Unfolding the color code

Aleksander Kubica, Beni Yoshida, Fernando Pastawski
2015 New Journal of Physics  
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
more » ... e, 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.
doi:10.1088/1367-2630/17/8/083026 fatcat:yhsovdu7hvf5pis7bvnn4bp2ou