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.
%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
}