Mathematics on a Quantum Computer

Tien D. Kieu

We explore in the framework of quantum computation the notion of computability, which holds a central position in mathematics and theoretical computer science. A quantum algorithm that exploits the quantum adiabatic processes is considered for Hilbert's tenth problem, which is equivalent to the Turing halting problem and known to be mathematically non-computable. Generalized quantum algorithms are also considered for some other mathematical non-computables in the same and in different non-computability classes. The key element of all these algorithms is the measurability of both the values of physical observables and the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of mathematics, ought to be determined not solely by mathematics itself but also by physical principles.

Wolfgang Scherer

M. A. Bashar, Mahmudul Hasan, M. Syduzzaman M. A. Chowdhury

The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.Mathematics lies at the heart of quantum computation and it is needed in every aspect of the research and development of quantum computation.In this paper we have presented mathematical concepts that are closely related to quantum computation and their relation to quantum mechanics in a way which is innovative and very easy to comprehend.Most of the mathematical concepts presented in this paper are deduced from the postulates of quantum mechanics and linear algebra.

Liao Heng, Bill McColl

Tien D. Kieu

We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not