# PhD Opportunities

## PhD Project:Topological quantum systems: synthesis and applications

Supervisor: Prof. Jiannis Pachos

Topological phases of matter (Nobel Prize in Physics, 2016) is one of the most exciting topics of modern physics. This area of research investigates the novel properties of materials that are robust against deformations of their parameters. This robustness makes topological material of interest to quantum technologies that request fault-tolerance in order to be useful.

The PhD project will aim to employ a variety of techniques and approaches from mathematics and theoretical physics (e.g. topology, quantum field theory, quantum gravity) in order to diagnose the exotic properties quantum matter can have. A final goal is the application of these investigations to proposing topological quantum computation schemes that are robust against errors.

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## PhD Project: Computer Simulations of Biological Macromolecules

Supervisor: Dr. Sarah Harris

Computational models are invaluable for visualisation in molecular biology, as they employ our best quantitative physical understanding of biomolecules and their interactions to predict their dynamics, which is often missing from biophysical experiments. Now that biophysical techniques are revealing highly organised supermacromolecular architectures at the length-scale directly above that of single molecules, which was invisible until very recently, there is a need for new computational tools to intepret these experiments. We are developing two lines of research in supermacromoleular biology – one for DNA, and one for proteins.

While it is well known that DNA is the molecule of heredity and that the sequence of bases in DNA encodes the genetic information that defines an organism, the way in which genomes are regulated is not understood. Recent experimental data shows that the physical arrangement of DNA within the nucleus is critical to genetic control. We have developed a model system involving small DNA circles that we can analyse both by experimental methods and atomistic computer modelling using well established computer programs to understand how the packaging of DNA helps in the control and regulation of the cell, and how it influences recognition by other molecules, such as proteins and drug molecules.

We are also writing our own modelling software that provides a continuum mechanics description of proteins, and which uses experimental electron microscopy data as input to the calculations. The model uses the Finite Element algorithm that we have generalised to include thermal fluctuations, known as Fluctuating Finite Element Analysis (FFEA), we are using this program to model the action of molecular motors such as myosin and dynein, and are improving our physical description of biomolecules and their interactions by adding more accurate representations of the hydrodynamic environment

Our approach is highly multidisciplinary, and we can adapt projects to suit researchers with backgrounds as diverse as physics, maths, chemistry, biology and computer science. Collaborative projects including experimental work are also available.

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## PhD Project: Near-term development and implementation of quantum algorithms

Supervisor: Dr. David Jennings

The past few years have seen the start of a “quantum space race” that is rapidly gaining momentum, and providing significant technological and conceptual breakthroughs. An array of global research centres, universities and companies are currently competing to develop a full-scale quantum computer. At the heart of this endeavour is a deep statement about physics, namely: Information is physical, and is drastically more powerful in quantum mechanics than in classical mechanics. Despite quantum theory being almost 100 years old, we are only now being to unravel this statement and to build devices that exploit this profound distinction.

The PhD project will exploit recent techniques in quantum information theory and machine learning to develop novel quantum algorithms that could be realised on existing or near-term quantum devices (such as IBM’s superconducting qubits that are already accessible online). Along the way, the project will also engage questions such as “what constitutes a quantum programming language?” and “what physical quantum resources (like entanglement and coherence) are required to realise a quantum computation?”.

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## PhD Project: Quantum Many-Body Scars

Supervisor: Dr. Zlatko Papic

In recent years, investigations of out-of-equilibrium phenomena in quantum many-particle systems have become one of the most active research areas in physics. A recent state-of-the-art experiment at Harvard/MIT [1] has succeeded in assembling large chains of strongly-interacting Rydberg atoms, which allowed them to build an impressive 51-atom quantum simulator. This experiment might not only pave the way to important quantum technology applications, but it has already discovered a new physical phenomenon: when the simulator was driven away from its equilibrium configuration, the experiment observed enigmatic quantum oscillations that remained coherent for unusually long times.

In our recent work [2] (which was also featured in Press, see [3, 4, 5]) we provided an explanation of this intriguing phenomenon by introducing a new concept of quantum many-body scar. In typical systems, as explained by Boltzmann and Gibbs in the 19th century, dynamics is usually chaotic, which allows such systems to reach thermal equilibrium after long times. Our work shows that there are surprising exceptions to this behaviour when a “scar” forms: the quantum system can retain some memory of its initial condition even though its dynamics is chaotic, similar to what happens to a particle scattering in a chaotic billiard [6]. Our work thus introduced a new mechanism to “protect” coherent oscillations in a chaotic system, such as the one engineered by Harvard/MIT.

The goal of this PhD project is to develop a deeper understanding of quantum many-body scars as a new class of systems where ergodicity is weakly broken. One of the pressing questions is what kind of systems support scars. Currently, it is believed that kinetic constraints (such as strong nearest-neighbour interactions between the atoms) are essential to the formation of scars, thus one of the goals of the project would be to investigate more systematically other types of models with similar constraints. On the other hand, the project will aim to answer a fundamental question: what is the meaning of a periodic orbit in a quantum many-body system? Such orbits play a fundamental role in the theory of single-particle chaotic billiards [6], but their meaning for a quantum many-body system is currently an open problem. Finally, the project will also investigate possible practical applications of quantum scars. Since the scars effectively “shield” the system from thermal relaxation, this might allow for new mechanisms of storing and manipulating quantum information.

**References**

[1] *Probing many-body dynamics on a 51-atom quantum simulator*, H. Bernien et al., Nature 551, 579-584 (2017).

[2] *Quantum many-body scars*, C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papic, Nature Physics 14, 745 (2018).

[3] https://www.leeds.ac.uk/news/article/4231/insight_into_quantum_chaos_may_be_the_key_to_quantum_computers

[4] https://ist.ac.at/nc/news-media/news/news-detail/article/explanation-for-puzzling-quantum-oscillations-has-been-found/6/

[5] https://www.nature.com/articles/s41567-018-0157-1

[6] *Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits*, Eric J. Heller, Phys. Rev. Lett. 53, 1515 (1984).

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## PhD Project: Dynamics of Geometry in Topological Quantum Matter

Supervisor: Dr. Zlatko Papic

Materials like magnets or water can be understood by studying the individual atoms that form them. In the last three decades, other types of materials have been discovered which cannot be understood in this simple approach. In such materials, quantum mechanics and strong correlations force the particles to lose their identity and form collective quantum states that resemble complicated loops and braids. These “topological phases of matter” have profoundly enriched our understanding of quantum matter (which was also recognised by the 2016 Nobel physics prize [1]), and they are currently being utilised for practical applications in terms of new ways of storing and manipulating quantum information, which is protected from from many sources of errors [2].

One of the best studied examples of topological phases is the so-called fractional quantum Hall effect (FQHE). Under experimental conditions of the FQHE, electrons form exotic types of quantum liquids where they fractionalise into new kind of particles called anyons. The reason why this happens has to do with topology, which is experimentally imposed by an applied magnetic field. In recent years, from the work of Haldane [3] and others, it has been realised that topology does not fully describe FQHE phases – these phases also have emergent degrees of freedom which have geometric character. This means that their quantised excitations behave like an analog of the elusive graviton particle in theory of quantum gravity.

This PhD project will investigate dynamics of fractional quantum Hall phases, in particular focusing on their geometric degrees of freedom. While the equilibrium properties of the FQHE have been well understood due to major theoretical efforts of the past three decades, the study of non-equilibrium dynamics of FQHE phases is an uncharted territory. In our recent work [3], we have addressed this question for the first time and we have shown that FQHE phases have rich dynamical properties, in particular they allow us to probe the mentioned “graviton” excitation and observe its dynamics after the FQHE system is “quenched” (i.e., the direction of the external magnetic field is suddenly changed). One of the goals of the project will be to understand the dynamics in the so-called non-Abelian FQHE phases, whose underlying particles have exchange statistics which is fundamentally different from fermions and bosons. (It is precisely this type of statistics that allows to use such systems to perform “topological quantum computation” [2].) The second goal of the project would be to investigate the dynamics of higher-spin excitations in FQHE phases, which can be viewed as generalisations of the “graviton” particle (which carries spin-2). The study of such exotic excitations would not only shed light on the richness of structure in FQHE phases, but the insights gained from it might prove to be of interest in various other areas of theoretical physics which have focused on higher-spin symmetry (e.g., generalization of gauge/gravity dualities, large N gauge theory, etc.).

**References**

[1] *The Nobel Prize in Physics 2016 – Scientific background: Topological phase transitions and topological phases of matter*, http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced.html

[2] *A Short Introduction to Topological Quantum Computation*, Ville Lahtinen and J. K. Pachos, arXiv:1705.04103 (2017).

[3] *Geometric Description of the Fractional Quantum Hall Effect*, F. D. M. Haldane, Phys. Rev. Lett. 107, 116801 (2011).

[4] *Geometric quench and non-equilibrium dynamics of fractional quantum Hall states*, Zhao Liu, Andrey Gromov and Zlatko Papic, arXiv:1803.00030.

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**PhD Project: From free fermions to parafermions: how to build a universal topological quantum computer from free particles**

Supervisors: Dr. Jiannis Pachos Dr. Zlatko Papic

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 [1], 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 [2], 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 [3]. 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 [4].

**References**

[1] *Introduction to Topological Quantum Computation*, Jiannis K. Pachos, Cambridge University Press, 2012.

[2] *Topological phases with parafermions: theory and blueprints*, Jason Alicea and Paul Fendley, arXiv:1504.02476.

[3] *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.

[4]* 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.

[5] *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.

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**PhD Project: Measuring, understanding and manipulating interactions in quantum systems: a novel machine learning approach**

Supervisors: Dr Zlatko Papic Dr. Jiannis Pachos

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” [1]. 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 [1].

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 [2]. 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” [3], and explain the origin of intriguingly slow dynamical regimes that have been observed in a recent 51-atom quantum simulator at Harvard [4].

**Note: the project requires computational background (e.g., Python/Matlab/Julia/C++…).**

**References**

[1] *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.

[2] *Machine learning: New tool in the box*, Nature Physics 13, 420–421, doi:10.1038/nphys4053.

[3] *Many body localization and thermalization in quantum statistical mechanics*, Rahul Nandkishore and David A. Huse, arXiv:1404.0686.

[4] *Probing many-body dynamics on a 51-atom quantum simulator*, H. Bernien *et al*, arXiv:1707.04344.