Frequency-Bin Dual-Rail Encoding

• Frequency-bin dual-rail encoding is a quantum information method that maps qubit logical states onto two well-separated frequency channels.
• It leverages techniques such as dense wavelength-division multiplexing and precise spectral control to achieve robust storage, retrieval, and manipulation of photonic qubits.
• Experimental implementations in optical, telecom, and microwave platforms demonstrate its potential for scalable quantum communication, quantum memory, and advanced quantum processing.

Frequency-bin dual-rail encoding is a method for photonic quantum information encoding in which quantum states are mapped onto the occupation of two discrete, well-separated frequency modes––the “rails”––serving as the logical basis states of a qubit. This structure parallels the dual-rail encoding used in spatial or polarization degrees of freedom, with the essential distinction that information is encoded in the frequency domain. The approach leverages the compatibility of frequency-bin encoding with dense wavelength-division multiplexing (DWDM), resilience to certain noise channels, and the ability to implement large-scale, high-dimensional quantum protocols. Recent advances have made frequency-bin dual-rail encoding experimentally accessible across optical, telecom, and even microwave photonic platforms. Its technical implementation and performance are crucial for quantum memory, quantum communication, and scalable quantum information processing.

1. Fundamental Architecture of Frequency-Bin Dual-Rail Encoding

Frequency-bin dual-rail encoding represents logical quantum states—such as |0⟩ and |1⟩—using the presence of a photon in one of two distinct frequency channels. Mathematically, a general dual-rail encoded qubit can be represented as

$$|\psi\rangle = \alpha |1_{\omega_0} 0_{\omega_1}\rangle + \beta |0_{\omega_0} 1_{\omega_1}\rangle,$$

where $|1_{\omega_0} 0_{\omega_1}\rangle$ denotes a single photon in frequency $\omega_0$ and none in $\omega_1$ (logical |0⟩), and vice versa for logical |1⟩. Superpositions and entangled states can be constructed analogously, enabling coherent operations and measurements in the frequency domain.

This encoding takes particular advantage of the spectral separability of the rails, allowing robust path multiplexing, resistance to certain types of cross-talk, and native compatibility with telecom infrastructures.

2. Physical Realizations and Underlying Mechanisms

Optical Gradient Echo Memory (GEM) Approach

A canonical realization in the optical domain uses Zeeman-split Raman absorption lines in cold atomic ensembles to provide two broad, addressable absorption features. A bias magnetic field $B_0$ introduces Zeeman splitting, with the two absorption lines displaced by $\pm \delta_0$ ($\delta_0 \approx 1.4$ MHz/G $\times B_0$). Applying a magnetic field gradient $B_z(z)$ produces linewidth broadening that enables efficient storage and retrieval of the two frequency rails via the gradient echo memory protocol (Higginbottom et al., 2016).

The coupled Maxwell–Bloch equations for storage and retrieval, incorporating the effective coupled mode for each frequency channel, are:

$$\begin{align*} \partial_t \sigma_1 &= -(\gamma_0' + i\delta') \sigma_1 + i g_1^{(cp)} \mathcal{E}_1^{(cp)}, \ (\partial_t + c \partial_z) \mathcal{E}_1^{(cp)} &= i g_1^{(cp)} N \sigma_1, \end{align*}$$

with analogous equations for the second frequency rail. Simultaneous application yields a dual-rail frequency memory.

Electro-Optic and Photonic Integration

Other platforms for frequency-bin dual-rail encoding use cascaded electro-optic modulators (EOMs) and pulse shapers to generate, manipulate, and analyze frequency-bin qubits. In such processors, arbitrary single-qubit rotations are performed by applying tailored spectral phase masks and periodic time-domain phase modulations to control hopping and mixing between frequency modes, enabling full SU(2) operations in the dual-rail basis (Lu et al., 2020). Cluster state generation and the implementation of multiqubit gates (e.g., CNOT) in the frequency domain have also been achieved (Lu et al., 2018, Wang et al., 14 Aug 2025).

In the microwave domain, superconducting circuit QED systems employ multiple levels of transmon qubits and engineered Raman transitions to generate dual-rail encoded microwave photons with frequencies $\omega_1$ and $\omega_2$ (Wang et al., 14 Aug 2025). This dual-rail mapping supports photon loss (erasure) detection since a logical qubit is always associated with single-photon occupation in one of the frequency rails.

A summary of representative platforms is given below:

Platform	Realization Mechanism	Reference
Cold atom GEM	Zeeman-split Raman lines, field gradients	(Higginbottom et al., 2016)
Photonic integration	EOMs + spectral pulse shapers	(Lu et al., 2020)
Microwave circuit QED	Raman transitions with transmons/resonator	(Wang et al., 14 Aug 2025)

3. Key Performance Metrics and Experimental Benchmarks

Performance of frequency-bin dual-rail encoding schemes is evaluated by several metrics:

• Memory efficiency: Fraction of photons successfully retrieved after storage (e.g., 35% overall for dual-rail GEM (Higginbottom et al., 2016)).
• Interference fringe visibility: Quality of the preserved superposition, e.g., 82% in GEM protocol, approaching the theoretical limit set by polarisation mode overlap.
• Phase stability: Standard deviation of output phase; phase noise due to environmental fluctuations directly impacts qubit fidelity (as low as 6° achieved with active stabilization).
• Gate fidelity: In frequency-bin CNOT gates, fidelity as high as $0.91 \pm 0.01$ (quantum operation), compared to $\sim 0.995$ inferred classically (Lu et al., 2018).
• State transformation fidelity: Single-qubit control with measured mode transformation and process fidelities exceeding $0.98$ (Lu et al., 2020).
• Bell state and entanglement fidelity: Frequency-bin Bell states with density matrix fidelities $\geq 97\%$ (Seshadri et al., 2022).

Factors limiting these metrics include polarization-mode mismatch and frequency-dependent polarization rotation, ambient magnetic field fluctuations (in atomic systems), photon loss, and pulse duration-limited decoherence in superconducting systems.

4. Error Mechanisms and Their Mitigation

Two key fidelity-limiting mechanisms in optical implementations are:

A. Frequency-Dependent Polarization Rotation:

Linear polarization of control and signal fields leads to slightly different coupled polarization modes for each frequency rail, limiting their mode overlap and hence, maximal fringe visibility (theoretical overlap $\sim$0.88 observed in (Higginbottom et al., 2016)). This limitation is mitigated by switching to circularly polarized fields, trading off balanced optical depths for eliminating coupled-mode mismatch.

B. Phase Noise from Environmental Fluctuations:

Ambient magnetic field fluctuations introduce uncontrolled phase drifts between the two rails, effectively decohering quantum superpositions. Techniques such as synchronizing operation to the AC mains cycle or using magnetic shielding have been shown to reduce phase instability significantly (from $\sim$15° to 6°) (Higginbottom et al., 2016).

In superconducting microwave circuit implementations, photon loss (e.g., to the environment or resonator decay) takes the logical state out of the dual-rail code space and is detectable via joint photon-number parity checks. The error thus becomes a heralded “erasure” instead of an undetected Pauli error, simplifying correction and improving overall error thresholds (Teoh et al., 2022). Gate operations are constructed to commute with error operators (e.g., transmon decay structure), further suppressing unheralded Pauli errors by at least an order of magnitude.

5. Applications Across Quantum Technologies

Quantum Memory and Quantum Networking

Dual-rail frequency-bin encoding enables parallel high-fidelity storage and recall of optical frequency-encoded qubits, addressing scalability bottlenecks in multi-channel quantum memories (Higginbottom et al., 2016). The protocol is naturally suited to multiplexed quantum repeaters and interfaces heterogenous quantum network nodes, including conversion from subcarrier (SCW) encoding to dual-rail or polarization encoding with fidelities of 95% (Melnik et al., 2022).

Quantum Gate and Cluster-State Protocols

Frequency-bin dual-rail gates (e.g., CNOTs) implemented through EOMs and pulse shaping support universal logic and programmable frequency manipulation with fidelities above 0.9 (Lu et al., 2018). Multi-photon and cluster-state resources for one-way quantum computing have been generated in the frequency-bin dual-rail basis, with demonstrated state fidelities exceeding 50% for up to four logical qubits, and loss-robust entanglement persisting up to chains of eleven logical qubits after erasure correction (Wang et al., 14 Aug 2025).

Quantum Key Distribution (QKD) and Multiparty Protocols

Frequency-bin dual-rail encoding is employed in entanglement-based QKD protocols, with quantum bit error rates stabilized via real-time adaptive phase rotation and secure key rates demonstrated over 26 km fiber at $\geq 4.5$ bits/s (Tagliavacche et al., 12 Nov 2024). Multiplexed quantum secret sharing protocols exploit independent frequency channels—each a dual-rail pair—supporting scalable, wavelength-multiplexed multi-user sessions with individual state fidelities above 90% (Cabrejo-Ponce et al., 5 Aug 2025).

6. Scalability, Integration, and Outlook

Frequency-bin dual-rail encoding is compatible with integrated photonic circuits. Microring-based sources, arrayed waveguide gratings for pulse shaping, and modern thin-film lithium niobate EOMs enable integration of both state generation and arbitrary frequency-domain manipulation (Myilswamy et al., 23 Dec 2024).

The main challenges to scaling remain the coherent manipulation of many frequency channels with minimal loss, the requirement for ultra-high pump suppression in integrated nonlinear sources, and the need for heterogeneous integration of best-in-class pulse shapers and modulators. Multiplexed encoding schemes, employing large numbers of frequency bins, offer both dense channel capacity for quantum networks and resource-efficient implementation of advanced protocols (e.g., high-dimensional entanglement, hyperentanglement, and fault-tolerant quantum error correction) (Myilswamy et al., 23 Dec 2024, Morrison et al., 2022, Lu et al., 27 Nov 2024).

The dual-rail frequency-bin paradigm combines resilience (photon-loss/erasure detection), telecom compatibility, and large Hilbert space access, positioning it as a key enabler for scalable quantum communication, distributed computation, and photonic quantum error correction in forthcoming quantum technologies.