Quantum Information Processing, Vol. 6, No. 6, December 2007 (© 2007)
Is Grover’s Algorithm a Quantum Hidden Subgroup
Samuel J. Lomonaco Jr.
and Louis H. Kauffman
Received May 13, 2007; accepted July 7, 2007; Published online: November 21, 2007
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.
KEY WORDS: Grover’s algorithm; Shor’s algorithm; quantum algorithms;
hidden subgroup algorithms; hidden subgroup problems.
2000 MATHEMATICS SUBJECT CLASSIFICATIONS: Primary 81P68;
2007 PACS CLASSIFICATION: Primary N 03.67.Ac; Secondary S 03.67.Lx.
Is Grover’s algorithm a quantum hidden subgroup (QHS) algorithm?
We do not completely answer this question. Instead, we show that
Grover’s algorithm is a QHS algorithm in the sense that it can be rephra-
sed as a quantum algorithm which solves a non-abelian hidden subgroup
problem (HSP) on the symmetric group S
. But we then go on to show
that the standard non-abelian QHS algorithm cannot solve the Grover
This leaves unanswered an intriguing question:
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To whom correspondence should be addressed.
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