paper

Unfolding the color code

arXiv:1503.02065 · doi:10.1088/1367-2630/17/8/083026

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 -dimensional closed manifold is equivalent to multiple decoupled copies of the -dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for , 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 -dimensional color code with boundaries of distinct colors, we find that the code is equivalent to multiple copies of the -dimensional toric code which are attached along a -dimensional boundary. In particular, for , we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the -dimensional toric code admits logical non-Pauli gates from the -th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the -qubit control- logical gate can be fault-tolerantly implemented on the stack of copies of the toric code by a local unitary transformation.

46 pages, 15 figures

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