Twisted XZZX toric code

Twisted XZZX toric code[1]

Alternative names: 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-tolerant BP (FTBP) decoder [7].

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].

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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Twist-defect surface codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based QECC Quantum

XZZX surface codeCDSC Qubit Lattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

XZZX surface code

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].

Cyclic quantum codeQECC Quantum

Twisted XZZX toric code

Children

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; Exam. 11 and Fig. 3][10; Exam. 3][4; Fig. 1]. Doubling the five-qubit code via the formalism of Ref. [4] yields a \([[10,2,3]]\) code.

\([[13,1,5]]\) cyclic code

The \([[13,1,5]]\) cyclic code is a small twisted XZZX toric code [1; Exam. 11 and Fig. 3].

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, C. Granade, S. D. Bartlett, and S. T. Flammia, “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
[3]: Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, “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]: K.-Y. Kuo and C.-Y. Lai, “Fault-Tolerant Belief Propagation for Practical Quantum Memory”, (2024) arXiv:2409.18689
[8]: 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
[9]: 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
[10]: 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 (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

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