Unfolding the color code

By Aleksander Kubica, B. Yoshida, Fernando Pastawski. 2015

185 citations recorded by the source corpus.

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 dth level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d-qubit control-Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7a7493b2c690b061f6f47039587403dbc2f1319f

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Citations in the corpus

The future of quantum computing with superconducting qubits (2022, methodology): “This approach relies on the fact that a 3D version of the surface code enables an easy (transversal) implementation of a logical CCZ 70,71 and a clever embedding of the 3D surface code into a 2+1 dimensional space-time.”
Magic state cultivation: growing T states as cheap as CNOT gates (2024, methodology): “In prior work, turning color codes into surface codes has been done by local unitary transformations [KYP15; Ito+24] and by lattice surgery [PFB17; SC22; Ito+24].”
Logical quantum processor based on reconfigurable atom arrays (2023, background): “Often portrayed in a triangular geometry, the colour codes used here are a tiling of three colours of weight-4 and weight-6 stabilizers, with X L and Z L operator strings running along the boundary of the code 91 .”