Contextual bound states for qudit magic state distillation

Abstract

Identifying necessary and sufficient conditions for universal quantum computing is a long-standing open problem for which contextuality is, perhaps, the only promising solution [Howard et al. Nature 510, 351 (2014)]. To justify this conjecture, Howard et al. showed that contextuality is equivalent to Wigner negativity for qudits, and is therefore necessary and possibly sufficient for qudit magic state distillation. Here, we reformulate magic state distillation in the language of discrete phase space, to show that any finite qudit magic state distillation routine contains bound states outside the Wigner polytope -these states exhibit contextuality but are useless for magic state distillation.

Qutrit and Ququint Magic States

“…In [18], it was also shown that negativity of the Wigner function is equivalent to contextuality with respect to stabilizer measurements. It was also recently shown that no finite magic state distillation routine can distill states tight to the boundary of any facet of the Wigner polytope [50], generalizing a well-known result for qubits [51,52]. (See also [53,19,49,54] for related work.…”

Magic state distillation with the ternary Golay code

“…has a threshold to depolarizing noise of 3/4. While it can be shown that no magic state distillation routine based on a finite stabilizer code can achieve this threshold [14,47], the possibility remains that a sequence of stabilizer codes exist that distil the strange state, whose threshold approaches 3/4. Of course, the ternary Golay code is an extremely special error-correcting code, and there is no reason to expect that one can generalize it to obtain such a sequence of codes.…”

“…We use the algorithm of [14], reviewed in §2, to simulate both projection onto the stabilizer code and subsequent decoding. d out Figure 4.…”

“…Our distillation routine takes 11 qutrits in the state ρ( 0 , S ) ⊗11 and outputs a single qutrit in the state ρ( 0 , S ) and is thus characterized by the two functions 0 ( 0 , S ), and S ( 0 , S )). We obtained these expressions, which are presented in appendix B, using the simulation algorithm of [14], as in the previous section. For small 0 and S , these can be expressed as follows: By iterating this procedure many times, we numerically determined the region of state space that distils to the Norell state.…”

“…Magic state distillation [6][7][8][9] is a leading approach to fault-tolerant quantum computing. In the past few years, magic state distillation for qudits of (typically odd prime) dimensions other than two has attracted some interest [10][11][12][13][14] and notably has been used to identify contextuality as an essential resource for universal quantum computation [15]. However, for the most part, qudit fault-tolerant quantum computing [16] appears relatively unexplored although attractive experimental realizations of qutrits do exist, e.g.…”