Abstract
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.
Highlights
Mathematical concepts pave the way for improvements in technology
We propose an alternative to anyon-based universal quantum computation (UQC) thanks to three-dimensional topology
Such manifolds carry a quantum geometry corresponding to quantum computing and positive operator-valued measure (POVM) identified in our earlier work [14,15,16]
Summary
Mathematical concepts pave the way for improvements in technology. As far as topological quantum computation is concerned, non-abelian anyons have been proposed as an attractive (fault-tolerant) alternative to standard quantum computing which is based on a universal set of quantum gates [2,3,4,5]. We propose an alternative to anyon-based universal quantum computation (UQC) thanks to three-dimensional topology. Irrespectively of the dimension of the Hilbert space where the quantum states live, a non-stabilizer pure state was called a magic state [13] An improvement of this concept was carried out in [14,15] showing that a magic state could be at the same time a fiducial state for the construction of an informationally complete positive operator-valued measure, or IC-POVM, under the action on it of the Pauli group of the corresponding dimension. UQC in this view happens to be relevant both to such magic states and to IC-POVMs. In [14,15], a d-dimensional magic state follows from the permutation group that organizes the cosets of a subgroup H of index d of a two-generator free group G.
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