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

涌现的动态平移对称性破缺作为局部化和拓扑转变的秩序原理

Emergent Dynamical Translational Symmetry Breaking as an Order Principle for Localization and Topological Transitions

本研究提出动态平移对称性破缺作为统一局部化和拓扑转变的秩序原理，通过时间平均局部平移对比度定量诊断相变过程。

On the Local Equivalence of 2D Color Codes and Surface Codes with Applications

In recent years, there have been many studies on local stabilizer codes. Under the assumption of translation and scale invariance Yoshida classified such codes. His result implies that translation invariant 2D color codes are equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and Poulin showed that a local translation invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. In this paper we focus on 2D topological color codes and relax the assumption of translation invariance. Using a linear algebraic framework …

Transversal Clifford gates on folded surface codes

Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and correct errors, while color codes have transversal Clifford gates and better code efficiency in the number of physical qubits needed to achieve a given code distance. A formal equivalence exists between color codes and folded surface codes, but it does not guarantee the transferability of any of these favorable properties. However, the equivalence does imply the…

Unfolding the color code

The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for …