Division of Applied Mathematics, Mälardalen University, Västerås, Sweden

Biography: Dr Richard Bonner is Associate Professor at the Division of Applied Mathematics, Mälardalen University, Västerås, Sweden. After completing his PhD in Mathematics (systems of differential and convolution equations in several variables) at the University of Stockholm in 1984, he held a two-year postdoc position at the Mittag-Leffler Institute, Stockholm, and spent the following two years as researcher at the Centre for Mathematical Analysis, Australian National University, Canberra, and the University of Queensland, Brisbane. His interests then shifted towards border areas of Computing and Economics, spending eight years as senior lecturer in Economic Informatics at Griffith University, Brisbane, focusing on the (mathematical) methods of Artificial Intelligence. After returning to Sweden seventeen years ago to take up his current position, he became involved in the theories of quantum computation and headed international research projects that effectively initiated the area of quantum learning.

Speech Title: ON QUANTUM LEARNING: THE PLAY AND THE REQUISITES
Abstract: We attempt a bird’s view on where the theory of quantum learning would be standing today, other than at the crossroads of computer science, cognitive science, and quantum mechanics. All three roads are inherently mathematical, so our account must be too, even when not worded fully formally.
The formal structures representing knowledge and therefore standard in mathematical logic, such as semi-lattices and pre-sheaves, recall, are all in particular partially ordered sets. The partial order encodes the intuition that any state of knowing implies the lower states, be it at a cost, for the theories of computation to determine (the states being in practise effectively presented). The generic name for the processes that increase knowledge is learning. The problem of learning is then inverse to deduction: to move, from low states of knowing, to succinct higher states. In a pre-sheaf on a topology, for example, this may formally amount to gluing sections. Theories of computation now enter again, so states may be effectively arrived at, along with the standard questions of mathematical analysis about how well the global (typically infinite) objects would be determined by the local (typically finite) data.
We follow in spirit the well-known paper by Cucker and Smale (2001, Bulletin of the American Mathematical Society, Vol. 39, No. 1, pp. 1–49), outlining the analytical tools that the mathematical scientist may consider using for the learning problem in classical (Boolean) settings. The word quantum, which I add, independently qualifies the device representing the knowledge (the register) and the device performing the computation (the processor). This formally produces three classes of quantum models; the justifiable questions of their physical feasibility are for the theory of quantum information to answer.
We then briefly survey, with focus on the logical and the analytical aspects, the limited writings on quantum learning. In particular, we trace the role of the Hilbert space structure, always basic for quantum physics, but also, since quite recently, for classical data analysis.