IQIM Postdoctoral and Graduate Student Seminar
Abstract: The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. In the talk, I will start with introducing these two models. Then, I will show that the color code in d dimensions is equivalent to multiple decoupled copies of the d-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. This finding generalizes the previous results for two-dimensional systems to higher dimensions and to systems without translation invariance. I will also analyze the case of codes with boundaries and explain how one can attach d+1 copies of the d-dimensional toric code in order to obtain the d-dimensional color code. In particular, for d=2, I show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. The last result concerns implementability of a logical non-Pauli gate from the d-th level of the Clifford hierarchy in the d-dimensional color code. In particular, I present how 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, saturating the bound by Bravyi and Koenig.
This is a joint work with Beni Yoshida and Fernando Pastawski.