Abstract
Magic state distillation is a critical component in leading proposals for fault-tolerant quantum computation. Relatively little is known, however, about how to construct a magic state distillation routine or, more specifically, which stabilizer codes are suitable for the task. While transversality of a non-Clifford gate within a code often leads to efficient distillation routines, it appears to not be a necessary condition. Here we have examined a number of small stabilizer codes and highlight a handful of which displaying interesting, albeit inefficient, distillation behaviour. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered.
Highlights
Most efforts towards building a large-scale quantum computer use error-correcting codes to protect the quantum information
P(sk), n where N is the total number of iterations needed to obtain pout starting at initial error rate p, and p(sk) is the probability of success on the kth iteration. This quantity relates to the efficiency/resource overhead of a magic state distillation routine
Magic state distillation routines are described in terms of stabilizer error correcting codes, which in turn are described by a set of generators {Gi} consisting of Pauli operators
Summary
Most efforts towards building a large-scale quantum computer use error-correcting codes to protect the quantum information. The most promising technique for circumventing this issue is to supplement the non-universal gate set with a supply of special resource states, known as magic states. Efficient codes for magic state distillation typically exhibit quadratic (p → O(p2)) or cubic (p → O(p3)) suppression of the error parameter p. A number of the codes presented below achieve tight distillation right up to the boundary of the stabilizer octahedron – the convex hull of Pauli eigenstates depicted in Figure 1 – whose interior contains states that are provably undistillable. K=1...N where N is the total number of iterations needed to obtain pout starting at initial error rate p, and p(sk) is the probability of success on the kth iteration This quantity relates to the efficiency/resource overhead of a magic state distillation routine. None of our codes require twirling (which diagonalizes the state in the {|H , |H⊥ } or {|T , |T ⊥ } basis) between rounds
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