Determining whether a given algebra is commutative or not is important in the study of these algebraic objects in general and in the classification of semifields in particular. The best classical (ie, nonquantum) algorithm for this task has a running time that is of order O(n3), where n is the dimension of the algebra. To reduce this cost, in this paper, we study an approach to test the commutativity of a finite-dimensional algebra using quantum adiabatic computing. Previous quantum algorithms solving the same problem were based on Grover's quantum search. The algorithm is built from a quantum oracle for the multiplication constants of the algebra. Results of the experiments carried out on a quantum computer simulator, based on two different annealing schedules, are presented, showing that a quantum adiabatic algorithm for the problem can determine the commutativity of finite-dimensional algebras with one-side bounded error with a running time of order , achieving a quadratic speedup over the classical case.
The search for ``a quantum needle in a quantum haystack'' is a metaphor for the problem of finding out which one of a permissible set of unitary mappings---the oracles---is implemented by a given black box. Grover's algorithm solves this problem with quadratic speedup as compared with the analogous search for ``a classical needle in a classical haystack.'' Since the outcome of Grover's algorithm is probabilistic---it gives the correct answer with high probability, not with certainty---the answer requires verification. For this purpose we introduce specific test states, one for each oracle. These test states can also be used to realize ``a classical search for the quantum needle'' which is deterministic---it always gives a definite answer after a finite number of steps---and 3.41 times as fast as the purely classical search. Since the test-state search and Grover's algorithm look for the same quantum needle, the average number of oracle queries of the test-state search is the classical benchmark for Grover's algorithm.
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring
R=∠r1,...,rk where
{r1,r2,...,rk} is an additive generating set for
R and a multilinear polynomial
f(x1,...,xm) over
R also accessed as a black-box function
f:Rm→R (where we allow the indeterminates
x1,...,xm to be commuting or noncommuting), we study the problem of testing if
f is an \emph{identity} for the ring
R. More precisely, the problem is to test if
f(a1,a2,...,am)=0 for all
ai∈R. We give a quantum algorithm with query complexity
O(m(1+α)m/2km+1m) assuming
k≥(1+1/α)m+1. Towards a lower bound, we also discuss a reduction from a version of
m-collision to this problem. We also observe a randomized test with query complexity
4mmk and constant success probability and a deterministic test with
km query complexity.
Abstract Quantum computing can solve some classes of problems faster than classical computing. Among the greatest quantum algorithms to showcase this benefit is Grover's search algorithm, which has a quadratic speed-up for unstructured search problems. In this study, we present an implementation of Grover’s algorithm on a large-scale dataset comprising approximately 131,072 (217) names. We contrast its efficiency with the classical linear search based on time complexity, scalability, and the likelihood of success. The experiment was conducted using Qiskit on a quantum simulator, where several query scenarios were implemented to verify the algorithm's efficiency and accuracy. Our findings highlight the theoretical advantage of Grover’s algorithm in situations involving large data sets, while also discussing real-world limitations such as gate noise, qubit number, and circuit depth as we move towards actual quantum hardware. This study serves as a step toward evaluating the real-world relevance of quantum search algorithms and provides information on current challenges and future directions in quantum information retrieval.
A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.
We develop a generic method for quantizing classical algorithms based on random walks. We show that under certain conditions, the quantum version gives rise to a quadratic speed-up. This is the case, in particular, when the Markov chain is ergodic and its transition matrix is symmetric. This generalizes the celebrated result of L. K. Grover (1996)and a number of more recent results, including the element distinctness result of Ambainis and the result of Ambainis, Kempe and Rivosh that computes properties of quantum walks on the d-dimensional torus. Among the consequences is a faster search for multiple marked items. We show that the quantum escape time, just like its classical version, depends on the spectral properties of the transition matrix with the marked rows and columns deleted.