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arXiv:1004.0776 · Journal of Mathematical Physics

Graph approach to quantum systems

Mladen Pavičić, Brendan D. McKay, Norman D. Megill, Krešimir Fresl

Published Oct 1, 2010• 24 citations
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Abstract

Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen–Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.

DOI
10.1063/1.3491766

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Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen–Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.
We consider algebras underlying Hilbert spaces used by quantum information algorithms. We show how one can arrive at equations on such algebras which define n-dimensional Hilbert space subspaces which in turn can simulate quantum systems on a quantum system. In doing so we use MMP diagrams and linear algorithms. MMP diagrams are tractable since an n-block of an MMP diagram has n elements while an n-block of a standard lattice diagram has 2^n elements. An immediate test for such an approach is a generation of minimal and arbitrary Kochen-Specker vectors and we present a minimal state-independent Kochen-Specker set of seven vectors from a Hilbert space with more than four dimensions.
Recently, quantum contextuality has been proved to be the source of quantum computation's power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen-Specker (KS) sets. In this paper, we show a method of vector generation that supersedes previous methods. It is implemented by means of algorithms and programs that generate hypergraphs embodying the Kochen-Specker property and that are designed to be carried out on supercomputers. We show that vector component generation of KS hypergraphs exhausts all possible vectors that can be constructed from chosen vector components, in contrast to previous studies that used incomplete lists of vectors and therefore missed a majority of hypergraphs. Consequently, this unified method is far more efficient for generations of KS sets and their implementation in quantum computation and quantum communication. Several new KS classes and their features have been found and are elaborated on in the paper. Greechie diagrams are discussed.
Quantum contextuality turns out to be a necessary resource for universal quantum computation and important in the field of quantum information processing. It is therefore of interest both for theoretical considerations and for experimental implementation to find new types and instances of contextual sets and develop methods of their optimal generation. We present an arbitrarily exhaustive hypergraph-based generation of the most explored contextual sets [Kochen-Specker (KS) ones] in 4, 6, 8, 16, and 32 dimensions. We consider and analyze 12 KS classes and obtain numerous properties of theirs, which we then compare with the results previously obtained in the literature. We generate several thousand additional types and instances of KS sets, including all KS sets in three of the classes and the upper part of a fourth set. We make use of the McKay-Megill-Pavi\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} (MMP) hypergraph language, algorithms, and programs to generate KS sets strictly following their definition from the Kochen-Specker theorem. This approach proves to be particularly advantageous over the parity-proof-based ones (which prevail in the literature) since it turns out that only a very few KS sets have a parity proof (in six KS classes 0.01% and in one of them 0%). MMP hypergraph formalism enables a translation of an exponentially complex task of solving systems of nonlinear equations, describing KS vector orthogonalities, into a statistically linearly complex task of evaluating vertex states of hypergraph edges, thus exponentially speeding up the generation of KS sets and enabling us to generate billions of novel instances of them. The MMP hypergraph notation also enables us to graphically represent KS sets and to visually discern their features.
Algorithms for finding arbitrary sets of Kochen‐Specker (KS) qubits as well as all the remaining vectors in a space of an arbitrary dimension are presented. The algorithms are based on linear MMP diagrams which generate orthogonalities of KS qubits, on an algebraic definition of states on the diagrams, and on nonlinear equations corresponding to MMP diagrams whose solutions are either KS qubits or the remaining vectors of a chosen space depending on whether the diagrams allow 0–1 states or not. The complexity of the algorithms is polynomial. New results obtained with the help of the algorithms are presented.
We find a new highly symmetrical and very numerous class (millions of nonisomorphic sets) of 4-dim Kochen–Specker (KS) vector sets. Due to the nature of their geometrical symmetries, they cannot be obtained from previously known ones. We generate the sets from a single set of 60 orthogonal spin vectors and 75 of their tetrads (which we obtained from the 600-cell) by means of our newly developed stripping technique. We also consider critical KS subsets and analyze their geometry. The algorithms and programs for the generation of our KS sets are presented.
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.