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Journal of Physical Mathematics

The Quantum Sieve of Eratosthenes

Artur Sowa

Published Jan 1, 2016• 1 citation
Publisher

Abstract

We introduce and examine quantum states of a special kind, referred to as E-states, whose properties are both structurally and functionally analogous to the sieve of Eratosthenes. More broadly, the concept of an E-state is related to a certain noncommutative extension of the Dirichlet ring, also discussed here for the first time. Furthermore, we demonstrate that E-states can be implemented on a universal quantum computer and, as a particular application, we construct an algorithm which implements the Dirichlet multiplication of sequences on a quantum computer. We also discuss the potential applicability of E-states to the problem of integer factorization although, we haste to add, we are not aware at present of the possibility of using this approach to obtain algorithms of sub-exponential complexity.

DOI
10.4172/2090-0902.1000180

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We introduce and examine quantum states of a special kind, referred to as E-states, whose properties are both structurally and functionally analogous to the sieve of Eratosthenes. More broadly, the concept of an E-state is related to a certain noncommutative extension of the Dirichlet ring, also discussed here for the first time. Furthermore, we demonstrate that E-states can be implemented on a universal quantum computer and, as a particular application, we construct an algorithm which implements the Dirichlet multiplication of sequences on a quantum computer. We also discuss the potential applicability of E-states to the problem of integer factorization although, we haste to add, we are not aware at present of the possibility of using this approach to obtain algorithms of sub-exponential complexity.
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer NNN is achieved in a number of steps equal to the number kkk of its prime factors, -- e.g., if NNN is the product of two primes, two quantum measurements are enough, regardless of the number of digits nnn of the number NNN. Since kkk is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.
Quantum computing is a winsome field that concerns with the behaviour and nature of energy at the quantum level to improve the efficiency of computations. In recent years, quantum computation is receiving much attention for its capability to solve difficult problems efficiently in contrast to classical computers. Specifically, some well-known public-key cryptosystems depend on the difficulty of factoring large numbers, which takes a very long time. It is expected that the emergence of a quantum computer has the potential to break such cryptosystems by 2020 due to the discovery of powerful quantum algorithms (Shor's factoring, Grover's search algorithm and many more). In this paper, we have designed a quantum variant of the second fastest classical factorization algorithm named "Quadratic Sieve". We have constructed the simulation framework of quantized quadratic sieve algorithm using high-level programming language Mathematica. Further, the simulation results are performed on a classical computer to get a feel of the quantum system and proved that it is more efficient than its classical variants from computational complexity point of view.
Quantum Information and Quantum Computation are rapidly evolving domains in their theoretical foundations, implementations and applications. The main scope of the present book is to give the basic ideas of this field to undergraduate students in Physics. A rigorous approach is used to provide the key concepts and methods. Many examples are discussed. To be self-contained, the principles of Quantum Mechanics are reviewed. The discussion is aimed to the specific, but paradigmatic, factorization problem of integers, dealt by the Quantum Shor’s Algorithm. Thus, notions of the Number Theory are given in the first Chapter, while the exposition of the algorithm and its implementation is reserved to the last Section.
Abstract We report the experimental realization of the prime number quantum potential VN(x), defined as the potential entering the single-particle Schrödinger Hamiltonian with eigenvalues given by the first N prime numbers. Using computer-generated holography, we create light intensity profiles suitable to optically trap ultracold atoms in these potentials for different N values. As a further application, we also implement a potential whose spectrum is given by the lucky numbers, a sequence of integers generated by a different sieve than the familiar Eratosthenes’s sieve used for the primes. Our results pave the way toward the realization of quantum potentials with arbitrary sequences of integers as energy levels and show, in perspective, the possibility to set up quantum systems for arithmetic manipulations or mathematical tests involving prime numbers.
The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.