Quantum Computing and Post-Quantum Cryptography

The advent of quantum computers will have a big impact on cryptography. For example, Shor's algorithm for factoring integers will render most number-theoretic cryptosystems obsolete. Postquantum cryptography is aimed to address this issue by developing cryptographic protocols that will remain secure in the quantum era. Lattice-based constructions seem to be the most promising out of many proposals, not only in terms of security, but also in terms of simplicity and efficiency. These cryptosystems rely on the computational hardness of the shortest vector problem (SVP) and its variants. So far, no efficient quantum algorithms are known for these problems, but this is an open research question. In this thesis, a quantum annealing approach is developed for finding the minimum distance on a lattice and solving SVP. The necessary background in cryptography and quantum information theory is introduced in the first chapter, using the RSA cryptosystem and Shor's factorisation algorithm as examples, to highlight the interplay between these two closely related disciplines. Additionally an overview of the Bose-Hubbard model for cold molecules in an optical lattice is given, which will serve as the basis for the quantum annealing algorithm. The second chapter describes the construction of an algorithm that solves SVP and CVP using quantum annealing and discusses adiabatic evolution and its most important features. In chapter three this algorithm is tested by performing simulations for small lattices, trying to highlight different aspects of the system and understand its behaviour. The ultimate goal of this chapter is to investigate the scaling of the speed of the algorithm with system size. This is done for lattice sizes up to ve lattice sites. The scaling appears to be exponential, although more simulations are needed to determine this relationship accurately

The main impact of the proposed Hub will be in training quantum engineers with a skillset to understand cutting-edge quantum research and a mindset toward developing this innovation, and the entrepreneurial skills to lead the market. This will grow the UK capacity in quantum technology. Through our programme, we nurture the best possible work force who can start new business in quantum technology. Our programme will provide multi-level skills training in quantum engineering in order to enhance the UK quantum technologies landscape at several stages. Through the training we will produce quantum engineers with training in innovation and entrepreneurship who will go into industry or quantum technology research positions with an understanding of innovation in quantum technology, and will bridge the gap between the quantum physicist and the classical engineer to accelerate quantum technology research and development. Our graduates will have to be entrepreneurial to start new business in quantum technology. By providing late-stage training for current researchers and engineers in industry, we will enhance the current landscape of the quantum technology industry. After the initial training composed of advanced course works, placements and short projects, our students will act as a catalyzer for collaboration among quantum technology researchers, which will accelerate the development of quantum technology in the UK. Our model actively encourages collaboration and partnerships between Imperial and national quantum tehcnology centres and we will continue to maintain the strong ties we have developed through the Centre for Doctoral Training in order to enhance our on-going training provisions. The Hub will also have an emphasis on industrial involvement. Through our new partnerships students will be exposed to a broad spectrum of non-academic research opportunities. An important impact of the Hub is in the research performed by the young researchers, PhD students and junior fellows. They will greatly enhance the research capacity in quantum technology. Imperial College has many leading engineers and quantum scientists. One of the important outcomes we expect through this Hub programme is for these academics to work together to translate the revolutionary ideas in quantum science to engineering and the market place. We also aim to influence industry and policy makers through our outreach programme in order to improve their awareness of this disruptive technology.