My work is on high-level mathematical structures for fundamental phenomena in computer science and physics. There is a lot in common between these fields, which both study the laws and emergent features of systems with local interactions. Particular areas of recent interest include the structure of quantum algorithms, a new higher-categorical language for physics and computer science, and the classification of topological quantum field theories.
Here are some things I've been working on:
- Globular has launched, a web-based proof assistant for semistrict higher category theory; now with 4-category support. It's been used 5000 times in the first 3 months since launch! (With Krzysztof Bar and Aleks Kissinger.)
- A new preprint introduces a 3d notation for proofs in multiplicative linear logic with units, with better locality properties than proof nets, and a more powerful notion of equivalence. The main technical results are coherence theorems for swallowtailators and Frobenius pseudomonoids. (With Lawrence Dunn.)
- QIP 2016: my paper with Ben Musto on quantum Latin squares has been accepted. We show how you can use them to make completely new sorts of teleportation protocol, which can't be constructed from ordinary techniques, like nice error bases or Werner's shift-and-multiply procedure.
- Topological quantum field theory (TQFT) is a simple type of quantum field theory in which diffeomorphic manifolds are assigned equivalent invariants. In two papers on the arXiv (first, second), we show that 3d TQFTs are classified by modular tensor categories equipped with extra data. (Joint with Bruce Bartlett, Chris Douglas, and Chris Schommer-Pries.)
- LICS 2013: Work on quantum algorithms uses a topological formalism to give a new description of the Deutsch-Jozsa, single-shot Grover and hidden subgroup algorithms. This high-level approach gives rise to new proofs of correctness, and new 'multicoloured' quantum search procedures which are more powerful than the traditional Grover algorithm and its existing generalizations.
- LICS 2012, MFPS 2013: I've been working on a higher-categorical theory of information flow, which uses topological diagrams involving points, lines and surfaces to represent informatic procedures such as encryption, quantum measurement and controlled operations. Equations between diagrams specify procedures such as quantum teleportation and encrypted communication, and solutions to these equations in monoidal bicategories correspond to real-world implementations of the procedures. Papers are available here and here. (Some of this is joint work with Mike Stay.)
If you're interested in these ideas, please get in touch! I am always interested in giving talks, meeting other scientists with similar interests, and starting new collaborations.
Here are some places I will in 2016.
- On 26-29 April I will be at the 2016 Workshop on Informatic Processes in Fontainebleau, near Paris.
- On 25-29 May I will be at the Georgia Topology Conference organized by David Gay, in Athens, Georgia.
- On 2-5 June I will be at Foundational Methods in Computer Science 2016 at the University of British Colombia in Vancouver, giving a tutorial on Globular, and giving a contributed talk on linear logic.
- Between 11 July and 5 August I will be giving a lecture course on Categories, Quantum Computation and Topology at the 2016 AARMS Summer School, at Dalhousie University, Halifax.
- I will be at CT2016 in the week 8-13 August, also at Dalhousie University, Halifax.
- Between 14 August and 16 October I will be visiting the Simons Institute at Berkeley, California for their programme on Logical Structures in Computation.
Here's some other things I'm doing at the moment:
- I co-organize the Oxford Advanced Seminar in Informatic Structures (OASIS), with Chris Heunen, Clare Horsman, Aleks Kissinger and Sam Staton.
- I'm part of the Centre for Quantum Mathematics and Computation (QMAC), an interdisciplinary collaboration between the Oxford's Computer Science, Mathematics and Physics departments.
- I maintain two popular mailing lists, one for international quantum foundations announcements with over 800 subscribers, and another for quantum computing and quantum foundations events taking place in Oxford with around 100 subscribers. Please get in touch if you'd like to be added to either of these.
- Since 2012, I've been lecturing a course on Categorical Quantum Mechanics with my colleague Chris Heunen, and we're also writing a book that will come out soon with OUP.
- I frequently supervise DPhil, MSc and undergraduate students (and sometimes school students too!) on topics related to the mathematical foundations of quantum computing. If you'd like to work on a project with me, get in touch.
Groupoid Semantics for Thermal Computing
Krzysztof Bar and Jamie Vicary
Wormholes and entanglement
John Baez and Jamie Vicary
Topological Structure of Quantum Algorithms
In Proceedings of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science. Pages 35. 2013.