Title: The scum at CQC Oxford

Abstract: Besides sitting in our dedicated basement bar, what else are we about. This will be a bit of a quantum group history lesson, and a bit about current quantum industry landscape.

Title: Categories of Formal and Cognitive Concepts

Abstract: I'll give an overview of some of my recent work on compositional semantics based on formal notions of concepts from cognitive science and computer science. If I have time I may also mention work on formal models of consciousness.

Title: Grammar Circuits

Abstract: More compositional natural language semantics, directed by natural language syntax. This time the semantics are in a more computer-friendly format.

Title: Implementing "from words to meaning"

Abstract: The Discocat model (Coecke et al.) and its implementations ( discopy ) propose a way of generating a sentence meaning via grammatical composition of it words meanings. Grammatical composition is often obtained exploiting reduction rules of the compact closed category of Pregroups. A prerequisite for such composition is the assignment of pregroup types to the words of the sentence. Currently, this assignment process is performed manually. This yields several limitations, including the difficulty of scaling up the model and testing it extensively on real data. Another aspect of Discocat that needs further exploration regards semantic representation of functional words (e.g transitive verbs ). Functional words are represented in Discocat - following a widely accepted approach (see e.g. CCG grammars) - as processes inputting and outputting other parts of speech. Discocat initially referred to distributional compositional categorical model of meaning, aiming at composing distributional semantic models. Distributional models represent word meaning according to their distribution in a Corpus with respect to a set of context words. In such framework - as well as the state-of-the-art learnt semantic vectorial embedding (e.g. Word2Vect, BERT etc) - the representation of functional words remains non trivial. Several theoretical options suitable for the Discocat model have been proposed (density matrices, tensors etc) but scaled-up implementations are currently missing. We aim to discuss the plan for a model (or two separate ones) that is (are) capable of performing the following tasks: 1) Learning pregroup type assignments via supervised methods; 2) learning semantic representations of functional words via unsupervised methods. The aim of the model(s) is (are) to provide a full implementation of Discocat, able to effectively go from real text to a computable sentence meaning, trained for a chosen semantic task.

Title: Concrete Diagrams

Abstract: String diagrams serve as a graphical language for tensor networks. Rewrite rules can be used to reason about a wide range of problems from different disciplines when they are cast as tensor networks. Examples include model-counting, partition functions, link invariants, quantum amplitudes, and meaning of sentences in natural language.

Title: Title not available: talk is in the future.

Abstract: Do you ever have that strange feeling that something is about to happen but you don't quite know what? Did you ever wonder why? This talk will have the answer: it's called "causality", and we know nothin... *coughs* all about it.

Title: Routed quantum circuits

Abstract: We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

Title: Testing gravitational decoherence through the heating of a mechanical resonator

Abstract: The theory of classical channel gravity models gravitational interactions as classical measurement channels. These channels are a source of decoherence even if the results of the measurements are never recorded in a lab and thus the gravitational interaction can be thought of as having the same effect as an observer. This leads to two potentially observable effects – decoherence in the position basis and a density dependent heating effect. We have set up an experiment to test for the latter, using a cavity optomechanical setup at cryogenic temperatures to measure the mode heating of a silicon nitride membrane.

Title: On the Operational Nature of Causality

Abstract: When considering the information-theoretical structure of spacetime, it is of a fundamental importance to stick to an operational perspective, within which the notion of `information' can be consistently and univocally defined. In this talk, we discuss a mathematical framework enabling us to rigorously describe definite and indefinite causality from such an operational perspective: whether a given scenario satisfies specific causal assumptions will be directly detectable in the empirical correlations between inputs and outputs, in a theory-independent way.

Title: Higher Order Process Theories

Abstract: We'll see some results of imposing physical conditions at the level of supermaps, to do so we will need to present the appropriate mathematical background for their discussion, phrased only in the language of string diagrams.

Title: Quantum Origami: CPM Constructions, Galois Theory and Decoherence

Abstract: In this talk I will give an overview of some recent work in collaboration with Stefano, connecting generalised CPM constructions with Galois theory and decoherence. In doing so we are able to construct an infinite family of probabilistic quantum-like theories which exhibit rich decoherence structures in bijection with the lattice of subgroups of a given Galois group.

Title: What can a ZX normal form buy for you

Abstract: In this talk I will first give an introduction to a normal form of arbitrary vectors with size a power of 2 which was formed in algebraic ZX-calculus. Then I will describe the usefulness of such a normal form from different aspects, including its applications to the proof of completeness of qudit ZX-calculus and to the derivation of generalised Euler decomposition in terms of qudit Z and X phases.

Title: Diagrammatic Differentiation

Abstract: We introduce the notion of gradient for parametrised tensor diagrams. It's simple: the gradients of composition and tensor are both given by the product rule. In particular, this allows to compute the gradient of quantum circuits and ZX diagrams in a diagrammatic way. It also applies to the parametrised functors involved in the definition of QNLP algorithms. We spice things up by introducing a diagrammatic representation of non-linearities, which we call bubbles. These bubbles allow to encode both a quantum circuit and its classical post-processing in the same diagram. We can compute their gradients using a diagrammatic chain rule.

Joint work with Richie Yeung and Giovanni de Felice.

Soon on the arXiv, already on github.com/oxford-quantum-group/discopy

Title: Editing ZX-diagrams in PyZX

Abstract: I'll give a demo of the diagram editor in PyZX I've been slowly working on over the last 6 months.

Title: ZX-calculus quantum compilation to the superconducting gateset

Abstract: TBA

Title: Relating MBQC patterns to circuits via Pauli flow and Pauli Dependency DAGs

Abstract: Causal flow and generalised flow have already been shown as sufficient for circuit extraction, with efficient algorithms for identifying these properties and performing the extraction. Pauli flow is a slightly weaker set of conditions that extends gflow to exploit the knowledge that some vertices are measured in a Pauli basis. We show that Pauli flow can similarly be identified efficiently and that any MBQC pattern whose underlying graph admits a Pauli flow can be efficiently transformed into a gate-based circuit without using ancilla qubits. We then use this relationship to examine relations between circuit rewriting techniques using the ZX Calculus (see arXiv:1903.10477) and Pauli Dependency DAGs (see T-graphs in arXiv:1903.12456).

Title: So What Now?

Abstract: As grandpa Bob and his CQC scum move down the road, I'll meditate a bit on the future: open problems, some cool stuff on the horizon, going mainstream, and staying weird.

Title: Introduction to Spoiler-Duplicator game comonads

Abstract: Model theorists care about structures not "as they really are", but stuctures up to definability in a logic. In recent research conducted by this group and others, several logics have been shown to have an associated indexed comonad. Morphism in categories associated with the indexed comonad capture equivalence in the corresponding logic, the existential-positive fragment of the logic, and the logic extended with counting quantifiers. Additionally, these comonads provide a unifying perspective for many concepts in finite model theory: providing coalgebraic semantics for graph-theoretic parameters, parameterized Chandra-Merlin correspondences, and restricted class Lovasz results. I will go over these comonads, and how this framework can be used to study the complexity of constraint satisfaction problems..

Title: Counting homomorphisms

Abstract: Lovász (1967) showed that two graphs A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any graph C. I will present a categorical generalisation of this result and explain how it can be used, in combination with the game comonads introduced by Abramsky et al., to obtain homomorphism counting results in finite model theory.

This is joint work with Anuj Dawar and Tomáš Jakl.

Title: Automatic Differentiation on a Higher Order Language

Abstract: I present semantic correctness proofs of Automatic Differentiation (AD). I will consider a forward-mode AD method on a higher order language with algebraic data types, and characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. I describe a rich semantics for differentiable programming, based on diffeological spaces. I show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. I show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces.

Title: An Categorical Perspective on Conditioning

Abstract: Conditioning on equality of continuous random variables is generally ill-defined, as the exact observation may have probability zero; this is Borel's paradox. I'm giving a categorical perspective on what conditioning is, using the formalism of Markov categories. I use this to construct a programming language for conditional probability, verify properties about it, and derive an algebraic presentation of conditioning.

Title: Markov categories: randomness and information flow

Abstract: Markov categories are a new and promising framework to describe phenomena involving randomness, noisy channels, and information flow. In particular, they seem to provide a sort of internal language for probability theory, similar to how simply typed lambda calculus is the internal language of cartesian closed categories.

The research on Markov categories is still ongoing, and largely unexplored. In this short talk we give the main definitions and ideas, and state some of the results that have been proven so far.

Joint research with Tobias Fritz, Tomas Gonda, and Eigil Rischel.

Title: What the heck is the reverse derivative?

Abstract: There are two types of derivative operations used in machine learning: the forward derivative and the reverse derivative. Everyone is familiar with the forward derivative, it’s the standard derivative from differential calculus. But what is the reverse derivative? From the programmer’s perspective, it is much more common for the reverse derivative to play the central role due to its increased efficiency and improved accuracy when computing with real smooth functions. In this talk, I will give an introduction to the reverse derivative operator, talk about the reverse chain rule and reverse product rule, and discuss the categorical axiomatizations of the reverse derivative. I will also discuss how from the reverse derivative one can build a forward derivative and transpose operation, and vice-versa.

Title: Categorical semantics of the ZX-calculus

Abstract: In this talk, I will go over various fragments of the ZX-calculus which have semantics in terms of spans of finite sets. I will start with simple fragments, adding more generators yielding new props via distributive law and pushout.

Title: Profunctor string diagrams

Abstract: TBA

Title: Unknotting knots in braided monoidal categories

Abstract: Can we automatically detect if two string diagrams in braided monoidal categories are equal? I am not aware of any known algorithm for this, so I don't even know if it is decidable! Crazy, right? In this presentation I want to show a hardness result for this problem: that it is at least as difficult as unknotting a knot. This latter problem has been extensively studied and we do not have any polynomial algorithm for it, suggesting that the word problem for braided monoidal categories is genuinely difficult.

Title: “Towards a QNLP-Inspired Music Processing”

Abstract: Like natural language, music follows rules for combining symbols sequentially and simultaneously. Whereas the symbols of language are words, the symbols of music are sounds; e.g. musical notes. In this sense, even spoken words can (part of) music. Here meaning is just structure because the concept of a dictionary with word-definitions does not exist in music. But there are dictionary of processes: that is, things that you are allowed to do compose. Can we adapt QNLP to process music? In this talk I will present some initial ideas and show some diagrams that I am sketching to represent musical processes.