The Effect of Sample Property on Optimum Search by Quantum Computing Hamed Edalati Fard, Majid Vafaei Jahan, Mehrdad Jalali Department of Computer Engineering Mashad Branch, Islamic Azad University Mashad, Iran hamed.edalati@gmail.com, VafaeiJahan@mshdiau.ac.ir, mehrdadjalali@ieee.org Abstract—Quantum computers are designed based on quantum mechanics. They have special features such as entanglement and parallelism, which do not exist in classic mechanicsbased computers. Therefore, quantum algorithms have their own privilege for solving some problems compare to classic ones such as finding the minimum value of a function in optimization problems. For instance, finding the minimum of N elements in quantum method is faster than classic method. In this case, having information about N elements of distribution does not reduce the cost of finding the minimum value in classic method due to linear search of each element. But in quantum method, all elements are simultaneously considered as well as distribution information, which is related to the whole elements. This distribution information effectively influences on finding the minimum value. Numerical simulations show having mean and variance of N elements can reduces the cost of minimum finding through quantum method by %40. Furthermore, it is shown the greater variance causes less cost. Keywords optimization; quantum search; adaptive search; sample distribution; mean; variance. I. INTRODUCTION The concept of quantum computers was presented in early 1980s. These types of computers are like classic computers with this difference that their base of working is quantum mechanics instead of classic one . In late 1980s and early 1990s, it was shown that quantum computer power in solving some specific problems is higher than classic computers. In 1994, Shor showed that a quantum computer can solve the known problem of “decomposition of an integer number N to prime factors” in a time order of polynomial log N. Whereas, for this problem about classic computers, there is no efficient known algorithm [1]. In 1996, Grover presented an algorithm could find an element among N unordered elements with the time order of O(√N). The equivalent classic algorithm of this action is of order O(N) [2]. By using this capability, a quantum algorithm for finding the minimum element between N elements is presented that is of order O(√N) versus time order O(N) of its equivalent classic algorithm. The main core of this algorithm is Grover's Search (GS) which in proper times is called in this algorithm. In this text, it is shown that by knowing mean and variance of N elements and using it in quantum algorithm of finding minimum, it could reduce search cost. In section II, some primal concepts of quantum computations are presented. GS which has an extensive application in most quantum computation has been completely described in section III. GS application in optimization problems is studied in section IV. In section V, it is shown that by applying mean and variance of sample's distribution, it is possible to reduce the cost of finding minimum. Section VI is composed of simulation results of a quantum algorithm of finding minimum. Conclusion is presented in section VII. II. PRIMAL CONCEPTS OF QUANTUM COMPUTATIONS In this section, some primal concepts of quantum computations are presented [4]. The basic unit of information in quantum computing is called qubit, which is the abbreviation of quantum bit. A bit can be 0 or 1 in a usual computer. A qubit can also be in or . Furthermore, it can take a state called superposition. This state is a linear combination of states and . If this state is called y , a superposition is written as: æ + æ = æ | | | b a y (1) Here α and β are complex numbers such that: = + b a (2) Since, a qubit can be a superposition of states and , whenever a measurement to be done, the same result wouldn’t be reached. In fact, when a qubit is measured, only it can be found in one of the states or . Quantum mechanic laws tell us absolute squares of α and β in (1) ICCKE2011, International Conference on Computer and Knowledge Engineering Oct. 13-14, 2011, Ferdowsi University of Mashhad, Mashhad, Iran 16