# Dual-rail quantum code[1]

## Description

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.

## Protection

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].

## Encoding

Optimal control pulses [2]

## Gates

General gates are performed using beamsplitters [3] and Kerr nonlinearities. Universal quantum computing can be achieved using the KLM protocol [4] 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 [5].Error-detecting \(CCZ\) and \(cSWAP\) gates using three-level ancilla [6].

## Fault Tolerance

## Threshold

Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when repeatedly concatenating with the Steane code [9].

## Realizations

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 [10–12] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [3]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [2] and Amazon Web Services [13] using transmon qubits, following earlier theoretical proposals [8,14]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [15].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [16].

## Parents

- One-hot quantum code
- Two-mode binomial code — The two-mode binomial code for \(S=N=0\) reduces to the dual-rail code.

## Cousins

- Concatenated bosonic code — The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail code with a stabilizer code [4]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [17] that protects against \(d-1\) AD errors [18]. Concatenating the outer dual-rail code with an inner single-mode bosonic code yields several gates that are independent of the inner code [19].
- Single-mode bosonic code — Concatenating the outer dual-rail code with an inner single-mode bosonic code yields several gates that are independent of the inner code [19].
- Amplitude-damping (AD) code — Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [7,8]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [17] that protects against \(d-1\) AD errors [18].
- Quantum parity code (QPC) — An \([[8,1,2]]\) QPC correcting a single AD 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 [17,20,21]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [18].
- \([[4,2,2]]\) Four-qubit code — An \([[8,1,2]]\) QPC correcting a single AD 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 [17,20,21]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [18].
- Cluster-state code — The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [22–24]; see review [12].
- Cluster-state code — The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [22–24].
- Fusion-based quantum computing (FBQC) code — FBQC resource states are concatenated with dual-rail codes to increase loss detection.

## References

- [1]
- I. L. Chuang and Y. Yamamoto, “Simple quantum computer”, Physical Review A 52, 3489 (1995) DOI
- [2]
- A. Koottandavida et al., “Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity”, (2023) arXiv:2311.04423
- [3]
- Y. Lu et al., “High-fidelity parametric beamsplitting with a parity-protected converter”, Nature Communications 14, (2023) DOI
- [4]
- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
- [5]
- 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
- [6]
- T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
- [7]
- A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052
- [8]
- A. Kubica et al., “Erasure qubits: Overcoming the \(T_1\) limit in superconducting circuits”, (2022) arXiv:2208.05461
- [9]
- 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
- [10]
- C. R. Myers and R. Laflamme, “Linear Optics Quantum Computation: an Overview”, (2005) arXiv:quant-ph/0512104
- [11]
- P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
- [12]
- S. Slussarenko and G. J. Pryde, “Photonic quantum information processing: A concise review”, Applied Physics Reviews 6, (2019) arXiv:1907.06331 DOI
- [13]
- H. Levine et al., “Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons”, Physical Review X 14, (2024) arXiv:2307.08737 DOI
- [14]
- J. D. Teoh et al., “Dual-rail encoding with superconducting cavities”, Proceedings of the National Academy of Sciences 120, (2023) arXiv:2212.12077 DOI
- [15]
- K. S. Chou et al., “Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements”, (2023) arXiv:2307.03169
- [16]
- K. Alexander et al., “A manufacturable platform for photonic quantum computing”, (2024) arXiv:2404.17570
- [17]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [18]
- R. Duan et al., “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [19]
- H.-K. Lau and M. B. Plenio, “Universal Quantum Computing with Arbitrary Continuous-Variable Encoding”, Physical Review Letters 117, (2016) arXiv:1605.09278 DOI
- [20]
- 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
- [21]
- T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
- [22]
- N. Yoran and B. Reznik, “Deterministic Linear Optics Quantum Computation with Single Photon Qubits”, Physical Review Letters 91, (2003) arXiv:quant-ph/0303008 DOI
- [23]
- M. A. Nielsen, “Optical Quantum Computation Using Cluster States”, Physical Review Letters 93, (2004) arXiv:quant-ph/0402005 DOI
- [24]
- D. E. Browne and T. Rudolph, “Resource-Efficient Linear Optical Quantum Computation”, Physical Review Letters 95, (2005) arXiv:quant-ph/0405157 DOI

## Page edit log

- Yinchen Liu (2024-03-15) — most recent
- Esha Swaroop (2024-03-14)
- Victor V. Albert (2021-12-18)
- Dhruv Devulapalli (2021-12-17)
- Aniket Maiti (2024-02-08)

## Cite as:

“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_rail