Dual-rail quantum code[1] 


Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used.


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 [2] and Kerr nonlinearities. Universal quantum computing can be achieved using the KLM protocol [3] with only linear optical elements and photon detectors.A probabilistic CZ gate via a non-linear sign-shift gate, which transforms the Fock states \( \alpha|0\rangle+\beta|1\rangle+\gamma|2\rangle\) into \(\alpha|0\rangle+\beta|1\rangle-\gamma|2\rangle \), followed by measurement [4].Error-detecting \(CCZ\) and \(cSWAP\) gates using three-level ancilla [5].

Fault Tolerance

Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise [6].


Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when repeatedly concatenating with the Steane code [7].


The dual-rail code is ubiquitous in linear-optical quantum devices and is behind the KLM protocol, one of the first proposals for fault-tolerant computation. See reviews [8,9] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [2]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [10] and Amazon Web Services [11] using transmon qubits, following earlier theoretical proposals [6,12]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [13].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [14].


For Deutsch's problem [15] 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 \(\frac{1}{2} (1- \text{sech} \gamma/2)\).



  • Concatenated quantum code — The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail with a stabilizer code [3]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [16].
  • Quantum parity code (QPC) — An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [1618].
  • \([[4,2,2]]\) Four-qubit code — An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [1618].
  • Cluster-state code — The KLM protocol can be combined cluster states in various ways [1921].
  • Cluster-state code — MBQC can be achieved with dual-rail codes using linear optical elements and photon detectors [21].
  • Fusion-based quantum computing (FBQC) code — FBQC resource states are concatenated with dual-rail codes to increase loss detection.


I. L. Chuang and Y. Yamamoto, “Simple quantum computer”, Physical Review A 52, 3489 (1995) DOI
Y. Lu et al., “High-fidelity parametric beamsplitting with a parity-protected converter”, Nature Communications 14, (2023) DOI
E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
E. Knill, “Bounds on the probability of success of postselected nonlinear sign shifts implemented with linear optics”, Physical Review A 68, (2003) arXiv:quant-ph/0307015 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
M. Silva, M. Rötteler, and C. Zalka, “Thresholds for linear optics quantum computing with photon loss at the detectors”, Physical Review A 72, (2005) arXiv:quant-ph/0502101 DOI
C. R. Myers and R. Laflamme, “Linear Optics Quantum Computation: an Overview”, (2005) arXiv:quant-ph/0512104
P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
A. Koottandavida et al., “Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity”, (2023) arXiv:2311.04423
H. Levine et al., “Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons”, Physical Review X 14, (2024) arXiv:2307.08737 DOI
J. D. Teoh et al., “Dual-rail encoding with superconducting cavities”, Proceedings of the National Academy of Sciences 120, (2023) arXiv:2212.12077 DOI
K. S. Chou et al., “Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements”, (2023) arXiv:2307.03169
K. Alexander et al., “A manufacturable platform for photonic quantum computing”, (2024) arXiv:2404.17570
“Rapid solution of problems by quantum computation”, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992) DOI
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
N. Yoran and B. Reznik, “Deterministic Linear Optics Quantum Computation with Single Photon Qubits”, Physical Review Letters 91, (2003) arXiv:quant-ph/0303008 DOI
M. A. Nielsen, “Optical Quantum Computation Using Cluster States”, Physical Review Letters 93, (2004) arXiv:quant-ph/0402005 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

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“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_rail
@incollection{eczoo_dual_rail, title={Dual-rail quantum code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dual_rail} }
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“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_rail

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