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Nanosystems Physics Chemistry Mathematics

Stability of Grover’s algorithm in respect to perturbations in quantum circuit

Kseniya Gubaidullina, Sergey A. Chivilikhin

Published Apr 20, 2017• 4 citations
Publisher

Abstract

Grover's algorithm is a quantum algorithm for searching specified elements in an unsorted list.It has many valuable applications.The utilization of Grover's algorithm, to adapt it to accelerate the works of well-known classical algorithms, is very promising, and it is one of the fastest algorithms to solve such problems like global optimization and graph coloring.In this regard, it is very important to study the stability of the Grover's algorithm, to know how distortion of the circuit's elements affects on it results.This work presents the results of the simulation of Grover's algorithm, research of its stability with respect to perturbations of quantum logic circuit elements and its dependencies from the number of qubits, used in quantum circuit.Another part of this research was realized on IBM quantum processor and shows the stability of the 2-qubit Grover's algorithm.

DOI
10.17586/2220-8054-2017-8-2-243-246

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Grover's algorithm is a quantum algorithm for searching specified elements in an unsorted list.It has many valuable applications.The utilization of Grover's algorithm, to adapt it to accelerate the works of well-known classical algorithms, is very promising, and it is one of the fastest algorithms to solve such problems like global optimization and graph coloring.In this regard, it is very important to study the stability of the Grover's algorithm, to know how distortion of the circuit's elements affects on it results.This work presents the results of the simulation of Grover's algorithm, research of its stability with respect to perturbations of quantum logic circuit elements and its dependencies from the number of qubits, used in quantum circuit.Another part of this research was realized on IBM quantum processor and shows the stability of the 2-qubit Grover's algorithm.
Algorithms and data structures are important in information technology (particularly in 6G terahertz wireless communications), logistics, transportation, and route planning in medicine. One of the fundamental algorithms that find application in many practical problems is the shortest path algorithm. The shortest path algorithm is used to find the optimal route between two vertices while minimizing the length or number of intermediate nodes. This problem is one of the 21 NP-hard problems formulated by Richard Karp in 1972. It still poses a serious challenge to science, since there is no polynomial algorithm that can provide an exact solution to this problem. The article considers approaches to solving the problem of finding the shortest path on a graph, and discusses their advantages and disadvantages. Various algorithms, such as the Bellman-Ford algorithm, Dijkstra, and the A* algorithm, are considered; their efficiency and applicability in various situations are analyzed. Issues of optimization of shortest path algorithms are discussed, such as the use of heuristics, graph preprocessing, and other methods that can improve the performance and accuracy of algorithms. The article proposes an optimization of Dijkstra’s algorithm with quantum acceleration. The stability of the quantum algorithm and the magnitude of distortions in a circuit with a different number of qubits are estimated. It is shown that with an increase in the intensity of distortions, the probability of detecting the desired element decreases, and the time spent on its search increases. When using Grover’s algorithm to optimize a path in a quantum network, these interferences cause an increase in the time it takes to find the optimal path and an increase in the probability of choosing an incorrect route. It is shown that an increase in the number of qubits and, accordingly, the number of iterations makes the algorithm more stable. It is demonstrated that the stability of Grover’s algorithm for circuits with a large number of qubits has a similar nature. The work of quantum circuits on 9, 10 and 11 qubits, the difference in the probabilities of correct operation differs by less than 10–3. Thus, from the obtained dependencies, it is possible to predict the nature of such an influence on circuits with an even larger number of qubits, the stability of which cannot be studied due to the insufficient computing power of classical computers. The most efficient way to modify Dijkstra’s algorithm is to replace the priority queue in the classical version of the algorithm with some quantum data structure that finds the minimum element for the next iteration of the algorithm. Despite the fact that quantum search does not provide a potential gain in asymptotics, another quantum modification of Dijkstra’s algorithm can be used. Namely, instead of maintaining and searching for a minimum in the priority queue, use the quantum algorithm immediately to search for a vertex that needs to be added to the shortest path tree, which will additionally acceleration the graph traversal by Dijkstra’s algorithm.
Quantum computing is a quickly growing field of research thanks to recent hardware advances. The quantum mechanical properties of quantum computers allow them to solve certain families of problems faster than classical computers. A quantum algorithm solving such a problem is Grover's algorithm, which finds an element in an unordered set faster than any classical search algorithm. In this paper an implementation of a 4-qubit Grover's algorithm for the IBM Q computer ibmqx5 is presented. Executing the implementation on an ibmqx5 simulator yield results in line with the theoretically optimal results. The accuracy of the ibmqx5 simulation results compared to the ibmqx5 execution results suggests that current hardware is not yet suitable,due to required complexity of the circuits for a 4-qubit Grover implementation.
Grover’s algorithm is a quantum search algorithm that makes a huge difference in finding an element of interest from unsorted and unstructured data. In the case of a classical search algorithm, testing O(N) elements within a dataset of size N is a significant number. Grover’s algorithm reduces this number to O(N)O(\sqrt N )O(N​) so the gain is obvious. This paper covers applied aspects of Grover’s algorithm, namely quantum circuit layout, efficiency of gates, fidelity of qubits, and the optimization of resources for searching unstructured data. Focus has been placed on quantum error correction: namely, planar surface code; resource costs involved have been quantified. The paper discusses the implications of quantum computing on data retrieval and how algorithms in quantum computing are crucial for solving issues presented by a changing environment of computation.
We propose the concepts of the explicit oracle and the implicit oracle for realizing quantum algorithms. Then, the quantum circuit of the well-known Grover algorithm is constructed with the explicit oracle to solve the Hamiltonian cycle problem for the complete graph. The quantum circuit has a quadratic speedup over the classical unstructured search algorithm for solving the same problem. The IBM quantum computer simulator is used to run the quantum circuit to validate that it can indeed derive the Hamiltonian cycle of the complete graph.
This study investigates the applications and adaptations of the Grover's search algorithm, an algorithm that uses quantum technologies to provide quadratic speedup while searching unsorted data. Through extensive literature review, the applications of Grover's algorithm such as database search, machine learning, cryptography, and optimization is explored in order to further understand areas open to alteration in the algorithm. Visuals are provided with the aim of enhancing the comprehension of Grover's algorithm. This study underscores how various components of Grover's search-like the oracle, diffusion operator, and iteration count-can be modified to solve complex problems more efficiently than classical search algorithms. A table generalizing aspects of modification and each of their benefits is given as an output. The growing importance of quantum technologies and the United Nations' selection of 2025 as the International Year of Quantum Science and Technology makes this research timely.