arXiv subjects

Mark Ettinger

Publications and source records attributed to Mark Ettinger.

4 recordsLinked to original sources

arXiv2004

The quantum query complexity of the hidden subgroup problem is polynomial

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers.

quant-ph↗

arXiv1999

Hidden Subgroup States are Almost Orthogonal

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.

quant-ph↗

arXiv1999

A Quantum Observable for the Graph Isomorphism Problem

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.

quant-ph↗

arXiv1998

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.

quant-ph↗