## Differentiating and Integrating ZX Diagrams

ZX-calculus has proved to be a useful tool for quantum technology with a wide range of successful applications. Most of these applications are of an algebraic nature. However, other tasks that involve differentiation and integration remain unreachable with current ZX techniques. Here we elevate ZX to an analytical perspective by realising differentiation and integration entirely within the framework of ZX-calculus. We explicitly illustrate the new analytic framework of ZX-calculus by applying it in context of quantum machine learning.

## A Quantum Natural Language Processing Approach to Musical Intelligence

A paper from our Compositional Intelligence team which uses QNLP techniques in musical composition. This paper is part of the work accompanying the world’s first ever symposium on Quantum Computing and Music. There has been tremendous progress in Artificial Intelligence (AI) for music, in particular for musical composition and access to large databases for commercialisation through the Internet.

## Representing Matrices Using Algebraic ZX-calculus

In this paper, we represent all the elementary matrices of size 2m × 2m using algebraic ZX-calculus, then show their properties on inverses and transpose using rewriting rules of ZX-calculus. As a result, we are able to depict matrices of this size by string diagrams without resorting to a diagrammatic normal form.

## lambeq – An Efficient High-Level Python Library for Quantum NLP

lambeq, the first high-level Python library for Quantum Natural Language Processing (QNLP), is presented in this paper. The open-source toolkit offers a detailed hierarchy of modules and classes implementing all stages of a pipeline for converting sentences to string diagrams, tensor networks, and quantum circuits ready to be used on a quantum computer.

## The Mathematics of Text Structure

In this paper, we give a mathematical foundation, referred to as DisCoCirc, for how sentences interact in texts in order to produce the meaning of that text. First we revisit DisCoCat. While in DisCoCat all meanings are fixed as states (i.e. have no input), in DisCoCirc, word meanings correspond to a type, or system, and the states of this system can evolve. Sentences are gates within a circuit which update the variable meanings of those words.

## Meaning Updating of Density Matrices

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.

## DisCoPy: Monoidal Categories in Python

We introduce DisCoPy, an open source toolbox for computing with monoidal categories. The library provides an intuitive syntax for defining string diagrams and monoidal functors. Its modularity allows the efficient implementation of computational experiments in the various applications of category theory where diagrams have become a lingua franca.

## Quantum Natural Language Processing on Near-Term Quantum Computers

In this work, we describe a full-stack pipeline for natural language processing on near-term quantum computers, aka QNLP. The language-modelling framework we employ is that of compositional distributional semantics (DisCoCat). Within this model, the grammatical reduction of a sentence is interpreted as a diagram, encoding a specific interaction of words according to the grammar. This interaction, together with a specific choice of word embedding, realises the meaning of a sentence.

## Grammar-Aware Question-Answering on Quantum Computers

We perform the first implementation of an NLP task on noisy intermediate-scale quantum (NISQ) hardware. Sentences are instantiated as parameterised quantum circuits. We encode word-meanings in quantum states and explicitly account for grammatical structure – which even in mainstream NLP is not commonplace – by faithfully hardwiring it as entangling operations.

## Foundations for Near-Term Quantum Natural Language Processing

We provide conceptual and mathematical foundations for near-term quantum natural language processing (QNLP) and do so in quantum computer scientist-friendly terms. We opted for an expository presentation style and provide references for supporting empirical evidence and formal statements concerning mathematical generality.