arXiv:quant-ph/9901034

Hidden Subgroup States are Almost Orthogonal

Mark Ettinger, Peter Høyer, Emanuel H. Knill

Published Jan 14, 1999• 40 citations

Abstract

It is well known that quantum computers can efficiently find a hidden subgroup $H$ of a finite Abelian group $G$. This implies that after only a polynomial (in $\log |G|$) number of calls to the oracle function, the states corresponding to different candidate subgroups have exponentially small inner product. We show that this is true for noncommutative groups also. We present a quantum algorithm which identifies a hidden subgroup of an arbitrary finite group $G$ in only a linear (in $\log |G|$) number of calls to the oracle function. This is exponentially better than the best classical algorithm. However our quantum algorithm requires an exponential amount of time, as in the classical case.

DOI: 10.48550/arxiv.quant-ph/9901034

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arXiv:quant-ph/9901034

Hidden Subgroup States are Almost Orthogonal

Mark Ettinger, Peter Høyer, Emanuel H. Knill

Published Jan 14, 1999• 40 citations

arXiv HTML PDF

Abstract

It is well known that quantum computers can efficiently find a hidden subgroup

H

of a finite Abelian group

G

. This implies that after only a polynomial (in

\log |G|

) number of calls to the oracle function, the states corresponding to different candidate subgroups have exponentially small inner product. We show that this is true for noncommutative groups also. We present a quantum algorithm which identifies a hidden subgroup of an arbitrary finite group

G

in only a linear (in

\log |G|

) number of calls to the oracle function. This is exponentially better than the best classical algorithm. However our quantum algorithm requires an exponential amount of time, as in the classical case.

DOI: 10.48550/arxiv.quant-ph/9901034

Related papers

Efficient quantum algorithm for identifying hidden polynomials

Thomas Decker, Jan Draisma, Paweł Wocjan

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$ -variate polynomial functions of total degree $n\geq 2$ . The problem of identifying hidden $m$ -variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$ .We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$-variate polynomial functions of total degree $n\geq 2$. The problem of identifying hidden $m$-variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$.

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Massoud Amini, Mehrdad Kalantar, Mahmood M. Roozbehani

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It is well known that quantum computers can efficiently find a hidden subgroup

H

of a finite Abelian group

G

. This implies that after only a polynomial (in

\log |G|

G

in only a linear (in

\log |G|

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with

d

elements. The hidden functions of the generalized problem are not restricted to be linear but can also be

m

-variate polynomial functions of total degree

n\geq 2

. The problem of identifying hidden

m

-variate polynomials of degree less or equal to

n

for fixed

n

and

m

is hard on a classical computer since

\Omega(\sqrt{d})

black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of

d

with constant probability and that has a running time that is only polylogarithmic in

d

\Z_{p^r} \rtimes \Z_{q^s}

, when $p^r/q= \up{poly}(\log p^r)$, where

p

q

are any odd prime numbers and

r, s

are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups.

S_n

would give rise to an efficient quantum algorithm for Graph Isomorphism Problem. We formulate a hidden sub-hypergroup problem for finite hypergroups and solve it for finite commutative hypergroups. The given algorithm is efficient if the corresponding QFT could be calculated efficiently.