Higher Algebra and Quantum Protocols
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
This research project will integrate expertise in the Oxford Mathematical Institute and the Birmingham School of Computer Science to build theoretical and practical tools for higher-dimensional algebraic computations in geometry and quantum information science.
Through a structured research agenda, we will formulate the first combinatorial language of higher categories that is amenable to computer implementation and admits a natural geometric calculus, providing a natural setting to investigate modern questions in higher representation theory, quantum algebra, and quantum computation. We will then design and build a computer implementation of this language and thereby create a research and proof assistant that natively allows the exploration and verification of complex manipulations of high-dimensional structures. Finally we will apply this computational tool to generalise structures in quantum algebra, yielding novel protocols in quantum information, such as quantum error correction of nonplanar states.
This research effort will be organised into four strands---Foundations, Computer Implementation, Graphical Calculus, and Quantum Protocols---and the ambitious, integrated research agenda will be achieved by uniting researchers across the two institutions and disciplines, and by actively engaging with the broader scientific community.
Through a structured research agenda, we will formulate the first combinatorial language of higher categories that is amenable to computer implementation and admits a natural geometric calculus, providing a natural setting to investigate modern questions in higher representation theory, quantum algebra, and quantum computation. We will then design and build a computer implementation of this language and thereby create a research and proof assistant that natively allows the exploration and verification of complex manipulations of high-dimensional structures. Finally we will apply this computational tool to generalise structures in quantum algebra, yielding novel protocols in quantum information, such as quantum error correction of nonplanar states.
This research effort will be organised into four strands---Foundations, Computer Implementation, Graphical Calculus, and Quantum Protocols---and the ambitious, integrated research agenda will be achieved by uniting researchers across the two institutions and disciplines, and by actively engaging with the broader scientific community.
Planned Impact
In recent years, research at the interface of higher algebra and modern homotopy theory and geometry has seen drastic advances, and has contributed immensely to various branches of mathematics and mathematical physics. Our research fulfills the quest for a uniform description of many of the individually observed phenomena at this interface. More generally, it builds a solid formal foundation for connecting and translating between the algebraic and geometric realms. We believe this research will have transformative impact on a range of subject areas, as it will enable the performing of complex geometric operations in a formally and computationally verified language.
Our project will provide the first language of higher categories that is amendable to computer implementation while also offering a direct geometric interpretation of its syntax. This geometric interpretation of a combinatorial language will, for instance, give new tools for the classification and construction of homotopical invariants. These tools will fuel developments in mathematical physics and quantum information by providing methods of describing topological field theories and other quantum structures and protocols.
The project's new tools and areas of research will impact the next generation of researchers. By incorporating the combinatorial geometry tools produced by this project into teaching and supervision, we intend to guide young researchers toward a new viewpoint on traditional and novel aspects of the higher algebra/geometry interface. In time, our methods can be used throughout the undergraduate and graduate curriculum, transforming and streamlining exposition and learning for students and researchers alike.
As the project will develop completely novel perspectives and tools, a key focus will be placed on dissemination and interaction with the broader research community. Due to the foundational nature as well as the breadth of application of the results of the project, we will be positioned to present the research not only at mathematical conferences, but also at interdisciplinary meetings in order to highlight and distribute its connections to computer science and physics. During our project we will foster collaborations with other UK researchers, and in this way strengthen the inter-university network in the UK. We will further facilitate expert visits from around the world to study with us and exchange knowledge. Finally, we will make our new diagrammatic tools available to practitioners in industry who could benefit from their use.
The economic impact of our proposal stems from its applications to quantum information theory. Our work on the construction of quantum error correcting codes and diagrammatic formulations of higher tensor networks can directly influence developments in practical quantum computation. Due to the inscrutability of quantum logic for many applications, easy frameworks for creating highly complex algorithms will be necessary for continued progress in quantum computer science.
Our project will provide the first language of higher categories that is amendable to computer implementation while also offering a direct geometric interpretation of its syntax. This geometric interpretation of a combinatorial language will, for instance, give new tools for the classification and construction of homotopical invariants. These tools will fuel developments in mathematical physics and quantum information by providing methods of describing topological field theories and other quantum structures and protocols.
The project's new tools and areas of research will impact the next generation of researchers. By incorporating the combinatorial geometry tools produced by this project into teaching and supervision, we intend to guide young researchers toward a new viewpoint on traditional and novel aspects of the higher algebra/geometry interface. In time, our methods can be used throughout the undergraduate and graduate curriculum, transforming and streamlining exposition and learning for students and researchers alike.
As the project will develop completely novel perspectives and tools, a key focus will be placed on dissemination and interaction with the broader research community. Due to the foundational nature as well as the breadth of application of the results of the project, we will be positioned to present the research not only at mathematical conferences, but also at interdisciplinary meetings in order to highlight and distribute its connections to computer science and physics. During our project we will foster collaborations with other UK researchers, and in this way strengthen the inter-university network in the UK. We will further facilitate expert visits from around the world to study with us and exchange knowledge. Finally, we will make our new diagrammatic tools available to practitioners in industry who could benefit from their use.
The economic impact of our proposal stems from its applications to quantum information theory. Our work on the construction of quantum error correcting codes and diagrammatic formulations of higher tensor networks can directly influence developments in practical quantum computation. Due to the inscrutability of quantum logic for many applications, easy frameworks for creating highly complex algorithms will be necessary for continued progress in quantum computer science.
People |
ORCID iD |
| Christopher Douglas (Principal Investigator) |
Publications
Bartels A
(2022)
Conformal Nets V: Dualizability
in Communications in Mathematical Physics
Bartels Arthur
(2019)
Conformal nets V: dualizability
in arXiv e-prints
C Dorn
(2021)
Framed Combinatorial Topology
C Douglas
(2023)
Fusion 2-categories and a state-sum invariant for 4-manifolds
Dorn C
(2021)
Framed combinatorial topology
Dorn Christoph
(2022)
Manifold diagrams and tame tangles
in arXiv e-prints
| Description | We developed a new mathematical theory called "framed combinatorial topology" in the mathematical fields of Algebraic and Geometric Topology. Our theory describes combinatorial phenomena arising at the intersection of stratified topology and higher algebra. The theory synthesizes notions from classical combinatorial topology with a novel combinatorial approach to framings. Our work, in particular, achieved the following. (1) We exhibited elements of a unifying language relating several mathematical subfields: - the theory of stratified manifolds (singularity theory, higher Morse theory, classification of smooth structures, etc.), - quantum algebra (TQFTs, knot invariants, knotted surfaces and higher knots, etc.), - and higher category theory (homotopy theory, higher operads, higher computads, etc.) (2) Through this unification, our research yields new perspectives and research methods for many of structures central in modern algebraic topology and mathematical physics. This, for instance, allows for combinatorial 'higher categorical' representations of smooth structures on manifolds, as well as the combinatorial study of classical differential singularities. For several examples, this methods generated new insights into and new approaches to study the deeper algebraic properties of these geometric structures. Concretely, we were able to show that higher knot diagrams admit canonical combinatorial representation in all dimensions. (3) Many new avenues for future research were opened in the process, and these are spread across many disciplines. Concretely, we made several conjectures in our work regarding the precise nature of the relation between combinatorial and differential singularity, as well as the relation between tangle diagrams and differentiable manifolds. Moreover, our work found surprising applications in computability theory, in that several questions that are impossible to computably answer in the setting of classical combinatorial topology do find computable answers in the setting of framed combinatorial topology. This leads to several follow-up questions about exhibiting the exact "frontier" of computability in framed combinatorial topology, which turns out to be closely intertwined with the aforementioned questions about understanding combinatorial and differential singularities. Finally, in applying the geometry of stratified manifolds to the description of higher-categorical structures, a new project arose aiming specify a "geometric higher type theory" in which types semantically correspond to non-groupoidal higher categories. (4) While framed combinatorial topology is a new mathematical subfield, we have strongly engaged in disseminating the new mathematical perspective provided by the theory. This has been achieved via providing introductory material in online "wiki" pages, via several talks at universities in the UK and abroad, and via the participation in community-led scientific projects (such as the nLab and the n-Category Cafe). |
| Exploitation Route | Due to its foundational nature, our research may have future applications in a broad range of areas in mathematics, mathematical physics, and computer science. This in particular includes the subfields of Cobordism theory, Quantum Topology, TQFTs, and Theoretical Condensed Matter Physics. Concretely, several other current EPSRC-funded projects are working on directly related questions, for instance, by studying connections of quantum field theory and enumerative geometry, or the connection of 5-dimensional field theories to complex 3-dimensional singularities. Our project relates to these endeavours by breaking down barriers between geometry and higher algebra, respectively between singularities and the combinatorics of dualizable algebraic structures, and putting their relation on a rigorous foundation. At the same time, our project also delivered computationally tractable combinatorial structures, and put this to use in building new tools at the interface of computer science and geometric topology, with the goal of filling persistent gaps in our knowledge of low-dimensional geometric structures that could so-far not be addressed with traditional frameworks. In the long run, one may envisage that the software tools developed in the process (in particular, in regards to type theory and higher knot theory) will find application in areas such as verification and cryptography. |
| Sectors | Digital/Communication/Information Technologies (including Software),Education,Other |
| URL | https://cxdorn.github.io/notes/#research |
| Description | The research conducted on this grant has lead to the nucleation of a new research area/field, namely that of "Framed Combinatorial Topology". This new area and its implications are described in more detailed in the Key Findings section. |
| Sector | Digital/Communication/Information Technologies (including Software),Education,Other |
| Description | Floer homotopy theory Research Professorship |
| Amount | $25,000 (USD) |
| Organisation | Mathematical Sciences Research Institute |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 08/2022 |
| End | 12/2022 |
| Description | Quantum symmetries Research Professorship |
| Amount | $25,000 (USD) |
| Organisation | Mathematical Sciences Research Institute |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 01/2020 |
| End | 05/2020 |
| Description | Cambridge research |
| Organisation | University of Cambridge |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Our research team has contributed the primary theoretical framework to this collaboration, and an understanding of the importance of developing a corresponding globular model, for applications, for instance, to higher algebraic and categorical comparisons, and perhaps to practical computer implementations as well. |
| Collaborator Contribution | Our partner at Cambridge (Lukas Heidemann) has developed a particularly attractive technical approximation framework which we are using to prove crucial genericity results for manifold diagrams. |
| Impact | We have begun drafting a paper ``Globular Manifold Diagrams" based on this collaboration, and expect it will be completed soon. |
| Start Year | 2021 |
| Description | Fusion 2-categories |
| Organisation | University of Hamburg |
| Department | Hamburg Observatory |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | PI Douglas and David Reutter, now at U. Hamburg, developed the novel notion of fusion 2-category and defined a new statesum 4-manifold invariant based on it, and have continued developing the theory during the period 2018--present. The notion of fusion 2-category has already achieved widespread recognition as a crucial concept in contemporary mathematical physics. |
| Collaborator Contribution | Collaborator Reutter contributed critically to every stage of the project, from theoretical foundations, to technical development, to writing and dissemination. |
| Impact | The primary paper output of this collaboration is the paper "Fusion 2-categories and a statesum invariant for 4-manifolds" and its revisions in 2020 and 2022, and its dissemination through numerous talks and seminars over the last few years. There is already a substantial literature (~40-80 papers, mostly in mathematical physics) based directly on the ideas from our paper. |
| Start Year | 2018 |
| Description | MSRI program "Floer homotopy theory" |
| Organisation | Mathematical Sciences Research Institute |
| Country | United States |
| Sector | Charity/Non Profit |
| PI Contribution | PI Douglas was a research professor in the Mathematical Sciences Research Institute (MSRI) program on "Floer homotopy theory". His PhD thesis was an early work in the field, and as such he was well positioned to engage in more recent developments. This particularly involved the continuation and development of some of his ideas, and ideas developed with Manolescu, on twisted Seiberg-Witten-Floer invariants of 3-manifolds. |
| Collaborator Contribution | This large program involved many researchers from around the world, building on one another's work; this included a number of researchers for instance developing concrete constructions of the aforementioned twisted SWF invariants. |
| Impact | As this program was recent, the outputs are still in development, particularly a paper on twisted SWF invariants. |
| Start Year | 2022 |
| Description | MSRI programs "Higher Categories and Categorification" and "Quantum Symmetries" |
| Organisation | Mathematical Sciences Research Institute |
| Country | United States |
| Sector | Charity/Non Profit |
| PI Contribution | PI Douglas was a research professor in the Mathematical Sciences Research Institute (MSRI) program on "Higher Categories and Categorification" (https://www.msri.org/programs/323) and in the MSRI program on "Quantum Symmetries" (https://www.msri.org/programs/312). PDRA Dorn also participated and was actively engaged in the first program. PI Douglas helped guide the evolving research concerns of these large collaborative programs, and ran one of the key research seminars for the Quantum Symmetries program. He also was the official mentor for a postdoctoral member in the Higher Categories program. |
| Collaborator Contribution | These large programs involved many researchers from around the world, communicating and sharing and jointly developing research directions for the fields. |
| Impact | Outputs include the book "Framed Combinatorial Topology", the research monograph "Manifold diagrams and tame tangles", and the development and revision of the research monograph "Fusion 2-categories and a state-sum invariant for 4-manifolds". |
| Start Year | 2020 |
| Title | Homotopy.io |
| Description | The Birmingham/Cambridge partners on the other portion of this joint grant developed and implemented the software tool homotopy.io, "a web-based proof assistant for finitely-presented globular n-categories, for arbitrary n". |
| Type Of Technology | Webtool/Application |
| Year Produced | 2019 |
| Impact | Use in formalization and investigations in low-dimensional geometric and algebraic topology, including knot-theory, coherence theory, singularity theory, handlebody theory. |
| URL | https://ncatlab.org/nlab/show/homotopy.io |
| Description | Dissemination and engagement activities -- 17 talks |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Seventeen research talks by PDRA Dorn at such institutions as Berkeley, Toronto, Lisbon, Hamburg, and Oxford. |
| Year(s) Of Engagement Activity | 2019,2020,2021,2022,2023 |
| Description | MSRI2020 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Engagement with MSRI Program on Quantum Symmetries, and MSRI Program on Higher Categories and Categorification. |
| Year(s) Of Engagement Activity | 2020 |
| Description | MSRI2022 |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | MSRI2022 program on Floer Homotopy Theory |
| Year(s) Of Engagement Activity | 2022 |
| Description | Research Visit (City University of New York) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | PDRA Dorn held several seminars on about ongoing research during a research visit from Oct - Nov 2019. |
| Year(s) Of Engagement Activity | 2019 |