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 di ﬀ erentiation and integration remain unreachable with current ZX techniques. Here we elevate ZX to an analytical perspective by realising di ﬀ erentiation 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 for the analysis of barren plateaus.

Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning

Diagrammatic representations of quantum algorithms and circuits oﬀer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for ﬁxed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry. We provide several new ZX-diagram rewrite rules and generalizations for this setting. In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coeﬃcients can be eﬀectively ignored. This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules. We demonstrate that the diagrammatic approach oﬀers useful insights into algorithm structure and performance by considering several ansätze from the literature including realizations of hardware-eﬃcient ansätze and QAOA. We ﬁnd that by employing a diagrammatic representation, calculations across diﬀerent ansätze can become more intuitive and potentially easier approach systematically than by alternative means. Fi-nally, we outline how diagrammatic approaches may aid in the design and study of new and more eﬀective quantum circuit ansätze.

Diagrammatic Analysis for Parameterized Quantum Circuits

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 di ﬀ erentiation and integration remain unreachable with current ZX techniques. Here we elevate ZX to an analytical perspective by realising di ﬀ erentiation 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.

Differentiating and Integrating ZX Diagrams

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

Addition and Differentiation of ZX-Diagrams

We present lambeq, the first high-level Python library for Quantum Natural Language Processing (QNLP). 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. lambeq supports syntactic parsing, rewriting and simplification of string diagrams, ansatz creation and manipulation, as well as a number of compositional models for preparing quantum-friendly representations of sentences, employing various degrees of syntax sensitivity. We present the generic architecture and describe the most important modules in detail, demonstrating the usage with illustrative examples. Further, we test the toolkit in practice by using it to perform a number of experiments on simple NLP tasks, implementing both classical and quantum pipelines.

lambeq: An Efficient High-Level Python Library for Quantum NLP.

We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this to ZX diagrams, we show how to calculate diagrammatically the gradient of a linear map with respect to a phase parameter. For diagrams of parametrised quantum circuits, we get the well-known parameter-shift rule at the basis of many variational quantum algorithms. We then extend our method to the automatic differentation of hybrid classical-quantum circuits, using diagrams with bubbles to encode arbitrary non-linear operators. Moreover, diagrammatic differentiation comes with an open-source implementation in DisCoPy, the Python library for monoidal categories. Diagrammatic gradients of classical-quantum circuits can then be simplified using the PyZX library and executed on quantum hardware via the tket compiler. This opens the door to many practical applications harnessing both the structure of string diagrams and the computational power of quantum machine learning.

Diagrammatic Differentiation for Quantum Machine Learning

While the DisCoCat model (Coecke et al., 2010) has been proved a valuable tool for studying compositional aspects of language at the level of semantics, its strong dependency on pregroup grammars poses important restrictions: first, it prevents large-scale experimentation due to the absence of a pregroup parser; and second, it limits the expressibility of the model to context-free grammars. In this paper we solve these problems by reformulating DisCoCat as a passage from Combinatory Categorial Grammar (CCG) to a category of semantics. We start by showing that standard categorial grammars can be expressed as a biclosed category, where all rules emerge as currying/uncurrying the identity; we then proceed to model permutation-inducing rules by exploiting the symmetry of the compact closed category encoding the word meaning. We provide a proof of concept for our method, converting "Alice in Wonderland" into DisCoCat form, a corpus that we make available to the community.

A CCG-Based Version of the DisCoCat Framework.

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. We are interested in further advancing this field, focusing on composition. In contrast to current black-box AI methods, we are championing an interpretable compositional outlook on generative music systems. In particular, we are importing methods from the Distributional Compositional Categorical (DisCoCat) modelling framework for Natural Language Processing (NLP), motivated by musical grammars. Quantum computing is a nascent technology, which is very likely to impact the music industry in time to come. Thus, we are pioneering a Quantum Natural Language Processing (QNLP) approach to develop a new generation of intelligent musical systems. This work follows from previous experimental implementations of DisCoCat linguistic models on quantum hardware. In this chapter, we present Quanthoven, the first proof-of-concept ever built, which (a) demonstrates that it is possible to program a quantum computer to learn to classify music that conveys different meanings and (b) illustrates how such a capability might be leveraged to develop a system to compose meaningful pieces of music. After a discussion about our current understanding of music as a communication medium and its relationship to natural language, the chapter focuses on the techniques developed to (a) encode musical compositions as quantum circuits, and (b) design a quantum classifier. The chapter ends with demonstrations of compositions created with the system.

Richie Yeung

Papers

Diagrammatic Differentiation for Quantum Machine Learning

Diagrammatic Differentiation for Quantum Machine Learning

A CCG-Based Version of the DisCoCat Framework.

lambeq: An Efficient High-Level Python Library for Quantum NLP.

A Quantum Natural Language Processing Approach to Musical Intelligence.