Description
Multimode Fock-state bosonic approximate code derived from a matrix model, i.e., a bosonic theory with a large non-Abelian 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 an energy penalty to scale as \(\log(N)\) in order to obtain a polynomial memory lifetime below a critical temperature.
Protection
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 by the approximate error-correction conditions. For sufficiently low temperature, the memory time scales as \(N^2\) when the model is subject to a thermal bath [2].Cousin
- 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 \(N\).
Primary Hierarchy
References
- [1]
- A. Milekhin, “Quantum error correction and large \(N\)”, SciPost Physics 11, (2021) arXiv:2008.12869 DOI
- [2]
- 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