A comprehensive introduction to the theory of word-representable graphs

Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy⋯ (of even or odd length) or a word yxyx⋯  (of even or odd length). A graph G=(V,E) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy ∈ E.   Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-representable graphs including the most recent developments in the area

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

Distribution of returned isomers.

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

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

Encoding of 2-methylbutane.

Representation of 2-methylbutane (an isomer of pentane C5H12) as a one hot encoded bit string, degree sequence, graph, and molecular graph (clockwise from the top left).</p

Graphs of QUBOS.

A: Butane (C4H10), B: Heptane (C7H16).</p

Number of runs to find all Heptane isomers.

Number of runs to find all Heptane isomers.</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