## Objectives and learning outcomes

The development of quantum information and computation entaisl the need for formal tools to model, verify and reason rigorously about quantum systems. This course explores the connection between (quantum) computation, (resource-sensitive) logic, and category theory, given by the Curry-Howard-Lambek correspondence, which identifies propositions with types and proofs with programs, and formalizes the model space as a (monoidal) category. Thus, in the end, students will be able to apply to the development of quantum programs a solid and operative knowledge in

• semantics and calculi for the quantum paradigm;
• fundations, methods and tools for the specification and verification of quantum programs.

## Syllabus

• Categories for quantum computing
• Categories, funtors, and natural transformations.
• Universal properties and adjunctions.
• Monoidal categories; Aplications: (categories of) relations, matricess, and Hilbert spaces.
• The Curry-Howard-Lambek correspondence.
• Introduction to intuitionistic and linear logica.
• Introduction to the lambda-calculus and its linear variant.
• The Curry-Howard-Lambek correspondence for classical computation.
• The Curry-Howard-Lambek correspondence for quantum computation.
• Diagramatic reasoning in monoidal categories
• Diagramatic representations of monoidal categories. String diagrams.
• Computational interpretation of quantum mechanics; associated categorical structures: monoidal (composition), compact closed (entanglement), adjunctions (internal product), biproduts (non deterministic branching).
• Quantum processes.

## Summaries (2019-20)

##### T Lectures
• Sep 9 (9-11h): Introduction to the course: objectives, learning outcomes, organisation.

Categories: definition and worked examples.

• Sep 16 (11-13h): Functors. Definition and worked examples.

##### P Lectures
• Sep 9 (11-13h): Familiar and less familiar examples of categories. Exercises on defining / checking the strucutre of a category.
• Sep 16 (14-16h) Exercises on rephrasing well-known, pointwise results in the language of categories: monos, epis and isos. Examples of functors.

## Support

##### Bibliography
• S. Abramsky and N. Tzevelekos. Introduction to categories and categorical logic. In B. Coecke, editor, New Structures for Physics, pages 3–94. Springer Lecture Notes on Physics (813), 2011.
• J. Baez and M. Stay. Physics, topology, logic and computation: A Rosetta stone. In B. Coecke, editor, New Structures for Physics, pages 95–172. Springer Lecture Notes on Physics (813), 2011.
• B. Coecke and A. Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning . Cambridge University Press, 2017.
##### Complementary Bibliography
• S. Awodey. Category Theory . Oxford Logic Guides. Oxford University Press, 2006.
• S. Abramsky and B. Coecke. Categorical quantum mechanics . In Kurt Engesser, Dov Gabbay, and Daniel Lehmann, editors, Handbook of Quantum Logic and Quantum Structures, pages 261–324. North-Holland, Elsevier, 2011.
• ... other references will be suggested according to the lecturing scheduling.