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Quantum Information Processing

Is Grover’s Algorithm a Quantum Hidden Subgroup Algorithm?

Samuel J. Lomonaco, Louis Hirsch Kauffman

Published Nov 20, 2007• 9 citations
Publisher
DOI
10.1007/s11128-007-0066-1

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It has been shown in recent years that quantum information has a topological nature (\cite{AC}, \cite{Co}, \cite{Co2}). In \cite{V}, Vicary undergoes the study of quantum algorithms using this new topological approach. The advantage of this new formalism is that it allows new proofs of correctness of the algorithms such as Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. It also provide more clear insight to the generalizations of these algorithms. In this paper, we consider (multi-step) Grover algorithm from this new perspective.

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The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm.
One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor's quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an "alphabetic building blocks approach," we use these primitives to form an "algorithmic toolkit" for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover's algorithm is most surprisingly "almost" a QHS algorithm, and how this result suggests the possibility of an even more complete "algorithmic tookit" beyond the QHS algorithms.
It has been shown in recent years that quantum information has a topological nature (\cite{AC}, \cite{Co}, \cite{Co2}). In \cite{V}, Vicary undergoes the study of quantum algorithms using this new topological approach. The advantage of this new formalism is that it allows new proofs of correctness of the algorithms such as Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. It also provide more clear insight to the generalizations of these algorithms. In this paper, we consider (multi-step) Grover algorithm from this new perspective.