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Workshop on Computation: Theory and Practice (WCTP)

An Oracle Design for Grover’s Quantum Search Algorithm for Solving the Exact String Matching Problem

Jeffrey A. Aborot

Published Dec 1, 2018• 3 citations
Publisher
DOI
10.1142/9789813279674_0003

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We propose new quantum circuit designs for the Grover oracle, i.e., the oracle of Grover’s algorithm, to solve the vertex cover problem (VCP) efficiently. Grover’s algorithm is an unstructured quantum search algorithm to search a target data item out of N data pieces by calling the oracle O(N)\mathrm{O}(\sqrt{N})O(N​) times. It represents a quadratic speedup compared to the classical unstructured data sequential search algorithm, which requires O(N) oracle calls. We are in the noisy intermediate-scale quantum (NISQ) era in which quantum computers have a limited number of qubits, short decoherence time, and low gate fidelity. It is thus desirable to design quantum circuits with three properties: (1) a reduced number of qubits, (2) shorter quantum depth, and (3) fewer gates. We created novel designs of a quantum counter and a quantum semaphore to be applied to the Grover oracle to have the three properties. Through the IBM Quantum Lab service, we executed Grover’s algorithm with the novel quantum circuit designs to solve the VCP to validate the design correctness.
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.
Abstract Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the algorithm, especially when Grover's algorithm is a subroutine in other algorithms. Grover's algorithm can be deterministic if the phase of the oracle or the diffusion operator is delicately designed. The precision of the phases could be a problem. A near‐deterministic quantum search algorithm without the phase design is proposed. The algorithm has the same oracle and diffusion operators as Grover's algorithm. One additional component is the rescaled diffusion operator. It acts partially on the database. The success probability of Grover's algorithm is improved by the partial diffusion operator in two different ways. The possible cost is one or two more queries to the oracle. The deterministic search algorithm is also designed when searching one out of eight, sixteen, and thirty‐two.