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
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
- Abelian quantum-double stabilizer code — Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [2].
- Generalized homological-product CSS code
Children
- Kitaev surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The qudit Shor code is a small qudit surface code on a Möbius strip with smooth boundary, which is obtained from removing a face of the tesselation of the projective plane \(\mathbb{R}P^2\) [7; Fig. 4].
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 [12–14,14] 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
Page edit log
- Victor V. Albert (2022-11-02) — most recent
- Victor V. Albert (2022-01-05)
- Ian Teixeira (2021-12-19)
Cite as:
“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface