Quantum technology is an emerging interdisciplinary field, where mathematicians, physicists, computer scientists and engineers combine their knowledge with the prospect of making revolutionary advances in science and engineering and solving important societal and industry challenges. When compared to classical computers, quantum computers will dramatically reduce processing time for some specific tasks and are capable of handling new and otherwise intractable problems. In particular, quantum technology will be crucial for the development of a new era of communication and cryptography.
However, before the potential of quantum computing can be achieved, major theoretical and technological obstacles need to be overcome. At center stage is the inherent problem that the components of a quantum computer inadvertently interact with each other and the environment. As a consequence, one can only perform short and imperfect calculations unless one finds ways to remediate this defect. The vulnerability to errors is currently the single biggest problem holding back quantum computing from realizing its great promise.
Our goal with this project is to obtain new results in quantum error correction having concrete applications for transmission of quantum and classical information under the presence of noise. Moreover, benefiting from added expertise through collaboration with the defense sector, our research will expectedly have impact within communication, cryptography, and cybersecurity. In particular, by using a group theory approach, we will develop new mathematical methods to detect, benchmark, mitigate and correct errors in quantum computations stemming from the interaction with the environment. We aim to find find new formulas to quantify the performance of encryption methods under the presence of noise, and study new ways of measuring the efficiency of quantum error-correcting processes.
Quantum technology is a very active and rapidly expanding field, that has recently attracted a lot of interest from physicists, mathematicians and computer scientists. The power of quantum computers comes from their ability to use quantum mechanical principles such as superposition, measurement, and entanglement. Arguably, one of the most attractive features of quantum computing is that quantum algorithms are conjectured to solve certain computational problems exponentially faster than any classical algorithm. On a practical level, all these new visions are based on the ability to control the quantum states of (a small number of) micro-systems individually and to use them for information transmission and processing. From a more fundamental point of view, quantum communication is based on the application of principles from quantum mechanics in an information theoretical context, formulated in the mathematical language of linear operators and vector spaces.
The goal of the project is to obtain new results in quantum error correction having applications within cryptography and quantum communication based on properties of projective group representations, and by studying new ways of measuring the efficiency of quantum error-correcting processes. In particular, our focus will be on two topics: we first study unitary error bases and frames for quantum systems from a group symmetry perspective, and develop techniques for constructing quantum error-correcting codes. We pay special attention to a class of unitary error bases called nice error bases that come from projective representations of finite groups. Next, we investigate new methods to measure error correction ability for quantum communication channels under the presence of noise. The ability to recover a code under a given noise, subject to various metrics, is then used to give information about the quality of the code.