# Physical, algebraic and geometric underpinnings of topological quantum computation

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

Conventional computer architecture is designed using an essentially classical physical model of the relationship between components and code (hardware and software), in which each `bit' holds a fixed value 0 or 1 until it is changed. Conventional components are large enough that this model is a good approximation to reality. On the other hand, for very small `components' we know that this is _not_ a good approximation. We know from experiment that they behave differently from the classical way (that our experience of the macroscopic world trains us to think about things). This difference can manifest itself as Heisenberg uncertainty, which is not something desirable in a computation. However it can also be thought of, loosely, as taking many values at once, like a hugely fast or massively parallel computer. If this aspect can be harnessed, the disadvantageous `quantum' phenomenon becomes advantageous --- perhaps revolutionarily so. In recent years computer scientists have shown in principle that the parallelism _can_ be harnessed for certain kinds of computation. The next challenge is to design (then build) a quantum computer.However, partly because the quantum model is not intuitive, the best design language is mathematics --- a language built on far fewer `dangerous' assumptions than conventional engineering design. And the good news is (i) that an encouraging basis of usable mathematics is being developed; and (ii) that this challenge is taking the mathematics in intrinsically interesting directions. Leeds University hosts a leading centre for research in quantum information, and also hosts research into some of the main types of mathematics that turn out to be needed: applied representation theory and integrable systems. This project uses expertise in linear category theory, quantum geometry, and related areas of representation theory and integrable systems to provide radically new models of quantum computation. The project interfaces this expertise with expertise on topological phases of matter, and expertise on practitioner constraints, in order to implement the models, ready for laboratory testing. An intriguing way to reinvent the error-robustness of classical digital computing is to work with topological characteristics of the `computer components' --- that is, characteristics that are invariant under small local distortions of the system (which are typically the main kind of error inducing `noise' present). This proposal is concerned, therefore, with the investigation of _topological_ systems that can support quantum information tasks, such as quantum memory, quantum computation and quantum cryptography. The goal is to propose small scale _topological_ models, amenable to laboratory simulations which would then test their feasibility as models for quantum computation. The physics behind the models may be described in terms of `anyon' particles which can be experimentally realized in topological insulators and in graphene carbon, and which can encode and manipulate quantum information error-robustly. The objective here is to develop the theoretical underpinnings of this technology by means of the relation to certain algebraic structures (realized by a topological diagram calculus) and corresponding problems in low-dimensional topology and representation theory. In particular, while guided firmly by the requirements of physical realizability, the project endeavours to deepen the understanding of numerically and analytically solvable models arising from theoretical constructs such as generalized Temperley-Lieb diagram categories, as well as novel models of quantum geometry developed through the theory of exactly integrable quantum systems.

### Planned Impact

Research:Our proposed research on TQC is strategically positioned at the interface between mathematics and quantum physics, enabling us to develop and exploit ideas from both areas, with research impact both ways: ``back" into basic mathematics and ``forward" towards future technological applications. We expect TQC to have research impact in algebra, particularly representation theory and invariant theory; Integrable Systems; and computational Statistical Mechanics. We expect new results on the theoretical underpinnings to provide a wider framework for TQC, leading to new potential physical implementations, continuing ``forward" with research impact into experimental physics. Successful experimental implementation of TQC will further lead to technological impact and exploitation for future IT and services.Technology and exploitation:The potential for wholly new technologies, for communications, sensing and computing, based on quantum physics, is now very well established. The grand vision for future IT is a combination of new quantum technologies working alongside evolved conventional information technologies, seamlessly networked together to provide capabilities and services well beyond what exists today. Topological QC represents a new and very promising route for QC. Rather than using concatenated encoding into many quantum bits, quantum information is instead encoded into distributed topological objects that can exist in certain physical systems-with protection against errors emerging naturally from the distributed topological nature of the objects. There is real potential for TQC as a way forward, providing a new exploitation path towards quantum technologies protected against errors without the need for a Moore's-law-like growth in practical qubit resources. Various stages of R&D are required to progress from where we are today to TQC applications and technologies. The research proposed here focuses on the next two stages, but is designed to lead smoothly towards future applications.i) Research on the underpinning mathematics and theoretical physics required for TQC. The results will be new candidate physical systems for TQC. Our proposal incorporates a strategic visitor programme, providing additional leverage from the visitor collaboration and maximising the impact of our work.ii) Work towards experimental implementation of TQC. Pachos and Spiller have a wide spectrum of experimental contacts within the QI and condensed matter experimental communities, and significant experience of direct collaboration with experimentalists. They are very well placed to deliver the theoretical output i) into the relevant experimental communities, again maximising the impact of our work. Our proposal incorporates an international workshop at Leeds, to add further leverage and impact to i) and ii), with specific effort being made to involve practitioners at the workshop. Following i) and ii), detailed experimental investigations will be required to test the emergent new ideas, firstly addressing the proof of principle for topological encoding of quantum information into relevant physical systems. The initial applications of TQC will likely be based on memory or communications, with quantum gates enabling universal QC as the final goal. Pachos and Spiller aim to provide partnership forexperimental collaborators in future proposals geared towards the ideas that emerge from this work, thus ensuring its wider impact. We shall pay particular attention to the new implementation ideas that emerge, for Intellectual Property and future commercial exploitation. Quantum technologies based upon TQC for our future IT will require major experimental R&D following the work we propose here. Our direct impact will be in the mathematical foundations and the relevant theoretical physics, devising implementations for TQC, and ensuring that these ideas make it to experiment, thus kick-starting the R&D path.

### Organisations

### Publications

Kadar Zoltan
(2014)

*Local representations of the loop braid group*in arXiv e-prints
Joshi C
(2017)

*Qubit-flip-induced cavity mode squeezing in the strong dispersive regime of the quantum Rabi model.*in Scientific reports
Fu Wei
(2016)

*Direct linearising transform for 3D discrete integrable systems: The lattice AKP, BKP and CKP equations*in ArXiv e-prints
Fu Wei
(2017)

*Linear integral equations, infinite matrices, and soliton hierarchies*in ArXiv e-prints
Fu W
(2018)

*Linear integral equations, infinite matrices, and soliton hierarchies*in Journal of Mathematical Physics
Fu W
(2017)

*Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations.*in Proceedings. Mathematical, physical, and engineering sciences
Fu W
(2017)

*On reductions of the discrete Kadomtsev-Petviashvili-type equations*in Journal of Physics A: Mathematical and Theoretical
Finch P
(2015)

*Induced topological phases at the boundary of 3D topological superconductors.*in Physical review letters
Estarellas MP
(2017)

*Topologically protected localised states in spin chains.*in Scientific reports
Estarellas M
(2016)

*Topologically protected localised states in spin chains*
Estarellas M
(2017)

*Robust quantum entanglement generation and generation-plus-storage protocols with spin chains*in Physical Review A
Delice N
(2015)

*On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants*in Journal of Physics A: Mathematical and Theoretical
De Lisle J
(2014)

*Detection of Chern numbers and entanglement in topological two-species systems through subsystem winding numbers*in New Journal of Physics
De Lisle J
(2016)

*Nested defects on the boundary of topological superconductors*in Physical Review B
Cirio M
(2014)

*( 3 + 1 ) -dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions*in Physical Review B
Bullivant Alex
(2015)

*Entropic Topological Invariants in Three Dimensions*in ArXiv e-prints
Bullivant A
(2017)

*Twisted quantum double model of topological order with boundaries*in Physical Review B
Bullivant A
(2017)

*Topological phases from higher gauge symmetry in 3 + 1 dimensions*in Physical Review B
Bullivant A
(2019)

*Representations of the Necklace Braid Group: Topological and Combinatorial Approaches*in Communications in Mathematical Physics
Bullivant A
(2016)

*Entropic manifestations of topological order in three dimensions*in Physical Review B
Bullivant A
(2016)

*Topological phases from higher gauge symmetry in 3+1D*
Bullivant A
(2019)

*Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry*in Reviews in Mathematical Physics
Brown Benjamin J.
(2014)

*Quantum memories at finite temperature*in ArXiv e-prints
Brown B
(2016)

*Quantum memories at finite temperature*in Reviews of Modern Physics
Brown B
(2014)

*Quantum memories at finite temperature*
Baur Karin
(2017)

*A generalised Euler-Poincaré formula for associahedra*in arXiv e-prints
Baur Karin
(2016)

*The fibres of the Scott map on polygon tilings are the flip equivalence classes*in arXiv e-prints
Baur K
(2018)

*The fibres of the Scott map on polygon tilings are the flip equivalence classes*in Monatshefte für Mathematik
Baur K
(2019)

*A generalised Euler-Poincaré formula for associahedra A GENERALISED EULER-POINCARÉ FORMULA FOR ASSOCIAHEDRA*in Bulletin of the London Mathematical Society
Barnett S
(2017)

*Journeys from quantum optics to quantum technology*in Progress in Quantum Electronics
Alsallami Shami A
(2017)

*Closed-form modified Hamiltonians for integrable numerical integration schemes*in ArXiv e-prints
Alba Emilio
(2015)

*Winding number order in the Haldane model with interactions*in ArXiv e-prints
Alba E
(2016)

*Winding number order in the Haldane model with interactions*in New Journal of Physics
Ahmed C
(2023)

*Tonal partition algebras: fundamental and geometrical aspects of representation theory*in Communications in Algebra
Ahmed C
(2021)

*On the number of principal ideals in d-tonal partition monoids*in Annals of CombinatoricsDescription | This project is concerned with the physical, algebraic and geometric underpinnings of topological quantum computation - an approach to quantum computation in which the fundamental challenge of error protection is met using topological ideas. Significant new knowledge has been generated in several aspects of this project. (In all cases below see the Publication Outputs for details.) For example mathematical models of physical systems have been studied and solved; and new mathematical models have been developed. For the physical aspects of topological quantum computation, we achieved a multitude of outcomes. These range from the modelling of topological systems and the experimental quantum simulation of Majorana-based quantum computation, to the theoretical understanding of topological models in one and two dimensions in terms of free-fermion models. To summarise these results we also published review articles in high profile journals, as well as pedagogical articles that are accessible to students and researchers that enter the field. There are several important developments along these mathematical modelling lines in the project, but one that neatly exemplifies the spirit of the project is the development of "higher gauge theory". Roughly speaking this arises from the application to gauge theory (a well established and powerful model in physics) of a radical new paradigm with its origins in pure mathematics, called higher category theory. This, together with our other approaches, not only offers the prospect of suggesting radical frameworks for topological quantum computation, but may also lead to important new mathematical developments, for example in representation theory and in the study of topological invariants. We have held several very successful international research workshops, leading to a number of international research collaborations, of world-wide scope. The two PhD students trained within the project represent a huge new resource for the research community, due to their great promise (already significantly fulfilled) as researchers and research leaders. |

Exploitation Route | As anticipated in our Pathways statement, our results are already being used heavily in the academic sector, both on the physics and mathematics sides (see for example many citations of our Publication Outputs). In due course it is also anticipated that our proposals will contribute to the development of practical quantum computation, leading to interest in the digital/IT sector. The "underpinnings" aspect of our project will lead to interest in the eduction sector; and, further along the line, the facilitation of quantum computation will lead to interest in the electronic, manufacturing and security sectors. |

Sectors | Digital/Communication/Information Technologies (including Software) Education Electronics Manufacturing including Industrial Biotechology Security and Diplomacy |

Description | The grant enabled both researchers and investigators on the project to develop high-value knowledge and skills of both direct and indirect benefit to the UK economy and society. For example at least one investigator has gone on to fill a key role in a technology area of national strategic importance; while at least one researcher is a strong role model for skilled and successful young researchers from diverse backgrounds. |

First Year Of Impact | 2014 |

Impact Types | Societal Economic |

Description | Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology |

Amount | £2,554,972 (GBP) |

Funding ID | EP/W007509/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 07/2022 |

End | 07/2027 |