Abstract
As quantum contextuality proves to be a necessary resource for universal quantum computation, we present a general method for vector generation of Kochen-Specker (KS) contextual sets in the form of hypergraphs. The method supersedes all three previous methods: (i) fortuitous discoveries of smallest KS sets, (ii) exhaustive upward hypergraph-generation of sets, and (iii) random downward generation of sets from fortuitously obtained big master sets. In contrast to previous works, we can generate master sets which contain all possible KS sets starting with nothing but a few simple vector components. From them we can readily generate all KS sets obtained in the last half a century and any specified new KS sets. Herewith we can generate sufficiently large sets as well as sets with definite required features and structures to enable varieties of different implementations in quantum computation and communication.
Highlights
It has recently been recognized that “contextuality [can serve] as a [quantum] computational resource” [1]
Further elaboration and future implementation of contextual sets within such a framework of providing a computational resource for quantum computation would require an optimal way for their massive generation with desired properties and structures
Most of the results in the paper are generated within our hypergraph language and its algorithms and programs written in C we developed in [21, 34, 36, 37, 45,46,47,48] and extended here, as well as the parity-proof algorithms and programs developed in [23, 35, 38, 42]
Summary
It has recently been recognized that “contextuality [can serve] as a [quantum] computational resource” [1]. For these there is the additional task of rejecting hypergraphs that do not admit a vector coordinatization, a problem whose existing algorithms are still computationally infeasible in some cases This approach remains the only deterministic and completely exhaustive generation method for obtaining KS sets. We show that sets of simple orthogonal vectors inherently lead to KS sets Such simple components we obtain either from the coordinatization of the master sets from the aforementioned polytope approach or directly from an automated computer search. It provides us with a uniform and general method for KS set generation and with a larger scope and a bigger, more thorough picture of quantum contextuality than any of the previous approaches.
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