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arXiv:0809.1538

Asymptotic Quantum Search and a Quantum Algorithm for Calculation of a Lower Bound of the Probability of Finding a Diophantine Equation That Accepts Integer Solutions

Rubens Viana Ramos, Juliana Loureiro de Oliveira

Published Sep 9, 2008• 0 citations
arXivHTMLPDF

Abstract

Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.

DOI
10.48550/arxiv.0809.1538

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Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.
We present a quantum version of the classical probabilistic algorithms as studied by Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the counting algorithm originally introduced by Brassard et al. One of the main features of our quantum probabilistic algorithm is its full unitarity and reversibility, which would make its use possible as part of larger and more complicated networks in quantum computers. As an example of this we describe polynomial time algorithms for studying some important problems in number theory, such as the test of the primality of an integer, the so called prime number theorem, and Hardy and Littlewood's conjecture about the asymptotic number of representations of an even integer as a sum of two primes.
We consider Grover's algorithm of quantum search for one or several integers out of N = 2n, where n is a number of quantum bits in the memory register. There is a black-box or subroutine containing information about hidden integers and it can easily recognize these integers but we do not know which ones out of N they are. To find the hidden items we can do no better with a classical computer than to apply the subroutine repeatedly to all possible integers until we hit on the special one and in the worst case we have to repeat this procedure N times. We have analyzed the Grover algorithm carefully and showed that it enables to speed up this search quadratically, although its realization requires to know a number of hidden items. The lower bound for the probability of successful solving the search problem has been obtained. The validity of the results was demonstrated by simulation of the Grover search algorithm using the package QuantumCircuit written in the Wolfram Mathematica language.
The nature of the search Suppose you know that exactly one n -bit integer satisfies a certain condition, and suppose you have a black-boxed subroutine that acts on the N = 2 n different n -bit integers, outputting 1 if the integer satisfies the condition and 0 otherwise. In the absence of any other information, to find the special integer you can do no better with a classical computer than to apply the subroutine repeatedly to different random numbers until you hit on the special one. If you apply it to M different integers the probability of your finding the special number is M/N . You must test ½ N different integers to have a 50% chance of success. If, however, you have a quantum computer with a subroutine that performs such a test, then you can find the special integer with a probability that is very close to 1 when N is large, using a method that calls the subroutine a number of times no greater than. This very general capability of quantum computers was discovered by Lov Grover, and goes under the name of Grover's search algorithm . Shor's period-finding algorithm and Grover's search algorithm, together with their various modifications and extensions, constitute the two masterpieces of quantum-computational software. One can think of Grover's black-boxed subroutine in various ways. The subroutine might perform a mathematical calculation to determine whether the input integer is the special one. Here is a simple example.
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.
Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer values for the unknown variables and check for equality. Grover’s algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the Diophantine equation, Grover’s algorithm can be used to find solutions in a brute force way more efficiently than classical methods. We present a hand-coded example for the simplest possible Diophantine equation, and results for a more complicated, but still simulable, equation encoded with a high-level quantum language.
We show that Durr and Hoyer's quantum algorithm of searching for an extreme point of an integer function cannot be sped up for functions that are chosen randomly. Any other algorithm acting in a substantially shorter time o(√2n) (n →α) gives an incorrect answer for the functions phi with the single point of maximum chosen randomly with probability Perr →1. The lower bound as Ω(√2n /b) is established for the quantum search for solution of the equation f (x) = 1, where f is a Boolean function with b such solutions chosen at random with asymptotic probability 1 (n→α).