A Modification of Grover's Quantum Search Algorithm

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Photonics and Optoelectronics

P&O

A Modification of Grover’s Quantum Search Algorithm I. Ashraf1, T. Mahmood2, V. Lakshminarayanan3 1

Department of Physics, Quaid-i-Azam University Islamabad, 45320, Pakistan 2 Department of Physics, Hazara University Mansehra, Pakistan 3 School of Optometry, Department of Physics and Electrical Engineering, University of Waterloo 200 University Avenue West, Waterloo, Ontario N2l 3G1Canada 1 drimrana@comsats.net.pk; 2tmabbasi2004@yahoo.com; 3 vengu@uwaterloo.ca

In this paper we propose a fast quantum search algorithm inspired by Grover’s search algorithm [11]. It has shown that using the same technique as Grover’s algorithm but by

Abstract-We propose a quantum search method based on Grover’s algorithm. This algorithm is described and we show that to search for a single marked element from an unsorted search space of N elements, the number of queries are required using this algorithm when compared to for the Grover algorithm.

dividing the register with sub-registers, we can find

Keywords-Quantum Search Algorithm; Grover’s Technique; Sub-register

([ ( π + 2 ) / 4 ] N ) 1

O

3

n  3 

a

elements into M = 2 marked element in

steps instead of O

( N ) as

in

Grover’s algorithm.

INTRODUCTION I. A computation is a physical process. It may be performed by a piece of electronics, on an abacus, or in our brain. But it is a process that takes place in nature and as such, it is subject to the laws of physics. Application of principles of quantum mechanics in the development of techniques for computation and information processing has given birth to the science of quantum computation. According to Feynman [1], the computers based on laws of quantum mechanics instead of classical physics could be used to model quantum mechanical systems and other large scale computations. Quantum computers are machines that rely on quantum phenomena such as quantum interference and quantum entanglement in order to perform the computation [2-4].

The contents of this paper can be summarized as follows. Section II describes the Grover’s quantum search algorithm for an unsorted database. In Sec. III, we give the details of our fast quantum search algorithm based on the idea of subregisters. In the last section, we will explain and discuss our results. II. GROVER’S QUANTUM SEARCH ALGORITHM Quantum mechanics can speed up range of search applications over unsorted data. Consider a search problem having an unsorted database containing N elements, out of which just one element satisfies a given condition that it is marked. The problem is to find this marked element. Once an element is examined, it is possible to tell whether or not it satisfies the condition in one-step. However, there do not exist any sorting on the database that would aid its selection. Classically, searching an unsorted database requires linear search, i.e, examine the items in database one by one. One has to keep track of the examined elements so that it is not checked again. To find marked elements with a probability of 50% any classical algorithm, deterministic or probabilistic, will need to access the database a minimum of 0.5 N times (and N times in the worst case). Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple elements; therefore, it can speed up the search. Grover’s quantum search algorithm shows that by using the same amount of hardware as in the classical case, but having input and output in superposition of states, we can

The quantum mechanically produced computers will speedup certain computations dramatically. But one of the main difficulties of quantum computers is that de-coherence destroys the information in the superposition of states contained in a quantum computer, thus making long computations impossible. It has been shown [5] how to reduce the effect of de-coherence for information stored in quantum memory. Quantum computers offer an essential speed advantage over classical computers [6]. Some recent reviews [7, 8] have attempted to explain how a quantum computer differs from a classical, conventional computer. In order to solve a particular problem, computers, be it classical ones or quantum, follow a precise set of instructions called an algorithm. But the kinds of search algorithms that can be run on a quantum computer are qualitatively different from those that run on classical computers [9].

find an element in O

( N)

quantum mechanical steps

instead of O ( N ) classical steps. It provides a quadratic speedup over its classical counterpart, which is considerable when N is large. Grover’s algorithm is probabilistic in the sense that it gives the correct answer with high probability.

Over the past years, several quantum algorithms have emerged. Some are exponentially faster than their best classical counterparts [10]; others are polynomial-faster [11]. While a polynomial speed up less than we would like ideally, quantum search has proven to be considerably more versatile than the quantum algorithms exhibiting exponential speedups.

III.

MODIFIED QUANTUM SEARCH ALGORITHM

P&O Volume 1, Issue 1 Apri 2012 PP. 20-24 www.jpo-journal.org○ C World Academic Publishing 20


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