Equivalence of 2D color codes (without translational symmetry) to surface codes

TL;DR

This paper demonstrates that any 2D color code can be mapped onto exactly two surface codes using a linear algebra approach, without requiring ancilla qubits, extending the understanding of topological code equivalences.

Contribution

It introduces a new linear algebra-based mapping from 2D color codes to two surface codes, removing the need for translation invariance and ancilla qubits.

Findings

Any 2D color code can be mapped onto two surface codes.

The surface code is induced directly from the color code.

The mapping does not require ancilla qubits.

Abstract

In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. For 2D color codes, Delfosse relaxed the constraint on translation invariance and mapped a 2D color code onto three surface codes. In this paper, we propose an alternate map based on linear algebra. We show that any 2D color code can be mapped onto exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes.