Unfolding the color code

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 dth level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d-qubit control-Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.