# Movassagh-Ouyang Hamiltonian code[1]

## Description

This is a family of codes derived via an algorithm that takes as input any binary classical code and outputs a quantum code (note that this framework can be extended to \(q\)-ary codes). The algorithm is probabalistic but succeeds almost surely if the classical code is random. An explicit code construction does exist for linear distance codes encoding one logical qubit. For finite rate codes, there is no rigorous proof that the construction algorithm succeeds, and approximate constructions are described instead.

This family strictly generalizes CSS codes (because CSS codes come only from linear or self orthogonal classical codes). These codes can be shown to be realized as a subspace of the ground space of a (geometrically) local Hamiltonian.

## Protection

## Rate

## Parents

- Qubit code
- Hamiltonian-based code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian. Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [1].

## Children

- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code — The \(((n,1,2))\) PI code is a Movassagh-Ouyang Hamiltonian code constructed from a binary code consisting of all codewords of weight 0, 2, or \(n\) [2; Appx. D].
- Qubit CSS code — Movassagh-Ouyang codes stem from a prescription that converts an arbitrary classical code into a quantum code.

## Cousins

- Qubit stabilizer code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
- Binary code — Movassagh-Ouyang codes are constructed from classical binary codes.
- Justesen code — Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [1].
- Spin code — Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [1].
- Frustration-free Hamiltonian code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian. Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [1].
- Codeword stabilized (CWS) code — The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other [1].

## References

- [1]
- R. Movassagh and Y. Ouyang, “Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians”, (2020) arXiv:2012.01453
- [2]
- S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332

## Page edit log

- Victor V. Albert (2021-12-16) — most recent
- Eric Kubischta (2021-12-15)

## Cite as:

“Movassagh-Ouyang Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/movassagh_ouyang