%0 Generic %1 kubica2015unfolding %A Kubica, Aleksander %A Yoshida, Beni %A Pastawski, Fernando %D 2015 %K codes color kubica %R 10.1088/1367-2630/17/8/083026 %T Unfolding the color code %U http://arxiv.org/abs/1503.02065 %X 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 König. 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.

@misc{kubica2015unfolding, 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.}, added-at = {2018-04-06T15:42:58.000+0200}, author = {Kubica, Aleksander and Yoshida, Beni and Pastawski, Fernando}, biburl = {https://www.bibsonomy.org/bibtex/2741cf0c58db7018759494c528e54fd73/dirsoares}, description = {Unfolding the color code}, doi = {10.1088/1367-2630/17/8/083026}, interhash = {169f4dcbdecd001406afc27dde992891}, intrahash = {741cf0c58db7018759494c528e54fd73}, keywords = {codes color kubica}, note = {cite arxiv:1503.02065Comment: 46 pages, 15 figures}, timestamp = {2018-04-06T15:42:58.000+0200}, title = {Unfolding the color code}, url = {http://arxiv.org/abs/1503.02065}, year = 2015 }