As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The minimum positive co-degree of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $\delta_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\Hos--Ko--Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.

Let us write ${\mathcal D}_{{\mathcal F}}(G)=\{F \in {\mathcal F}:F\cap G=\emptyset\}$ for a set $G$ and a family ${\mathcal F}$. Then a family ${\mathcal F}$ of sets is said to be ($\le l$)-almost intersecting ($l$-almost intersecting) if for any $F \in {\mathcal F}$ we have $|{\mathcal D}_{{\mathcal F}}(F)| \le l$ ($|{\mathcal D}_{{\mathcal F}}(F)|= l$). In this paper we investigate the problem of finding the maximum size of an ($\le l$)-almost intersecting ($l$-almost intersecting) family ${\mathcal F}$.

BACKGROUND: In rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system. RESULTS: For purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm. CONCLUSIONS: Hierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.