Matrix-model code[1,2] 


Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry.

A simple matrix-model code [2] consists of two spatially separated bosons with codewords \begin{align} |\mathcal{I}\rangle :=\prod_{(m,n)\in \mathcal{I} } \frac{\text{Tr}(a_1^{\dagger m}a_2^{\dagger n})}{N^{\frac{m+n}{2}}}|0\rangle_{12}~, \tag*{(1)}\end{align} where \(\cal I\) is some set of integer two-tuples, and \(n,m\geq 0\).

Gauge symmetry is assumed to be enforced in the above model. In other variants [2], gauge symmetry is enforced energetically, requiring a parameter to scale as \(\log(N)\) for polynomial memory lifetime. This translates to the bath coupling being suppressed as \(1/N\).


For the spatially separated boson code, logical errors stemming from gauge-invariant physical errors are suppressed polynomially with the number of modes \(N\), as shown per the error-correction conditions. Memory time scales as \(N^2\) when the model is subject to a thermal bath.



  • Self-correcting quantum code — Matrix-model codes are similar to self-correcting memories in the sense that memory time becomes infinite in the thermodynamic limit, but with corrections being polynomial in \(N\).


A. Milekhin, “Quantum error correction and large \(N\)”, SciPost Physics 11, (2021) arXiv:2008.12869 DOI
C. Cao, G. Cheng, and B. Swingle, “Large \(N\) Matrix Quantum Mechanics as a Quantum Memory”, (2022) arXiv:2211.08448
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Zoo Code ID: matrix_qm

Cite as:
“Matrix-model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_matrix_qm, title={Matrix-model code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Matrix-model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.