Gauss' law code[1,2] 

Description

An \([m+Dm,Dm,3]\) linear binary code for \(m\geq 3^D\), defined by the Gauss' law constraint of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory [2; Thm. 1]. The code can be re-phrased as a distance-one stabilizer code whose stabilizers consist of gauge-group elements. It can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [1,2].

Parents

Cousins

  • Abelian topological code — Gauge-group elements of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory form a single-error-correcting linear binary code [2; Thm. 1].
  • Fermion-into-qubit code — Gauge-group elements of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory form a single-error-correcting linear binary code [2; Thm. 1].
  • Concatenated quantum code — The Gauss' law code can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [1,2].

References

[1]
A. Rajput, A. Roggero, and N. Wiebe, “Quantum Error Correction with Gauge Symmetries”, (2022) arXiv:2112.05186
[2]
L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
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Zoo Code ID: gauss_law

Cite as:
“Gauss' law code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/gauss_law
BibTeX:
@incollection{eczoo_gauss_law, title={Gauss' law code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/gauss_law} }
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Cite as:

“Gauss' law code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/gauss_law

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/quantum_inspired/gauss_law.yml.