4 research outputs found
Simon's Algorithm, Clebsch-Gordan Sieves, and Hidden Symmetries of Multiple Squares
The first quantum algorithm to offer an exponential speedup (in the query
complexity setting) over classical algorithms was Simon's algorithm for
identifying a hidden exclusive-or mask. Here we observe how part of Simon's
algorithm can be interpreted as a Clebsch-Gordan transform. Inspired by this we
show how Clebsch-Gordan transforms can be used to efficiently find a hidden
involution on the group G^n where G is the dihedral group of order eight (the
group of symmetries of a square.) This problem previously admitted an efficient
quantum algorithm but a connection to Clebsch-Gordan transforms had not been
made. Our results provide further evidence for the usefulness of Clebsch-Gordan
transform in quantum algorithm design.Comment: 10 page
How a Clebsch-Gordan Transform Helps to Solve the Heisenberg Hidden Subgroup Problem
It has recently been shown that quantum computers can efficiently solve the
Heisenberg hidden subgroup problem, a problem whose classical query complexity
is exponential. This quantum algorithm was discovered within the framework of
using pretty-good measurements for obtaining optimal measurements in the hidden
subgroup problem. Here we show how to solve the Heisenberg hidden subgroup
problem using arguments based instead on the symmetry of certain hidden
subgroup states. The symmetry we consider leads naturally to a unitary
transform known as the Clebsch-Gordan transform over the Heisenberg group. This
gives a new representation theoretic explanation for the pretty-good
measurement derived algorithm for efficiently solving the Heisenberg hidden
subgroup problem and provides evidence that Clebsch-Gordan transforms over
finite groups are a new primitive in quantum algorithm design.Comment: 30 pages, uses qic.st
The Hidden Subgroup Problem
We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010,
including new results discovered since the survey of arXiv:quant-ph/0411037v1.
We recall how the problem provides a framework for efficient quantum algorithms
and present the standard methods based on coset sampling. We study the Dihedral
and Symmetric HSPs and how they relate to hard problems on lattices and graphs.
Finally, we conclude with the known solutions and techniques, describe
connections with efficient algorithms as well as miscellaneous variants of HSP.
We also bring various contributions to the topic. We show that in theory, we
can solve HSP over a given group inductively: the base case is solving HSP over
its simple factor groups and the inductive step is building efficient oracles
over a normal subgroup N and over the factor group G/N. We apply this analysis
to the Dedekindian HSP to get an alternative abelian HSP algorithm based on a
change of the underlying group. We also propose a quotient reduction by the
normal group obtained via Weak Fourier Sampling. We compute the exact
expression of Strong Fourier Sampling over the dihedral group, showing how the
previous reduction is natural and matches the standard one. We also give a
reduction of rigid graph isomorphism problem to HSP over the alternating group.
For this group and other simple groups, we propose maximal subgroup reduction
as a possible approach. We also analyse Regev's algorithm for the
poly(n)-uniqueSVP, prove how the degree of the polynomial is related to the
oracle complexity used and we suggest several variants.Comment: 99 pages, 13 figures. See also
http://www.maths-informatique-jeux.com/blog/frederic/?post/2010/07/30/Master-s-Project-The-Hidden-Subgroup-Proble
The central nature of the hidden subgroup problem
We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup for a large variety of groups. We also show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomial-time equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version. Topic Classification: Computational Complexity, Quantum Computing.