Mathematical aspects of quantum computing

Quantum technologies have gained immense technological importance as they possess the potential to revolutionise society within the next number of decades. The primary focus of this research is to partake in this revolution and assist in developing truly quantum processing. This PhD research is expected to highlight this fact on certain key fronts. The project aims to use a combination of information-theoretic and mathematics-motivated analytical methods; tools from linear algebra and probability measure theory (and in certain cases accompanied with numerical testing) will be employed. Our analysis of quantum systems will be done using results from measure theory and C*-algebras of Hilbert spaces.

The first challenge is to examine complementary approaches to quantum computation via the technique of path integration. This task is expected to demonstrate a biology-inspired approach to quantum computing, which is to be achieved by recreating well known quantum computing architectures through genetic algorithms. Underpinned by theoretical derivations, the task will involve a reexamination of the traditional Hamiltonian formalism of quantum mechanics in terms of Lagrangian path integrals. The Hamiltonian and Lagrangian approaches offer complementary views of the same quantum theory, but often problems which are very difficult to solve in one formalism are easy in the other; the path integral formalism is much better suited to calculating scattering amplitudes in high energy physics (with the help of Feynman diagrams), while the canonical formalism is far better for calculating energy levels of bound systems like the hydrogen atom.

The second challenge is centred on determining optimal constructions of quantum circuitry in the sense that for a particular network, one cannot accomplish the same task with fewer constituent quantum gates. One approach that may be exploited here is the use of integer sequences, which is a method recently pioneered by the group here at NTU. Interestingly, an extension of this method may allow the use of combinatorial hypergeometric series to be exploited in the overall quantum design set-up. Finally, methods governing the assessment of entropy and weak entanglement in quantum information processing will be based on recent developments on quantum information theory, especially, in entanglement theory.

This project will require a student with an interest in the investigation of quantum information theory, and the ability (and confidence) to adapt or combine existing mathematical techniques to generate new, more appropriate quantum designs.

Specific qualifications/subject areas required of the applicants for this project:
Applicants are expected to be highly motivated and creative individuals with strong academic records, and in receipt of a BSc Hons (2:1 or above) (or UK equivalent according to Naric) in mathematics or computer science. Experience in theoretical physics or quantum information is essential.