Movassagh-Ouyang Hamiltonian code[1] 


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.


Let \(C \subset \{0,1,\dots,q-1\}^n\) be a classical code with distance \(d_x\). Let \(d_z\) satisfy \(q^n > 2 V_q(d_z-1) -1\), where \(V_q(r)\) is the volume of the \(q\)-ary Hamming ball of radius \(r\). Then the algorithm produces a quantum code with distance \(d = \text{min}(d_x,d_z)\). Asymptotically, the distance scales linearly with \(n\).


The rate depends on the classical code, but distance can scale linearly with \(n\).




  • 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 \(n\)-qubit Movassagh-Ouyang Hamiltonian codes encoding one logical qubit with linear distance.
  • Codeword stabilized (CWS) code — The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other [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
S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
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Zoo Code ID: movassagh_ouyang

Cite as:
“Movassagh-Ouyang Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_movassagh_ouyang, title={Movassagh-Ouyang Hamiltonian code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Movassagh-Ouyang Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.