A large proportion of statistical inference tasks can be framed as either an optimisation or an integration problem. Markov chain Monte Carlo (MCMC) algorithms can be used to solve both, but are most commonly used for the latter. They have been successful in such important and diverse settings as the observation of gravitational waves, modelling the spread of infectious diseases, and predicting the results of elections from political polling data. MCMC algorithms are also popular outside of statistical inference, in particular their use is widespread for molecular dynamics simulations in statistical physics. Despite their numerous successes, current MCMC algorithms has some known drawbacks. A prominent example is their performance when model parameters vary over very different scales and exhibit multiple levels of inter-dependence (heterogeneous models). There is an increasingly urgent need to improve this performance as high volume and highly heterogeneous datasets become more and more available, and as researchers begin to ask progressively more nuanced questions from their data for which heterogenous models are needed. The standard approach to MCMC for heterogeneous models is through adaptive pre-conditioning of algorithms. Doing this naively in a high-dimensional setting comes at a significant cost (the required number of operations per algorithm step is often cubic in the number of model parameters, and the number of algorithmic tuning parameters to learn is quadratic). In addition, current state of the art algorithms such as Hamiltonian and Langevin Monte Carlo work particularly poorly in combination with the technique, as has recently been shown both theoretically and experimentally by myself and others. In this proposal I will attack this problem on two fronts. In the first work package I will develop and study a new suite of MCMC algorithms that are specifically tailored to heterogeneous models. I will do this by designing algorithms based on the recently derived class of Markov processes termed 'locally-balanced', for which there is considerable evidence of improved robustness to model heterogeneity. I will provide a rigorous foundation for this class of Markov processes, establish key theoretical properties on convergence to equilibrium and optimality, and then design new algorithms based on this class of processes, each tailored towards specific application areas of known interest. In the second work package I will develop new theoretically grounded methodology for scalable adaptive pre-conditioning of algorithms. I will do this in part by taking inspiration from the literature on sparse estimation of covariance matrices for high-dimensional datasets. I will design methods that are both scalable to high-dimensional settings and for which theoretical guarantees can be established, to provide a clear indication of expected performance gains. This should improve the applicability of existing state of the art methods such as Hamiltonian Monte Carlo to the high-dimensional and heterogeneous model setting. There is a keen focus on integrating new methods within widely used statistical software within the proposal. To this end, I have planned collaborations with the founding developers of the 'Stan' statistical programming language, which has over 100,00 users, as well as detailed plans to create bespoke open source packages in software such as R and Python. I also outline plans to work closely with data scientists to apply the new methodology in many prominent application areas.

The aim of this project is to form a network of international collaboration tasked with creating a new way to write software for quantum computers based upon using tensor networks. Tensor networks are amongst the very best ways to model quantum systems on a classical computer. The possibilities for a quantum system are so numerous that they cannot all be described on any classical computer. The problem is a profound one - to go from describing 30 quantum spins (around the limit for today's supercomputers) to 31 doubles the computational requirements, so evolving conventional hardware cannot keep up with the problems that we want to solve. Tensor networks get round this by focusing on the parts of the system that really matter so that we can get an approximate - but highly accurate - description that we can systematically improve as our computer gets better. Some of the most accurate predictions in quantum mechanics have been made using this approach. Tensor networks are also an excellent way of making use of the limited quantum resources available on near term intermediate-scale quantum (NISQ) computers. These computers have limited power - measured by a property of quantum systems known as entanglement - due to the degrading effect that the environment has on quantum correlations. Tensor networks use precisely this entanglement measure to determine how to approximate the most important properties of the quantum system, and it is for this reason that they are such a good way to programme quantum computers. This approach to quantum software has just begun to be developed, but already shows excellent promise. The International Quantum Tensor Network is designed to place the UK at the centre of an international push to further develop the approach. The applications of this quantum software will help to use quantum computers to simulate other quantum systems - with the promise ultimately to revolutionise quantum problems in chemistry and drug design - but also to solve a variety of classical problems including those in machine learning.