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
- XZZX surface code — Imposing twisted (a.k.a. shifted) boundary conditions on the toric XZZX code yields the twisted XZZX code [1; Ex. 11 and Fig. 3][4; Fig. 6].
- Cyclic quantum code
Children
- \([[13,1,5]]\) cyclic code — The \([[13,1,5]]\) cyclic code is a small twisted XZZX toric code [1; Ex. 11 and Fig. 3].
- Five-qubit perfect code — Twisted XZZX codes are 2D lattice extensions of the five-qubit perfect code. The five-qubit code is a small twisted XZZX toric code [1; Ex. 11 and Fig. 3][7; Ex. 3][4; Fig. 1]. Doubling the five-qubit code via the formalism of Ref. [4] yields a \([[10,2,3]]\) code.
Cousins
- \((5,1,2)\)-convolutional code — \((5,1,2)\)-convolutional codes (twisted XZZX toric codes) are 1D (2D) lattice extensions of the five-qubit perfect code.
- Asymmetric quantum code — Twisted XZZX codes perform well against biased noise [2,3,8]; see also Ref. [9].
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”, (2023) arXiv:2101.09349
- [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
- Victor V. Albert (2024-04-01) — most recent
Cite as:
“Twisted XZZX toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twisted_xzzx