The hidden subgroup problem (HSP) is a topic of research in
mathematics and
theoretical computer science
computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumscribe the ...
. The framework captures problems such as
factoring,
discrete logarithm,
graph isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H''
: f \colon V(G) \to V(H)
such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) ar ...
, and the
shortest vector problem In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: Lat ...
. This makes it especially important in the theory of
quantum computing because
Shor's quantum algorithm for factoring is an instance 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 group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
, and a set
, we say a function
hides the subgroup
if for all
if and only if
. Equivalently,
is constant on the
cosets
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of ''H'', while it is different between the different cosets of ''H''.
Hidden subgroup problem: Let
be a group,
a finite set, and
a function that hides a subgroup
. The function
is given via an
oracle, which uses
bits. Using information gained from evaluations of
via its oracle, determine a generating set for
.
A special case is when
is a group and
is a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
in which case
corresponds to the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
of
.
Motivation
The hidden subgroup problem is especially important in the theory of
quantum computing for the following reasons.
*
Shor's quantum algorithm for factoring and
discrete logarithm (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
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite se ...
for HSPs for certain
non-Abelian groups would imply efficient quantum algorithms for two major problems: the
graph isomorphism problem
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.
The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational compl ...
and certain
shortest vector problem In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: Lat ...
s (SVPs) in lattices. More precisely, an efficient quantum algorithm for the HSP for the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
would give a quantum algorithm for the graph isomorphism. An efficient quantum algorithm for the HSP for the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
would give a quantum algorithm for the
unique SVP.
Algorithms
There is a efficient
quantum algorithm for solving HSP over finite
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s in time polynomial in
.
Shor's algorithm
Shor's algorithm is a quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor.
On a quantum computer, to factor an integer N , Shor's algorithm runs in polynom ...
applies a particular case of this quantum algorithm.
For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle. However, the circuits that implement this may be exponential in
, making the algorithm overall not efficient; 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 group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s.
To solve the problem for finite
Abelian groups
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
,
the 'standard' approach to this problem involves: the creation of the quantum state
, a subsequent
quantum Fourier transform
In quantum computing, the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor' ...
to the left register, after which this register gets sampled. This approach has been shown to be insufficient for the hidden subgroup problem for the symmetric group.
[{{cite arXiv, eprint=quant-ph/0501056, author=Cristopher Moore, Alexander Russell, Leonard J. Schulman, title=The Symmetric Group Defies Strong Fourier Sampling: Part I, year=2005]
See also
*
Dual group (Quantum Computing)
*
Hidden shift problem
References
External links
Richard Jozsa: Quantum factoring, discrete logarithms and the hidden subgroup problemChris Lomont: The Hidden Subgroup Problem - Review and Open ProblemsHidden subgroup problem on arxiv.org
Group theory
Quantum algorithms