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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 .
Simon's problem considers access to a function : {,} {,}, as implemented by a black box or an oracle. This function is promised to be either a one-to-one function, or a two-to-one function; if is two-to-one, it is furthermore promised that two inputs and ′ evaluate to the same value if and only if and ′ differ in a fixed set of bits. I.e.,
The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...
The abelian hidden subgroup problem is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving Pell's equation, testing the principal ideal of a ring R and factoring. There are efficient quantum algorithms known for the Abelian hidden subgroup problem. [10]
Class function; Class of groups; ... Examples of groups; F. Factor system; Fibonacci group; List of finite simple groups; ... Hidden subgroup problem;
A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. ∀φ ∈ Inn(G)： φ(H) ≤ H. Since Inn(G) ⊆ Aut(G) and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic.
For example, consider the infinite cyclic group ℤ = b , embedded as a normal subgroup of the Baumslag–Solitar group BS(1, 2) = a, b . With respect to the chosen generating sets, the element b 2 n = a n b a − n {\displaystyle b^{2^{n}}=a^{n}ba^{-n}} is distance 2 n from the origin in ℤ , but distance 2 n + 1 from the origin in BS(1, 2) .
Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions. The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov 's notion of word growth .