# Dual-rail 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 \(|10\rangle\).

## 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.

## Notes

For Deutsch's problem [3] 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 \(\frac{1}{2} (1- \text{sech} \gamma/2)\).

## Parent

## Zoo code information

## 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]
- “Rapid solution of problems by quantum computation”, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992). DOI

## Cite as:

“Dual-rail code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_rail