Qudit Colour Codes and Gauge Colour Codes in All Spatial Dimensions

摘要

Two-level quantum systems, qubits, are not the only basis for quantumcomputation. Advantages exist in using qudits, d-level quantum systems, as thebasic carrier of quantum information. We show that color codes, a class oftopological quantum codes with remarkable transversality properties, can begeneralized to the qudit paradigm. In recent developments it was found that inthree spatial dimensions a qubit color code can support a transversalnon-Clifford gate, and that in higher spatial dimensions additionalnon-Clifford gates can be found, saturating Bravyi and K\"onig's bound [Phys.Rev. Lett. 110, 170503 (2013)]. Furthermore, by using gauge fixing techniques,an effective set of Clifford gates can be achieved, removing the need for statedistillation. We show that the qudit color code can support the qudit analoguesof these gates, and show that in higher spatial dimensions a color code cansupport a phase gate from higher levels of the Clifford hierarchy which can beproven to saturate Bravyi and K\"onig's bound in all but a finite number ofspecial cases. The methodology used is a generalisation of Bravyi and Haah'smethod of triorthogonal matrices [Phys. Rev. A 86 052329 (2012)], which may beof independent interest. For completeness, we show explicitly that the quditcolor codes generalize to gauge color codes, and share the many of thefavorable properties of their qubit counterparts.