## Description

Qudit-into-oscillator code whose protection against AD noise (i.e., photon loss) stems from the use of disjoint sets of Fock states for the construction of each code basis state. The simplest example is the dual-rail code, which has codewords consisting of single Fock states \(|10\rangle\) and \(|01\rangle\). This code can detect a single loss error since a loss operator in either mode maps one of the codewords to a different Fock state \(|00\rangle\). More involved codewords consist of several well-separated Fock states such that multiple loss events can be detected and corrected.

## Protection

Code distance \(d\) is the minimum distance (assuming some metric) between any two labels of Fock states corresponding to different code basis states. For a single mode, \(d\) is the minimum absolute value of the difference between any two Fock-state labels; such codes can detect up to \(d-1\) loss events. Multimode distances can be defined analogously; see, e.g., Chuang-Leung-Yamamoto codes. There are tradeoffs in how well a Fock-state code protects against loss/gain errors and dephasing noise [1].

## Parents

- Qudit-into-oscillator code
- Amplitude-damping (AD) code — Fock-state codes are designed to protect against bosonic AD noise.

## Children

- Bosonic \(q\)-ary expansion — The bosonic \(q\)-ary expansion allows one to map between prime-dimensional qudit states and a Fock subspace of a single mode.
- \(\chi^{(2)}\) code
- Chuang-Leung-Yamamoto (CLY) code — Chuang-Leung-Yamamoto code are multi-mode Fock-state codes.
- Ouyang-Chao constant-excitation PI code
- Very small logical qubit (VSLQ) code
- Matrix-model code — Matrix-model logical states are lie in a low-energy Fock-state subspace.
- Pair-cat code
- Bosonic rotation code — Single-mode Fock-state codes are typically rotationally invariant.

## Cousin

- Binary code — Fock-state code distance is a natural extension of Hamming distance between binary strings.

## References

- [1]
- Y. Ouyang and E. T. Campbell, “Trade-Offs on Number and Phase Shift Resilience in Bosonic Quantum Codes”, IEEE Transactions on Information Theory 67, 6644 (2021) arXiv:2008.12576 DOI

## Page edit log

- Victor V. Albert (2021-12-30) — most recent

## Cite as:

“Fock-state bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/fock_state