Simon's algorithm can determine whether the function f is bijective or periodic by using the polynomial-function evaluations. Simon's algorithm is more efficient than classical algorithms which require exponential function evaluations. In this paper, we assume that the range of the periodic function is a known subgroup. We further find an integer b orthogonal to the range. According to b, we can construct a quantum algorithm to determine whether f is bijective or periodic in only one function evaluation.
Simon's algorithm can determine whether the function f is bijective or periodic by using the polynomial-function evaluations. Simon's algorithm is more efficient than classical algorithms which require exponential function evaluations. In this paper, we assume that the range of the periodic function is a known subgroup. We further find an integer b orthogonal to the range. According to b, we can construct a quantum algorithm to determine whether f is bijective or periodic in only one function evaluation.
Daniel Simon's 1994 discovery of an efficient quantum algorithm for finding “hidden shifts” of Z 2 n provided the first algebraic problem for which quantum computers are exponentially faster than their classical counterparts. In this article, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G , this is the problem of recovering an involution m = ( m 1 ,…, m n ) ∈ G n from an oracle f with the property that f ( x ⋅ y ) = f ( x ) ⇔ y ∈ {1, m }. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G n , where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G n have a simple product structure, they share important representation--theoretic properties with the symmetric groups S n , where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called “standard method” requires highly entangled measurements on the tensor product of many coset states. In this article, we provide quantum algorithms with time complexity 2 O (√ n ) that recover hidden involutions m = ( m 1 ,… m n ) ∈ G n where, as in Simon's problem, each m i is either the identity or the conjugate of a known element m which satisfies κ( m ) = −κ(1) for some κ ∈ Ĝ . Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the “missing harmonic” approach of Moore and Russell. These are the first nontrivial HSP algorithms for group families that require highly entangled multiregister Fourier sampling.
Abstract Simon’s problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor’s algorithm. The generalized Simon’s problem is a natural extension of Simon’s problem, and also a special hidden subgroup problem: Given a function f : {0, 1} n → {0, 1} m , with the property that for any x, y ∈ {0, 1} n , there is some unknown hidden subgroup S ≤ Z n 2 such that f (x) = f (y) iff x ⊕ y ∈ S, where |S| = 2 k for some 0 ≤ k ≤ n (m ≥ n − k). The goal of generalized Simon’s problem is to find the hidden subgroup S. In this paper, we present two key contributions. Firstly, we characterize the structure of the generalized Simon’s problem in distributed scenario and introduce a corresponding distributed quantum algorithm. Secondly, we refine the algorithm to ensure exactness due to the application of quantum amplitude amplification technique. Our algorithm offers exponential acceleration compared to the distributed classical algorithm. When contrasted with the centralized quantum algorithm for the generalized Simon’s problem, our algorithm’s oracle requires fewer qubits, thus making it easier to be physically implemented. Particulary, the exact distributed quantum algorithm we develop for the generalized Simon’s problem outperforms the best previously proposed distributed quantum algorithm for Simon’s problem in terms of generalizability and exactness.
The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.
We study quantum computational complexity of several problems connected to the Hidden Subgroup Problem. This problem drew substantial attention when a polynomial time quantum algorithm for it was found. The algorithm generalized the quantum algorithms for factoring integers and finding discrete logarithms. We consider the the problems in the context of quantum ordered read-once decision diagrams. Our presentation starts with some fundamental functions related to the Hidden Subgroup Problem. These functions include Equality, Periodicity and simplified "Simon". We show linear upper and lower bounds on the width of quantum OBDD representations of these functions. In the second part of the research we show upper and lower bounds for the Hidden Subgroup Test.
We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.
We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upper-bounded by a polynomial in the worst case. We show that a natural generalization of Simon's problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worst-case running time. This is achieved by generalizing both Simon's and Grover's algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box.