Hidden subgroup problem on arvau

Wikipedia / Hidden subgroup problem

The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. It is a generalization of problems including factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor's algorithms for factoring and finding discrete logarithms in quantum computing are instances of the hidden subgroup problem for finite abelian groups, while the other problems correspond to finite groups that are not abelian.

Problem statement

Given a group, a subgroup hides is constant on each coset of H, while it is different between the different cosets of H.

Hidden subgroup problem is given via an oracle via its oracle, determine a generating set.

is a group homomorphism corresponds to the kernel.

Motivation

The hidden subgroup problem is especially important in the theory of quantum computing for the following reasons.

Shor's algorithm for factoring and for finding discrete logarithms (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite abelian groups.
The existence of efficient quantum algorithms for HSPs for certain non-abelian groups would imply efficient quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems (SVPs) in lattices. More precisely, an efficient quantum algorithm for the HSP for the symmetric group would give a quantum algorithm for the graph isomorphism. An efficient quantum algorithm for the HSP for the dihedral group unique SVP.

Quantum algorithms

There is an efficient quantum algorithm for solving HSP over finite abelian groups. For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle., making the algorithm not efficient overall; efficient algorithms must be polynomial in the number of oracle evaluations and running time. The existence of such an algorithm for arbitrary groups is open. Quantum polynomial time algorithms exist for certain subclasses of groups, such as semi-direct products of some abelian groups.

Algorithm for abelian groups

The algorithm for abelian groups uses representations, the general linear group over the complex numbers. A representation is irreducible if it cannot be expressed as the direct product. For an abelian group, all the irreducible representations are the characterss, which are the representations of dimension one; there are no irreducible representations of larger dimension for abelian groups.

Defining the quantum fourier transform

The quantum fourier transform, the additive cyclic group.

Procedure

called the dual group is.

The algorithm is as follows:

.
.
.
.
.
.
) is determined.

For the last equality, we use the following identity: is exponential in size.

Instances

Many algorithms where quantum speedups occur in quantum computing are instances of the hidden subgroup problem. The following list outlines important instances of the HSP, and whether or not they are solvable.