Modular-qudit surface code[13] 

Also known as \(\mathbb{Z}_q\) surface code.

Description

Extension of the surface code to prime-dimensional [1,2] and more general modular qudits [3]. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators.

Decoding

Renormalization group decoder [4,5].

Realizations

State preparation, anyon creation, anyon fusion, and transfer of entanglement between anyons and defect in a 24 qubit trapped ion device by Quantinuum [6].

Notes

The simplest Decodoku game is based on the qudit surface code with \( q=10\). See related Qiskit tutorial.

Parents

Children

Cousins

  • Hopf-algebra quantum-double code — The modular-qudit surface code can be generalized to a Hopf-algebra quantum-double code whose ground states remain the same but whose excitations are based on quasitriangular semisimple Hopf algebras of \(\mathbb{Z}_q\) [8].
  • Compactified \(\mathbb{R}\) gauge theory code — The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [9] of the qudit surface code as a bosonic stabilizer code.
  • Analog surface code — The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(\mathbb{R}\) oscillator limit [9] of the qudit surface code as a bosonic stabilizer code.
  • Tiger surface code — The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [9] of the qudit surface code as a tiger code.
  • Twist-defect surface code — Twist-defect surface codes have been extended to prime-dimensional qudits [10].
  • Double-semion stabilizer code — The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\), but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [11] or by gauging [1214,14] the one-form symmetry associated with said anyon [11; Footnote 20].
  • \(\mathbb{Z}_q^{(1)}\) subsystem code — The \(\mathbb{Z}_q^{(1)}\) subsystem code can be obtained from the \(\mathbb{Z}_q\) square-lattice surface code by gauging out the anyon \(e^{-1} m\) and applying single-qubit Clifford gates [11; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [11; Fig. 12].

References

[1]
A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
[2]
S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007) arXiv:quant-ph/0609070 DOI
[3]
H. Watanabe, M. Cheng, and Y. Fuji, “Ground state degeneracy on torus in a family of ZN toric code”, Journal of Mathematical Physics 64, (2023) arXiv:2211.00299 DOI
[4]
H. Anwar, B. J. Brown, E. T. Campbell, and D. E. Browne, “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
[5]
F. H. E. Watson, H. Anwar, and D. E. Browne, “Fast fault-tolerant decoder for qubit and qudit surface codes”, Physical Review A 92, (2015) arXiv:1411.3028 DOI
[6]
M. Iqbal et al., “Qutrit Toric Code and Parafermions in Trapped Ions”, (2024) arXiv:2411.04185
[7]
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
[8]
A. Conlon, D. Pellegrino, and J. K. Slingerland, “Modified toric code models with flux attachment from Hopf algebra gauge theory”, Journal of Physics A: Mathematical and Theoretical 56, 295302 (2023) arXiv:2210.07909 DOI
[9]
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
[10]
M. G. Gowda and P. K. Sarvepalli, “Quantum computation with generalized dislocation codes”, Physical Review A 102, (2020) DOI
[11]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[12]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[13]
L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[14]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
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Zoo Code ID: qudit_surface

Cite as:
“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface
BibTeX:
@incollection{eczoo_qudit_surface, title={Modular-qudit surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_surface} }
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“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/qudit_surface.yml.