From free fermions to parafermions: how to build a universal topological quantum computer from free particles
Topology plays a prominent role in describing quantum phenomena such as the quantum Hall effect and topological insulators. This burgeoning field of research, also recognised by the 2016 Nobel physics prize, promises practical applications in terms of new ways of storing and manipulating quantum information , which is protected from decoherence (see Figure). A fundamental ingredient of such topological quantum computation is the quasiparticles with non-Abelian exchange statistics, called anyons. In recent years, there has been much effort to experimentally realise the simplest kind of anyon – a Majorana fermion – and use them to build topological qubits. However, the relatively simple physics of Majorana fermions also places limitations on the type of quantum gates that can be simulated. Other types of anyons, like parafermions , which occur in more strongly interacting systems, have richer properties and can perform
more powerful (“universal”) quantum computation.
This project will study the fundamental properties of quantum systems that host parafermion quasiparticles. In contrast to Majorana fermions, which are well understood due to the analogies with topological superconductors, there is still little knowledge about parafermions. The main objective of this project is to understand the intrinsically interacting nature of parafermion states by using the new concept of “interaction distance” we recently introduced in . This allows us to approximate quantum states in a new way that generalises traditional methods, e.g., mean-field theory. Applying the interaction distance measure to parafermion states will give us new insights into the microscopic building blocks of parafermion states, which are reflected in their “entanglement spectra” and other properties that can be diagnosed using quantum information tools .
 Introduction to Topological Quantum Computation, Jiannis K. Pachos, Cambridge University Press, 2012.
 Topological phases with parafermions: theory and blueprints, Jason Alicea and Paul Fendley, arXiv:1504.02476.
 Optimal free models for many-body interacting theories, Christopher J. Turner, Konstantinos Meichanetzidis, Zlatko
Papic, Jiannis K. Pachos, Nature Communications 8, 14926 (2017); arXiv:1607.02679.
 Free-fermion descriptions of parafermion chains and string-net models, K. Meichanetzidis, C. J. Turner, A. Farjami, Z.
Papic, Jiannis K. Pachos, arXiv:1705.09983.
 Simulating the exchange of Majorana zero modes with a photonic system, Jin-Shi Xu, Kai Sun, Yong-Jian Han,
Chuan-Feng Li, Jiannis K. Pachos, Guang-Can Guo, Nature Communications 7, 13194 (2016), arXiv:1411.7751.
Measuring, understanding and manipulating interactions in quantum systems: a novel machine learning approach
The notion of a free particle is the backbone of entire physics. Free particle systems are easy to understand because they can studied via numerous theoretical techniques or simulated in a laboratory. Luckily, nature is “not just a sum” of free particles, as there are many remarkable phenomena where interactions between particles give rise to far more complex phenomena, such as quantum entanglement or exotic phases of matter (high-temperature superconductors, spin liquids and topological insulators). However, quantifying and understanding the interaction effects in quantum systems remains a challenge because describing such systems is exponentially hard due to the very rapid increase of their Hilbert spaces.
In this project, you will develop a new approach to quantify the effect of interactions in quantum systems based on our recent idea of “interaction distance” . Interaction distance measures the distance of any quantum state from the “closest” state of any free-particle system. This new tool allows to identify the effective free-particle description of a given quantum system based on specific patterns in its entanglement. Because of this novel point of view, we have already discovered surprising examples of free descriptions for systems which are naively expected to be strongly interacting .
Simply put, interaction distance allows us to map out the landscape of all quantum states in terms of the complexity of interaction effects in them (see figure). Apart from fundamental importance in quantum information, condensed matter physics and high-energy physics, interaction distance also provides a physical link with the recent approaches based on machine learning to describe quantum systems . Therefore, the second strand of this project is to investigate how to improve and physically benchmark these machine-learning methods for quantum many-body systems using interaction distance.
The fundamental understanding of interaction effects will be applied to several concrete problems, in particular how to use interactions to suppress dynamics in quantum systems, thereby extending robustness of encoded quantum information. More specifically, you will
investigate the possibility of extending topological quantum memories to arbitrary temperature due to the mechanism of “many-body localisation” , and explain the origin of intriguingly slow dynamical regimes that have been observed in a recent 51-atom quantum simulator at Harvard .
Note: the project requires computational background (e.g., Python/Matlab/Julia/C++…).
 Optimal free models for many-body interacting theories, Christopher J. Turner, Konstantinos Meichanetzidis, Zlatko Papic, Jiannis K. Pachos, Nature Communications 8, 14926 (2017); arXiv:1607.02679.
 Machine learning: New tool in the box, Nature Physics 13, 420–421, doi:10.1038/nphys4053.
 Many body localization and thermalization in quantum statistical mechanics, Rahul Nandkishore and David A. Huse, arXiv:1404.0686.
 Probing many-body dynamics on a 51-atom quantum simulator, H. Bernien et al, arXiv:1707.04344.
Light driven molecular motion
Supervisor: Dr. Arend Dijkstra
Absorption of light can cause large changes in molecular structure. This is important for biological function, for example in the first steps in vision. These processes happen on very fast time scales, and only recent experiments have been able to observe them in detail. Man made molecules can use similar principles to function as ultrafast photo switches.
This project aims at developing new theory to model motion in a molecule after light absorption. Detailed models of the interaction of an electronic state with a complex quantum mechanical environment will be made and their predictions will be compared with experiments. The project combines fundamental theory development in open quantum systems dynamics with practical applications.
Photoactive molecular complexes
In our group, we use models to understand how molecular systems use light to function. These models are compared with state of the art optical experiments, which allow us to probe the fundamental motions of electrons and nuclei that take place on femtosecond to picosecond time scales. Inspiration for our work comes from biological systems. Our work uses mathematics and computer programming. The projects are suitable for chemistry, physics and mathematics graduates. The first project is about photosynthesis. How is the energy that is collected by plants and bacteria from sunlight transported? It turns out that answering this question requires a detailed description of the pigment molecules that interact with the light, as well as of the protein and solvent environment. In this project, you will build a new model of the energy transport mechanism. The model will be based on quantum mechanics of an electronic system interacting with vibrations. A main goal of the project is to accurately determine the parameters that describe real systems, from either simulation or comparison to experiment.
The second project is about photo switching. Some of the fastest events in biology occur within the eye. As in photosynthesis, electrons are excited by light absorption, However, in the primary step in vision the nuclear motion induced by electronic excitation is very large. Cis-trans isomerization in the rhodopsin molecule completely changes the structure. The system clearly explores parts of the potential energy surface far away from equilibrium, such that a harmonic description is completely invalid. This is also the case in man-made photo switches. This is a challenging regime for models that treat both the electronic and the nuclear motion under the influence of the protein environment. This project aims at developing a new theory to describe quantum decoherence and friction outside the harmonic approximation.
 Dijkstra, A. G. and Tanimura. Y., New J. Phys. 14, 073027 (2012);
 Prokhorenko, V. I., Picchiotti, A., Pola, M., Dijkstra, A. G. and Miller, R. J. D., J. Phys. Chem. Lett. 7, 4445 (2016);
 Dijkstra, A. G., Wang, C., Cao, J. and Fleming, G. R., J. Phys. Chem. Lett. 6, 627 (2015).
Quantum Dynamics and Many-Body Localisation via Tensor Networks
Supervisor: Dr Zlatko Papic
Disorder and interactions between particles are common features of most physical systems. What are the possible classes of behaviour that a closed quantum system can have in the presence of both disorder and interactions? Although easy to formulate, this fundamental question of quantum statistical physics remains open.
Most systems in nature reach thermal equilibrium during the course of their evolution because their microscopic dynamics is chaotic. However, this is not the only possibility. Recently, many-body localization  — which arises due to quenched disorder and interactions – has come to attention as a generic mechanism that breaks ergodicity and prevents the system from thermalizing. This is in contrast to Anderson localization and integrable models, which also break ergodicity but in a non-generic way (i.e., they require either an absence of interactions or finely tuned coupling constants).
Many-body localized systems are promising for applications in quantum computing because they are able to “escape” thermalization even at “infinite” temperature. Because of this, their states can retain quantum coherence and are a promising venue to realize quantum order even at high temperature. In this sense, because they avoid dissipation, many-body localized systems can act as protected “quantum memories” for extended periods. Alternatively, many-body localization could boost the stability of topological phases of matter, which are usually fragile and realized only at low temperatures. The exotic non-local correlations in these phases, enhanced by many-body localization, would give another route to constructing protected quantum memory.
The implications of many-body localization are far-reaching: non-ergodic dynamics, the existence of novel phases and phase transitions that are forbidden by traditional statistical mechanics, possibility of designing robust quantum computing schemes, etc. Apart from many exciting theoretical prospects, recent experiments  have detected first evidence of many-body localization in optical lattices, while many other other experiments are currently under way.
Project. The goal of this project is to use quantum information techniques to study properties of many-body localized systems and, more generally, the dynamics of interacting disordered systems far from equilibrium. The project will focus on quantum entanglement in such systems and will include the development of numerical algorithms inspired by entanglement, such as tensor networks . In addition to the theoretical understanding of many-body localized systems, we will identify routes by which complex entanglement structures and their dynamical evolution can be probed and possibly protected from decoherence in experiments. We will also investigate how intricate types of order, like topological order, can be made more robust due to many-body localization.
In the initial phase, we will set up a foundation for the project based on the theoretical progress on many-body localization since 2010. As the toy model, we will focus on the Heisenberg model of spins-1/2 in a random field. We will study the dynamics of entanglement in this model when the system is driven out of equilibrium. The characteristic logarithmic-in-time growth of entanglement will help us arrive at the effective theory of many-body localized phases which was established in 2013. This will allow us to understand some of the universal properties of localized phases that have been observed in experiment . The second goal of the first phase of the project is a numerical implementation of a tensor-network algorithm describing a one-dimensional spin chain (e.g., the transverse-field Ising model). In doing so, we will learn how to study entanglement, dynamics and phase transitions using variational tensor network simulations.
Note: the project is suitable for students with background (or strong interest) in the numerics (C++/Python…).
 Rahul Nandkishore and David Huse, Many body localization and thermalization in quantum statistical mechanics, arXiv:1404.0686.
 Michael Schreiber et al., Observation of many-body localization of interacting fermions in a quasi-random optical lattice, Science 349, 842 (2015).
 Roman Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals of Physics 349, 117-158 (2014).