# Matrix-model code[1]

## Description

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 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, 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 \(1/N\).

## Protection

## Parents

- Fock-state bosonic code — Matrix-model logical states are lie in a low-energy Fock-state subspace.
- Approximate quantum error-correcting code (AQECC) — Matrix-model codes approximately protect against gauge-invariant errors in the large-mode limit.
- Hamiltonian-based code — Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.

## Cousins

- Holographic code — Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous non-Abelian gauge theories with large gauge groups [1].
- 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 \(N\).

## References

- [1]
- C. Cao, G. Cheng, and B. Swingle, “Large \(N\) Matrix Quantum Mechanics as a Quantum Memory”, (2022) arXiv:2211.08448

## Page edit log

- Victor V. Albert (2022-12-27) — most recent

## Cite as:

“Matrix-model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/matrix_qm