Unfolding the color code

TL;DR

This paper demonstrates the equivalence between the topological color code and multiple copies of the toric code in any dimension, providing explicit methods for decoupling and implementing fault-tolerant logical gates.

Contribution

It generalizes the known 2D equivalence to higher dimensions and offers explicit procedures for decoupling color codes into toric codes, highlighting the role of colorability.

Findings

Color code in any dimension is equivalent to multiple toric codes up to local unitaries.

Explicit recipe for decoupling color code components.

Logical non-Pauli gates from the Clifford hierarchy are realizable on multiple toric codes.

Abstract

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 the $d$-dimensional color code with $d+1$ boundaries of $d+1$ distinct colors, we find that the code is equivalent to multiple copies of the $d$-dimensional toric code which are attached along a $(d-1)$-dimensional boundary. In particular, for $d=2$, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the $d$-dimensional toric code admits logical non-Pauli gates from the $d$-th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and K\"{o}nig. In particular, we show that the $d$-qubit control-$Z$ logical gate can be fault-tolerantly implemented on the stack of $d$ copies of the toric code by a local unitary transformation.