Twisted XZZX toric code[1]
Alternative Names: XZZX cyclic code, Cyclic toric code, Generalized toric code (GTC), Genus-one genon code.
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 equivalent to cyclic shifts of a particular weight-four \(XZZX\) Pauli string.
Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [2]. See Ref. [3] for a table of some of these for small instances, where they are called genus-one genon codes.
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 \(d=a+b\) (\(d=\max(a,b)\)) for odd \(n\) (even \(n\)). The subfamily \([[d^2+1,2,d]]\) (i.e., \(a=1,b=d\) for odd \(d\)) includes \([[10,2,3]]\), \([[26,2,5]]\), \([[50,2,7]]\), \(\ldots\) [5; Exam. 1]. The subfamily \([[t^2+(t+1)^2,1,2t+1]]\) (i.e., \(a=t,b=t+1\)) includes \([[5,1,3]]\), \([[13,1,5]]\), \([[25,1,7]]\), \(\ldots\) [5; Exam. 4]. Small instances are tabulated in Ref. [3] as genus-one genon codes. Other types of distances have been considered for this code [2].Decoding
Fault-tolerant syndrome extraction circuits using flag qubits [2].AMBP4, a quaternary version [6] of the MBP decoder [7].Fault-tolerant BP (FTBP) decoder [8].Fault Tolerance
Fault-tolerant syndrome extraction circuits using flag qubits [2].Code Capacity Threshold
Depolarizing noise: \(17.5\%\) under AMBP4 decoding for the \([[(m^2+1)/2,1,m]]\) family [6; Fig. 10].Biased noise: between \(20\%\) and \(45\%\) at noise bias ranging from 1 to 10 under MWPM [2; Fig. 5].Threshold
Phenomenological noise: between \(3\%\) and \(10\%\) at noise bias ranging from 1 to 4 under MWPM [2; Fig. 5].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 (AQC)— Twisted XZZX codes perform well against biased noise [2,9,10]; see also Ref. [11].
- Bipartite cyclic cluster (BCC) code— The \([[d^2+1,2,d]]\) twisted XZZX toric codes (parameters \(a=1,b=d\)) are Clifford-equivalent to BCC codes for odd \(d\) [12].
Member of code lists
- 2D stabilizer codes
- Asymmetric quantum codes and friends
- Cyclic quantum codes
- Lattice qubit stabilizer codes
- Quantum codes
- Quantum codes with code capacity thresholds
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Qubit stabilizer codes (non-CSS)
- Surface code and friends
Primary Hierarchy
Parents
Imposing twisted (a.k.a. shifted) boundary conditions on the toric XZZX code yields the twisted XZZX code [1; Exam. 11 and Fig. 3][4; Fig. 6].
Twisted XZZX toric code
Children
This is the \(d=3\) instance of the \([[d^2+1,2,d]]\) family of twisted XZZX toric codes (parameters \(a=1\), \(b=3\)), presented in its CSS form [12].
The \([[13,1,5]]\) twisted toric code is a small twisted XZZX toric code [1; Exam. 11 and Fig. 3].
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; Exam. 11 and Fig. 3][13; Exam. 3][4; Fig. 1]. Its genus-one double cover is a \([[10,2,3]]\) toric code [3][13; Exam. 3]. The base code’s transversal \(SH\) gate lifts to a logical \(CX \cdot SWAP\) gate on that double cover [3].
The \([[7,1,3]]\) XZZX cyclic code is a cyclic non-CSS code whose generators are the cyclic shifts of the weight-four Pauli string \(XZIZXII\), which has \(XZZX\) support.
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]
- Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, “Tailored XZZX codes for biased noise”, Physical Review Research 5, (2023) arXiv:2203.16486 DOI
- [3]
- S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
- [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]
- A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings 348 (2012) arXiv:1202.0928 DOI
- [6]
- K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) 186 (2022) arXiv:2202.06612 DOI
- [7]
- 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
- [8]
- K.-Y. Kuo and C.-Y. Lai, “Fault-Tolerant Belief Propagation for Practical Quantum Memory”, (2024) arXiv:2409.18689
- [9]
- A. Robertson, C. Granade, S. D. Bartlett, and S. T. Flammia, “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
- [10]
- J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
- [11]
- B. Röthlisberger, J. R. Wootton, R. M. Heath, J. K. Pachos, and D. Loss, “Incoherent dynamics in the toric code subject to disorder”, Physical Review A 85, (2012) arXiv:1112.1613 DOI
- [12]
- M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865
- [13]
- 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
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-04-01)
Cite as:
“Twisted XZZX toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/twisted_xzzx, arXiv:2606.11484