On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing

Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.

SimExact – An Efficient Method to Compute Function Similarity Between Proteins Using Gene Ontology

Background: The rapidly growing protein and annotation databases necessitate the development of efficient tools to process this valuable information. Biologists frequently need to find proteins similar to a given protein, for which BLAST tools are commonly used. With the development of biomedical ontologies, e.g. Gene Ontology, methods were designed to measure function (semantic) similarity between two proteins. These methods work well on protein pairs, but are not suitable for protein query processing. Objective: Our aim is to facilitate searching of similar proteins in an acceptable time. Methods: A novel method SimExact for high speed searching of functionally similar proteins has been proposed. Results: The experiments of this study show that SimExact gives correct results required for protein searching. A fully functional prototype of an online tool (www.datafurnish.com/protsem.php) has been provided that generates a ranked list of the proteins similar to a query protein, with a response time of less than 20 seconds in our setup. SimExact was used to search for protein pairs having high disparity between function similarity and sequence similarity. Conclusion: SimExact makes such searches practical, which would not be possible in a reasonable time otherwise.

A Novel Construction of Constrained Verifiable Random Functions

Constrained verifiable random functions (VRFs) were introduced by Fuchsbauer. In a constrained VRF, one can drive a constrained key skS from the master secret key sk, where S is a subset of the domain. Using the constrained key skS, one can compute function values at points which are not in the set S. The security of constrained VRFs requires that the VRFs’ output should be indistinguishable from a random value in the range. They showed how to construct constrained VRFs for the bit-fixing class and the circuit constrained class based on multilinear maps. Their construction can only achieve selective security where an attacker must declare which point he will attack at the beginning of experiment. In this work, we propose a novel construction for constrained verifiable random function from bilinear maps and prove that it satisfies a new security definition which is stronger than the selective security. We call it semiadaptive security where the attacker is allowed to make the evaluation queries before it outputs the challenge point. It can immediately get that if a scheme satisfied semiadaptive security, and it must satisfy selective security.