Non-CSS color codes on 2D lattices : Models and Topological Properties

TL;DR

This paper introduces non-CSS variants of the 2D color code that differ in local stabilizer violations but share long-range physics, enabling fault-tolerant quantum computation and potential noise protection.

Contribution

It constructs non-CSS 2D color code models with distinct local properties that are not equivalent to the original, while maintaining similar long-range physics and fault-tolerance features.

Findings

Non-CSS models violate different numbers of stabilizers per qubit.

Code spaces of CSS and non-CSS versions are related by local unitaries.

Non-CSS codes support transversal Clifford gates for fault-tolerance.

Abstract

The two-dimensional color code is an alternative to the toric code that encodes more logical qubits while maintaining crucial features of the $\mathbb{Z}_2\times\mathbb{Z}_2$ toric code in the long wavelength limit. However its short range physics include single qubit Pauli operations that violate either three or six stabilisers as opposed to the toric code where single qubit Pauli operations violate two or four stabilisers. Exploiting this fact we construct several non-CSS versions of the two-dimensional color code falling into two families - those where either three, four or five stabilisers are violated and those which violate exactly four stabilisers for all the three types of single qubit Pauli operations. These models are not equivalent to the original color code by a local unitary transformation. Nevertheless the code spaces of the CSS and non-CSS versions are related by local unitaries which reflects the fact that their long range physics coincide. This also implies that the non-CSS versions host transversal Clifford gates and hence support fault-tolerant computations. As a consequence of the non-CSS structure, the logical operators are of a mixed type which in some cases include all the three Pauli operators making them potentially useful for protection against biased Pauli noise.