D(4N) groups create phase gates with fewer qubits per edge

Alison Warman and Sakura Schäfer-Nameki of the Mathematical Institute, University of Oxford, have devised a method for creating topologically protected phase gates in two dimensions, bypassing limitations imposed by the Bravyi-König theorem. Their work encodes a logical qubit using the quantum double of the dihedral group D_(4N) on a triangular spatial patch, enabling constant-depth unitary gates at any level of the Clifford hierarchy. Specifically, the researchers realize the phase gate T^(1/N) = diag(1,e^(iπ/(4N))) in the logical Z basis, and demonstrate that for 8N = 2^n, the logical gate lies at the n-th level of the Clifford hierarchy and has a qubit-only realization; it can be constructed in terms of Clifford-hierarchy stabilizers for a code with…