Twisted XZZX toric code[1] 

Also known as XZZX cyclic code, Cyclic toric code, Generalized toric code (GTC).

Description

A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [2]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [3].

Protection

A family of \([[a^2+b^2,k,d]]\) cyclic codes exists for all \(b > a \geq 1\) such that \(\text{gcd}(a,b)=1\) [4; Thm. 3.9]. Here, \(k=1\) (\(k=2\)) and \(d=a+b\) (\(d=b\)) for odd \(n\) (odd \(n\)). Other types of distances have been considered for this code [3].

Decoding

Fault-tolerant syndrome extraction circuits using flag qubits [3].AMBP4, a quaternary version [5] of the MBP decoder [6].

Fault Tolerance

Fault-tolerant syndrome extraction circuits using flag qubits [3].

Code Capacity Threshold

Depolarizing noise: \(17.5\%\) under AMBP4 decoding for the \([[(m^2+1)/2,1,m]]\) family [5; Fig. 10].Biased noise: between \(20\%\) and \(45\%\) at noise bias ranging from 1 to 10 under MWPM [3; Fig. 5].

Threshold

Phenomenological noise: between \(3\%\) and \(10\%\) at noise bias ranging from 1 to 4 under MWPM [3; Fig. 5].

Parents

Children

Cousins

References

[1]
A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[2]
A. Robertson et al., “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
[3]
Q. Xu et al., “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
[4]
R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
[5]
K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) (2022) arXiv:2202.06612 DOI
[6]
K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
[7]
A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
[8]
J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
[9]
B. Röthlisberger et al., “Incoherent dynamics in the toric code subject to disorder”, Physical Review A 85, (2012) arXiv:1112.1613 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: twisted_xzzx

Cite as:
“Twisted XZZX toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twisted_xzzx
BibTeX:
@incollection{eczoo_twisted_xzzx, title={Twisted XZZX toric code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/twisted_xzzx} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/twisted_xzzx

Cite as:

“Twisted XZZX toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twisted_xzzx

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/non-css/twisted_xzzx.yml.