\(D\)-dimensional twisted toric code

\(D\)-dimensional twisted toric code[1–4]

Description

Extension of the Kitaev toric code to higher-dimensional lattices with regular or shifted (a.k.a. twisted) boundary conditions. Such boundary conditions yields quibit geometries that are tori \(\mathbb{R}^D/\Lambda\), where \(\Lambda\) is an arbitrary \(D\)-dimensional lattice. Picking a hypercubic lattice yields the ordinary \(D\)-dimensional toric code. It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [4].

Protection

In two dimensions, different choices for the periodic boundary conditions yield higher-rate codes with parameters \([[L^2+1,2,L]]\) for odd \(L\) [2], and \([[L^2,2,L]]\) for even \(L\) [3]. Some higher-dimensional toric codes protect against burst errors [5].

Gates

Higher-dimensional toric codes can admit a cup product structure and can thus have logical gates in the Clifford hierarchy implemented by constant-depth Clifford circuits [6].

Cousin

Quantum LDPC (QLDPC) code— It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [4].

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

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

Parents

Homological code

\(D\)-dimensional twisted toric code

Children

Toric code

The \(D\)-dimensional twisted toric code reduces to the toric code for \(D=2\) and a square lattice.

References

[1]: H. Bombin and M. A. Martin-Delgado, “Topological quantum error correction with optimal encoding rate”, Physical Review A 73, (2006) arXiv:quant-ph/0602063 DOI
[2]: H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0605094 DOI
[3]: C. D. Albuquerque et al., On Toric Quantum Codes, Int. J. Pure Appl. Math. 50, 221–-226 (2009).
[4]: M. B. Hastings, “Quantum Codes from High-Dimensional Manifolds”, (2016) arXiv:1608.05089
[5]: C. C. Trinca, J. C. Interlando, R. Palazzo Jr., A. A. de Andrade, and R. A. Watanabe, “On the Construction of New Toric Quantum Codes and Quantum Burst-Error Correcting Codes”, (2022) arXiv:2205.13582
[6]: N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2024) arXiv:2410.16250

Page edit log

Marcus P da Silva (2025-02-04) — most recent
Victor V. Albert (2025-02-04)
Victor V. Albert (2024-05-03)

Cite as:

“\(D\)-dimensional twisted toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/higher_dimensional_toric

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_d/higher_dimensional_toric.yml.