Bob Coecke

JAIR Journal 2023 Journal Article

QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer

• Robin Lorenz
• Anna Pearson
• Konstantinos Meichanetzidis
• Dimitri Kartsaklis
• Bob Coecke

Quantum Natural Language Processing (QNLP) deals with the design and implementation of NLP models intended to be run on quantum hardware. In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum (NISQ) computers for datasets of size greater than 100 sentences. Exploiting the formal similarity of the compositional model of meaning by Coecke, Sadrzadeh, and Clark (2010) with quantum theory, we create representations for sentences that have a natural mapping to quantum circuits. We use these representations to implement and successfully train NLP models that solve simple sentence classification tasks on quantum hardware. We conduct quantum simulations that compare the syntax-sensitive model of Coecke et al. with two baselines that use less or no syntax; specifically, we implement the quantum analogues of a “bag-of-words” model, where syntax is not taken into account at all, and of a word-sequence model, where only word order is respected. We demonstrate that all models converge smoothly both in simulations and when run on quantum hardware, and that the results are the expected ones based on the nature of the tasks and the datasets used. Another important goal of this paper is to describe in a way accessible to AI and NLP researchers the main principles, process and challenges of experiments on quantum hardware. Our aim in doing this is to take the first small steps in this unexplored research territory and pave the way for practical Quantum Natural Language Processing.

TCS Journal 2022 Journal Article

Kindergarden quantum mechanics graduates...or how I learned to stop gluing LEGO together and love the ZX-calculus

• Bob Coecke
• Dominic Horsman
• Aleks Kissinger
• Quanlong Wang

This paper is a ‘spiritual child’ of the 2005 lecture notes Kindergarten Quantum Mechanics Coecke (2005) [24], which showed how a simple, pictorial extension of Dirac notation allowed several quantum features to be easily expressed and derived, using language even a kindergartner can understand. Central to that approach was the use of pictures and pictorial transformation rules to understand and derive features of quantum theory and computation. However, this approach left many wondering ‘where's the beef? ’ In other words, was this new approach capable of producing new results, or was it simply an aesthetically pleasing way to restate stuff we already know? The aim of this sequel paper is to say ‘here's the beef! ’, and highlight some of the major results of the approach advocated in Kindergarten Quantum Mechanics, and how they are being applied to tackle practical problems on real quantum computers. Toward that end, we will focus mainly on what has become the Swiss army knife of the pictorial formalism: the ZX-calculus, a graphical tool for representing and manipulating complex linear maps on 2 N dimensional space. First we look at some of the ideas behind the ZX-calculus, comparing and contrasting it with the usual quantum circuit formalism. We then survey results from the past few years falling into three categories: (1) completeness of the rules of the ZX-calculus, (2) state-of-the-art quantum circuit optimisation results in commercial and open-source quantum compilers relying on ZX, and (3) the use of ZX in translating real-world stuff like natural language into quantum circuits that can be run on today's (very limited) quantum hardware. We also take the title literally, and outline an ongoing experiment aiming to show that ZX-calculus enables children to do cutting-edge quantum computing stuff. If anything, this would truly confirm that ‘kindergarten quantum mechanics’ wasn't just a joke.

FLAP Journal 2020 Journal Article

Meaning Updating of Density Matrices.

• Bob Coecke
• Konstantinos Meichanetzidis

The DisCoCat model of natural language meaning assigns meaning to a sentence given: (i) the meanings of its words, and, (ii) its grammatical structure. The recently introduced DisCoCirc model extends this to text consisting of multiple sentences. While in DisCoCat all meanings are fixed, in DisCoCirc each sentence updates the meanings of words. In this paper we explore different update mechanisms for DisCoCirc, in the case where meaning is encoded in density matrices—which come with several advantages as compared to vectors. Our starting point is two non-commutative update mechanisms, borrowing one from quantum foundations research [46, 47], and the other one that originally appeared in the area of our current interest, language meaning updating [15, 48]. Unfortunately, both of these lack key algebraic properties, nor are internal to the meaning category. Passing to double density matrices [3, 71] we do get an elegant internal diagrammatic update mechanism. We also show that (commutative) spiders can be cast as an instance of the update mechanism of [46, 47]. This result is of interest to quantum foundations, as it bridges the work in Categorical Quantum Mechanics (CQM) with that on conditional quantum states. Our work also underpins the implementation of text-level Natural Language Processing (NLP) on quantum hardware, for which exponential space-gain and quadratic speed-up have previously been predicted.

TCS Journal 2018 Journal Article

Generalized relations in linguistics & cognition

• Bob Coecke
• Fabrizio Genovese
• Martha Lewis
• Dan Marsden
• Alex Toumi

Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of phrases and sentences from the meanings of individual words. More recently, similar compositional techniques have been applied to conceptual space models of cognition. Compact closed categories, particularly the category of finite dimensional vector spaces, have been the most common setting for categorical compositional models. When addressing a new problem domain, such as conceptual space models of meaning, a key problem is finding a compact closed category that captures the features of interest. We propose categories of generalized relations as a source of new, practical models for cognition and NLP. We demonstrate using detailed examples that phenomena such as fuzziness, metrics, convexity, semantic ambiguity can all be described by relational models. Crucially, by exploiting a technical framework described in previous work of the authors, we also show how the above-mentioned phenomena can be combined into a single model, providing a flexible family of new categories for categorical compositional modelling.

I&C Journal 2016 Journal Article

A mathematical theory of resources

• Bob Coecke
• Tobias Fritz
• Robert W. Spekkens

Many fields of science investigate states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by these theories include: Which resources can be converted into which others? At what rate can many copies of one resource be converted into many copies of another? Can a catalyst enable a conversion? How to quantify a resource? We propose a general mathematical definition of resource theory. We prove general theorems about how resource theories can be constructed from theories of processes with a subclass of processes that are freely implementable. These define the means by which costly states and processes can be interconverted. We outline how various existing resource theories fit into our framework, which is a first step in a project of identifying universal features and principles of resource theories. We develop a few general results concerning resource convertibility.

I&C Journal 2016 Journal Article

Pictures of complete positivity in arbitrary dimension

• Bob Coecke
• Chris Heunen

Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPM-construction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction.

I&C Journal 2013 Journal Article

Information Security as a Resource

• Ed Blakey
• Bob Coecke
• Michael Mislove
• Dusko Pavlovic

CSL Conference 2010 Conference Paper

Environment and Classical Channels in Categorical Quantum Mechanics

• Bob Coecke
• Simon Perdrix

Abstract We present a both simple and comprehensive graphical calculus for quantum computing. We axiomatize the notion of an environment, which together with the axiomatic notion of classical structure enables us to define classical channels, quantum measurements and classical control. If we moreover adjoin the axiomatic notion of complementarity, we obtain sufficient structural power for constructive representation and correctness derivation of typical quantum informatic protocols.