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arXiv:quant-ph/9907020 · Physical Review A

Quantum probabilistic subroutines and problems in number theory

A. Carlini, Akio Hosoya

Published Aug 18, 2000• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.62.032312

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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 present a quantum version of the classical probabilistic algorithms a' la Rabin for the test of primality of a given integer. Our quantum algorithm is based on the use of Grover's unitary operator for searching an unstructured database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the quantum counting algorithm by Brassard et al. Our quantum probabilistic algorithm is fully unitary and reversible, and can be used as part of larger and more complicated quantum networks. Polynomial time algorithms for testing the primality of an integer, the ‘prime number theorem’ and a conjecture about the asymptotic number of representations of an even integer as a sum of two primes are also discussed.
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.
Abstract This chapter presents early quantum algorithms. We first describe Deutsch’s and Bernstein-Vazirani, some of the first quantum algorithms that showed a query advantage over their classical counterparts. We then present Grover’s algorithm, a quantum algorithm that finds an item in an unstructured database with a quadratic speed-up over classical algorithms. We explain the quantum Fourier transform, an important subroutine in various quantum algorithms. The chapter concludes with a detailed presentation of two of the most prominent applications of the quantum Fourier transform: the period finding algorithm and Shor’s algorithm for prime factorization.
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.
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.