{g1}$ and $\ket{h0}$--$\ket{e1}$ transitions. (f) Graph representation of the generated dual-rail frequency-bin cluster state. Here, as an example, we show a state with four dual-rail logical qubits. The colors of the vertices represent different frequency channels. The number on each mode corresponds to its order in the bra--ket representation. It can be regarded as a 1D cluster state (the mode order is notated using Roman numerals) under frequency-bin encoding.
• Figure 2: Spectra of the emitted photon, $f(\omega)$, under different initial qubit states. The blue line corresponds to the qubit state prepared at $(\ket{g}+\ket{f})/\sqrt{2}$ state, and the red line corresponds to the qubit state prepared at $(\ket{e}+\ket{h})/\sqrt{2}$ state. They demonstrate that the frequency of the generated photons under two simultaneous drives depends only on the initially prepared qubit state. The purple line is the spectrum when the qubit is prepared in $(\ket{g}+\ket{e}+\ket{f}+\ket{h})/2$. The black dashed line is the expected average of the blue and red lines. All spectra are normalized by the same factor. Details can be found in Appendix \ref{['app_sub_sec:photon-generation-cali']}.
• Figure 3: Reconstructed density matrix of the generated two-logical-qubit frequency-bin cluster state. (a) Density matrix of the reconstructed state in the bra--ket notation. The blue (red) colored modes correspond to the $\ket{\omega_1}(\ket{\omega_2})$ modes. Each adjacent pair (i.e., at positions 0 and 1, 2 and 3, etc.) corresponds to two modes within the same time bin. (b) Density matrix of the reconstructed state, projected onto the logical subspace spanned by $\{\ket{\omega_1},\ket{\omega_2}\}$. Absolute values of the matrix elements are plotted in the diagonal and the lower-left triangle, and the complex arguments are plotted in the upper-right triangle.
• Figure 4: Fidelity of the generated photonic states: (a) without and (b) with photon loss correction, respectively. The blue dots are the fidelity of generated photonic states, which is reconstructed via state tomography. The black dots and dashed line represent the fidelities calculated based on the experimentally measured Pauli transfer matrix (PTM). The gray colored areas present their standard deviations.
• Figure 5: Localizable entanglement (LE) and its standard deviation between different modes in the generated states. (a) LE in a four-logical-qubit state. The lower-left part of the figure shows the LE between the physical modes, while the upper-right part corresponds to the LE between the corresponding logical qubits (indicated as I, II, III, and IV). The number in parentheses indicates the one-standard-deviation statistical uncertainty, referring to the uncertainty in the last digit of the LE value. (b) LE between the 0th and other physical modes in two-, three-, and four-mode states. The bottom and top horizontal axes represent the physical and logical qubit distances, respectively, between the 0th mode and the target mode. The first data point corresponds to zero logical distance. (c) LE between the logical qubit I and the other logical qubits in two-, three-, and four-mode states. In (b) and (c), the black dots and dashed line represent the LE calculated based on the experimentally measured Pauli transfer matrix (PTM). The gray colored areas present their standard deviations.