# Dual-rail quantum code[1]

## Description

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.

## 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].

## Gates

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].

## Realizations

The dual-rail code is ubiquitous in linear optical quantum devices [5].

## Notes

For Deutsch's problem [6] 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.

## Parent

## Cousins

- Fusion-based quantum computing (FBQC) code — FBQC resource states are concatenated with dual-rail codes to increase loss detection.
- Cluster-state code — MBQC can be achieved with dual-rail codes using linear optical elements and photon detectors [7].

## References

- [1]
- I. L. Chuang and Y. Yamamoto, “Simple quantum computer”, Physical Review A 52, 3489 (1995) DOI
- [2]
- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
- [3]
- T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
- [4]
- A. Kubica et al., “Erasure qubits: Overcoming the \(T_1\) limit in superconducting circuits”, (2022) arXiv:2208.05461
- [5]
- P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
- [6]
- “Rapid solution of problems by quantum computation”, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992) DOI
- [7]
- 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

- Victor V. Albert (2021-12-18) — most recent
- Dhruv Devulapalli (2021-12-17)

## Cite as:

“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/dual_rail