Dual-rail quantum code[1] 


Two-mode code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|01\rangle\), while the logical-one state is represented by \(|10\rangle\). The two modes of the encoding can represent temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding.


This is an error-detecting code against one photon loss event; it is often used in photonic quantum devices because of its ease of realization. A single loss event can be detected because, after the loss occurs, the output state \(|00\rangle\) is orthogonal to the codespace. Recovery is not possible, so a successful run of a quantum circuit is conditioned on not losing a photon during the circuit.

For Deutsch''s problem specifically, this code protects against errors resulting in states that have the correct photon number, but in the wrong modes [1].


General gates are performed using beamsplitters and Kerr non-linearities. Universal quantum computing can be achieved with photons in dual-rail encodings using the KLM protocol [2] with only linear optical elements and photon detectors.Error-detecting \(CCZ\) and \(cSWAP\) gates using three-level ancilla [3].

Fault Tolerance

Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise [4].


The dual-rail code is ubiquitous in linear optical quantum devices [5].Superconducting circuit devices: error detection has been demonstrated by Quantum Circuits Inc. [6] and Amazon Web Services [7], following earlier theoretical proposals [4,8].


For Deutsch's problem [9] with optical qubits, error correction using photon number detection reduces the error probability from \(\frac{1}{4} (1+e^{-\gamma}-2e^{-3\gamma/2}) \) to \(\frac{1}{2} (1- \text{sech} \gamma/2)\).See review [5] for more details.




I. L. Chuang and Y. Yamamoto, “Simple quantum computer”, Physical Review A 52, 3489 (1995) DOI
E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
A. Kubica et al., “Erasure qubits: Overcoming the \(T_1\) limit in superconducting circuits”, (2022) arXiv:2208.05461
P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
K. S. Chou et al., “Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements”, (2023) arXiv:2307.03169
H. Levine et al., “Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons”, (2023) arXiv:2307.08737
J. D. Teoh et al., “Dual-rail encoding with superconducting cavities”, (2022) arXiv:2212.12077
“Rapid solution of problems by quantum computation”, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992) DOI
D. E. Browne and T. Rudolph, “Resource-Efficient Linear Optical Quantum Computation”, Physical Review Letters 95, (2005) arXiv:quant-ph/0405157 DOI
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Zoo Code ID: dual_rail

Cite as:
“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/dual_rail
  title={Dual-rail quantum code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/dual_rail

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/fock_state/constant_excitation/dual_rail.yml.