Quantum Computational Number Theory

Yoshitaka Sasaki

Prasanna Kumar K

This paper discusses the deep connection between Number Theory, Cryptography, and Quantum Computing techniques in an endeavor to understand their application in developing new cryptographic techniques, optimization problems, and computational complexity. Like modular arithmetic, which uses prime factorization, in most quantum algorithms like Shor’s and Grover's algorithms, number theory is one of the significant paradigms. This work gives a detailed analysis of some mathematical derivation, theore,m and proof of these algorithms to their practical use as well as computational efficiency. Discussed issues include scaling, correction of errors, and unsolved concerns applying number theory; all discussed from the perspective of existing quantum architecture. Addressing this new research directions of the field, the paper also presents further developments in quantum cryptography, topological quantum computing, and the algorithmic area. In highlighting both theoretical and computational challenges, this work has shown that the synergy between number theory and quantum computing has the opportunity to revolutionize cryptography and optimization and to lay the groundwork for future quantum advancements

Miklós Banai

In this letter we discuss computing in terms of Gaisi Takeuti's quantum set theory using an explicit example of a rigid body of cuboid form. The universe V(L) of Takeuti will be determined. A set of real numbers in this Universe are also explicitly described including a set of binary numbers. Thus we arrive at the foundations of von Neumann' theory of computing in terms of ordinary binary numbers. Then we see that this extension of computing to the Universe V(L) provides a sound, mathematically well defined theory of quantum computing.

Kenji Tokuo

Stefano Guerrini, Simone Martini, Andrea Masini

We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum Turing machine. The class of quantum computable functions is recursively enumerable, thus opening the door to a quantum computability theory which may follow some of the classical developments.