The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor's algorithm for factoring in quantum computing is an instance of the hidden subgroup problem for finite abelian groups, while the other problems correspond to finite groups that are not abelian.
Given a group , 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 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 which case corresponds to the kernel of .
The hidden subgroup problem is especially important in the theory of quantum computing for the following reasons.
There is an efficient quantum algorithm for solving HSP over finite abelian groups in time polynomial in . For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle.^{[3]} However, the circuits that implement this may be exponential in , 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.
The algorithm for abelian groups uses representations, i.e. homomorphisms from to , the general linear group over the complex numbers. A representation is irreducible if it cannot be expressed as the direct product of two or more representations of . For an abelian group, all the irreducible representations are the characters, which are the representations of dimension one; there are no irreducible representations of larger dimension for abelian groups.
The quantum fourier transform can be defined in terms of , the additive cyclic group of order . Introducing the characterthe quantum fourier transform has the definition ofFurthermore we define . Any abelian group can be written as the direct product of multiple cyclic groups . On a quantum computer, this is represented as the tensor product of multiple registers of dimensions respectively, and the overall quantum fourier transform is .
The set of characters of forms a group called the dual group of . We also have a subgroup of size defined byFor each iteration of the algorithm, the quantum circuit outputs a element corresponding to a character , and since for all , it helps to pin down what is.
The algorithm is as follows:
The state in step 5 is equal to the state in step 6 because of the following:For the last equality, we use the following identity:
Theorem —
This can be derived from the orthogonality of characters. The characters of form an orthonormal basis:We let be the trivial representation, which maps all inputs to , to getSince the summation is done over , also being trivial only matters for if it is trivial over ; that is, if . Thus, we know that the summation will result in if and will result in if .
Each measurement of the final state will result in some information gained about since we know that for all . , or a generating set for , will be found after a polynomial number of measurements. The size of a generating set will be logarithmically small compared to the size of . Let denote a generating set for , meaning . The size of the subgroup generated by will be doubled when a new element is added to it, because and are disjoint and because . Therefore, the size of a generating set satisfiesThus a generating set for will be able to be obtained in polynomial time even if is exponential in size.
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.
Problem | Quantum Algorithm | Abelian? | Polynomial time solution? |
---|---|---|---|
Deutsch's problem | Deutsch's algorithm; Deutsch-Jozsa algorithm | Yes | Yes |
Simon's problem | Simon's algorithm | Yes | Yes |
Order finding | Shor's order finding algorithm | Yes | Yes |
Discrete logarithm | Shor's algorithm § Discrete logarithms | Yes | Yes |
Period finding | Shor's algorithm | Yes | Yes |
Abelian stabilizer | Kitaev's algorithm^{[4]} | Yes | Yes |
Graph Isomorphism | None | No | No |
Shortest vector problem | None | No | No |