Related papers: Abelian Hypergroups and Quantum Computation
The hidden subgroup problem (HSP) provides a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an…
Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Z_2^n provided one of the first algebraic problems for which quantum computers are exponentially faster than their classical…
The state hidden subgroup problem (StateHSP) is a recent generalization of the hidden subgroup problem. We present an algorithm that solves the non-abelian StateHSP over $N$ copies of the dihedral group of order $8$ (the symmetries of a…
Hidden Subgroup Problem(HSP) seeks to identify an unknown subgroup H of a group G for a given injective function f defined on cosets of H. Here we present an initialization-free quantum algorithm for solving HSP in the case where G is a…
Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are…
Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while…
An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it…
The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a…
We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.
Symmetries of the finite Heisenberg group represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. As is well known, these symmetries are properly expressed in terms of certain normalizer. This…
Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in…
In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, for p and q prime. We first present a classification of these groups in five classes. Then, we…
We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective…
According to the Gottesman-Knill theorem, a class of quantum circuits, namely the so-called stabilizer circuits, can be simulated efficiently on a classical computer. We introduce a new algorithm for this task, which is based on the…
It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using…
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential…
We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For…
We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a…
Quantum algorithms for factoring and discrete logarithm have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden…
We propose a natural application of Quantum Linear Systems Problem (QLSP) solvers such as the HHL algorithm to efficiently prepare highly excited interior eigenstates of physical Hamiltonians in a variational and targeted manner. This is…