61 research outputs found
On Quantum Algorithms for Noncommutative Hidden Subgroups
Quantum algorithms for factoring and discrete logarithm have previously been
generalized to finding hidden subgroups of finite Abelian groups. This paper
explores the possibility of extending this general viewpoint to finding hidden
subgroups of noncommutative groups. We present a quantum algorithm for the
special case of dihedral groups which determines the hidden subgroup in a
linear number of calls to the input function. We also explore the difficulties
of developing an algorithm to process the data to explicitly calculate a
generating set for the subgroup. A general framework for the noncommutative
hidden subgroup problem is discussed and we indicate future research
directions.Comment: 13 pages, no figures, LaTeX2
Quantum Hidden Subgroup Algorithms: The Devil Is in the Details
We conjecture that one of the main obstacles to creating new non-abelian
quantum hidden subgroup algorithms is the correct choice of a transversal.Comment: 5 pages, to appear in April 2004 Proceedings of SPIE on Quantum
Information and Computatio
Improved Low-qubit Hidden Shift Algorithms
Hidden shift problems are relevant to assess the quantum security of various
cryptographic constructs. Multiple quantum subexponential time algorithms have
been proposed. In this paper, we propose some improvements on a polynomial
quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use
subset-sum algorithms to significantly reduce its complexity. We also propose
new tradeoffs between quantum queries, classical time and classical memory to
solve this problem
The Optimal Single Copy Measurement for the Hidden Subgroup Problem
The optimization of measurements for the state distinction problem has
recently been applied to the theory of quantum algorithms with considerable
successes, including efficient new quantum algorithms for the non-abelian
hidden subgroup problem. Previous work has identified the optimal single copy
measurement for the hidden subgroup problem over abelian groups as well as for
the non-abelian problem in the setting where the subgroups are restricted to be
all conjugate to each other. Here we describe the optimal single copy
measurement for the hidden subgroup problem when all of the subgroups of the
group are given with equal a priori probability. The optimal measurement is
seen to be a hybrid of the two previously discovered single copy optimal
measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe
Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of non-abelian Groups
We present a family of non-abelian groups for which the hidden subgroup
problem can be solved efficiently on a quantum computer.Comment: 16 pages, LaTeX2e, 3 figure
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