# Fock-state bosonic code

## Description

Qudit-into-oscillator code whose protection against amplitude damping (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].

## Parent

## Children

- \(\chi^{(2)}\) code
- Chuang-Leung-Yamamoto (CLY) code — Chuang-Leung-Yamamoto code are multi-mode Fock-state codes.
- Ouyang-Chao constant-excitation permutation-invariant 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.

## Cousins

- Binary code — Fock-state code distance is a natural extension of Hamming distance between binary strings.
- Qubit code — Fock-state code whose codewords are finite superpositions of Fock states with maximum occupation \(N\) can be mapped into a qubit code with \(n\geq\log_2 N\) by performing a binary expansion of the Fock-state labels \(n\) and treating each binary digit as an index for a qubit state. Pauli operators for the constituent qubits can be expressed in terms of bosonic raising and lowering operators [2]. However, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [3].

## 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
- [2]
- Victor V. Albert, private communication, 2016
- [3]
- S. M. Girvin, “Introduction to Quantum Error Correction and Fault Tolerance”, (2023) arXiv:2111.08894

## 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