Modular-qudit color code[1]
Description
An extension of color codes on lattices to modular qudits. Codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizer commute. This can be done by puncturing a hyperspherical lattice [2] or constructing a star-bipartition; see [1; Sec. III]. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present.
Transversal Gates
Some modular-qudit color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the qudit Clifford hierarchy [1].
Gates
Modular-qudit color codes whose \(X\)-type stabilizers are placed on cells of dimension \(\nu\) support transversal gates in the \(\nu\)th level of the qudit Clifford hierarchy as long as \(\nu! \neq 0\) modulo the qudit dimension [1; Thm. 1]. These codes saturate the Bravyi-Koenig bound. In particular, 3D color modular-qudit color codes admit a transversal modular-qudit \(T\) gate.Braiding twist defects of hexagonal lattices for qudits of odd prime dimension [3].
Parents
- Modular-qudit CSS code
- Lattice stabilizer code — Modular-qudit color codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizers commute [1; Sec. III].
- Generalized homological-product CSS code
Children
- 2D color code — Modular-qudit 2D color codes reduce to 2D color codes for \(q=2\).
- 3D color code — Modular-qudit 3D color codes reduce to 3D color codes for \(q=2\).
Cousins
- Modular-qudit subsystem color code
- Generalized 2D color code — The generalized color code for \(G=\mathbb{Z}_q\) reduces to the 2D modular-qudit color code.
- \(k\)-orthogonal code — The notion of \(k\)-orthogonality can be extended to modular-qudit codes and is known as \(k^{\star}\)-orthogonality [1; Def. 2]. Modular-qudit color codes defined on lattices in \(D\) spatial dimension whose \(X\)-type stabilizers are placed on cells of dimension \(\nu \leq D\) are \(k^{\star}\)-orthogonal for all \(k \leq \nu\) [1; Lemma 5].
References
- [1]
- F. H. E. Watson et al., “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
- [2]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [3]
- M. G. Gowda and P. K. Sarvepalli, “Quantum computation with charge-and-color-permuting twists in qudit color codes”, Physical Review A 105, (2022) arXiv:2110.08680 DOI
Page edit log
- Victor V. Albert (2024-03-03) — most recent
Cite as:
“Modular-qudit color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qudit_color