Quantum Isomer Search

Isomer search or molecule enumeration refers to the problem of finding all the isomers for a given molecule. Many classical search methods have been developed in order to tackle this problem. However, the availability of quantum computing architectures has given us the opportunity to address this problem with new (quantum) techniques. This paper describes a quantum isomer search procedure for determining all the structural isomers of alkanes. We first formulate the structural isomer search problem as a quadratic unconstrained binary optimization (QUBO) problem. The QUBO formulation is for general use on either annealing or gate-based quantum computers. We use the D-Wave quantum annealer to enumerate all structural isomers of all alkanes with fewer carbon atoms (n < 10) than Decane (C10H22). The number of isomer solutions increases with the number of carbon atoms. We find that the sampling time needed to identify all solutions scales linearly with the number of carbon atoms in the alkane. We probe the problem further by employing reverse annealing as well as a perturbed QUBO Hamiltonian and find that the combination of these two methods significantly reduces the number of samples required to find all isomers.Comment: 20 pages, 9 figure

A QUBO formulation for top-τ eigencentrality nodes

The efficient calculation of the centrality or “hierarchy” of nodes in a network has gained great relevance in recent years due to the generation of large amounts of data. The eigenvector centrality (aka eigencentrality) is quickly becoming a good metric for centrality due to both its simplicity and fidelity. In this work we lay the foundations for solving the eigencentrality problem of ranking the importance of the nodes of a network with scores from the eigenvector of the network, using quantum computational paradigms such as quantum annealing and gate-based quantum computing. The problem is reformulated as a quadratic unconstrained binary optimization (QUBO) that can be solved on both quantum architectures. The results focus on correctly identifying a given number of the most important nodes in numerous networks given by the sparse vector solution of our QUBO formulation of the problem of identifying the top-τ highest eigencentrality nodes in a network on both the D-Wave and IBM quantum computers.</jats:p

Distribution of returned isomers.

Average number of times each isomer was returned per 10,000 samples. Left: Butane (C4H10), Right: Heptane (C7H16).</p

Computing degree centrality and eigencentrality from exponential function [10].

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Number of results returned for each energy.

Left: using quantum annealing, Right: using simulated annealing.</p

Number of results returned for each energy.

Number of results out of 10,000 samples returned at each energy. Left: Butane (C4H10), Right: Heptane (C7H16).</p

Results for graph <i>M</i>.

The graph M showing the most central nodes using (a) the NetworkX EC algorithm, and ((b) and (C)) QUBO on the IBM-Q and D-Wave quantum computers for (b) τ = 1 and (c) τ = 10 with and P1 = 5n where n is the number of nodes of the graphs. For (a), the most central node(s) is(are) of brighter colors and are encircled by larger circles. In (b) and (c), the bright colored (yellow) nodes are the most central nodes whiles the dark colored (purple) nodes are the least central nodes relative to τ.</p

Results for challenging graphs.

NetworkX results for the challenging graphs encountered; (a) graph G1, (b) Karate Club graph, and (c) Barabasi-Albert graph, BA(50, 5) = G4. D-Wave results obtained for the QUBO in Eq 11 using and P1 = 5n for (d) graph G1 with τ = 6, (e) Karate Club graph with τ = 5, (f) BA(50, 5) = G4 with τ = 3, (g) G1 with τ = 1, (h) Karate Club graph with τ = 1 and (i) BA(50, 5) = G4 with τ = 1. The D-Wave results colors the top τ most important nodes yellow and the least central nodes purple. The NetworkX result identifies most central nodes with larger and brighter circles (yellow being the top) and least central nodes with smaller and darker circles (least being purple).</p

Sequential distributions of results.

Distributions of returned isomers of Pentane (C5H12) after each run of 10,000 samples. Left: Not using QUBO perturbation, Right: Using QUBO perturbation.</p