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The Longer Term: Quantum Information Processing and Communication
Published in Simon Deleonibus, Electronic Device Architectures for the Nano-CMOS Era, 2019
Integer factoring is a special case of a larger class of problems (called the “Hidden Subgroup Problem”) which can also benefit from an exponential speedup when an adequate form of Shor’s Quantum Fourier Transform exists. Such a QFT is known to exist when the problem domain is a commutative group (e.g. integers modulo P, for factoring). Research is currently very active for extending Shor’s approach to non commutative groups. A few very specific problems in this larger class have been solved polynomially, but no general solution is known yet. One of the challenges is finding a polynomial algorithm for graph isomorphism, which is a problem of very high interest in many domains.
Faster quantum concentration via Grover's search
Published in International Journal of Parallel, Emergent and Distributed Systems, 2023
The promise of quantum computing in speeding up computations continues to attract research into exploring quantum algorithms for problems that arise in computer science, mathematics, and other scientific fields of study beyond searching and factorization. Indeed, several quantum algorithms and circuits have been reported for a wide range of classical problems extending from algebraic computations to pattern matching and many problems in graph theory. See for example [1,2] for a survey of quantum algorithms for Abelian and non-Abelian discrete Fourier transform, hidden subgroup problem, computing discrete logarithms, and several other problems in number theory, cryptography, and group theory. Another article by Montanaro provides an overview of quantum algorithms, and in particular surveys the complexity of quantum searching and optimization algorithms [3]. In graph theory, Grover's search algorithm and quantum walk techniques have been used to solve matching and network flow problems [4–6] and graph traversals [7].