Quantum computing is a fast-growing research area. A quantum computer can solve problems which are out of reach of the classical computers. It is essential to understand the fundamental structures of quantum information processing in order to develop specific and effective tools. In this habilitation thesis we consider several examples of graphical approaches for quantum computing : first of all the graph state formalism, the measurement-based model of quantum computing and the quantum secret sharing protocol. We then explore in depth the ZX-calculus, a category theory based diagrammatic language for quantum reasoning. This language is equipped with an equational theory to transform any diagram into an equivalent diagram. The language is said to be complete if, for any pair of diagrams representing the same quantum evolution,there exists a transformation from one to the other using the rules of the language. We show the completeness of ZX-calculation for several fragments of quantum mechanics.
Quantum computing is a fast-growing research area. A quantum computer can solve problems which are out of reach of the classical computers. It is essential to understand the fundamental structures of quantum information processing in order to develop specific and effective tools. In this habilitation thesis we consider several examples of graphical approaches for quantum computing : first of all the graph state formalism, the measurement-based model of quantum computing and the quantum secret sharing protocol. We then explore in depth the ZX-calculus, a category theory based diagrammatic language for quantum reasoning. This language is equipped with an equational theory to transform any diagram into an equivalent diagram. The language is said to be complete if, for any pair of diagrams representing the same quantum evolution,there exists a transformation from one to the other using the rules of the language. We show the completeness of ZX-calculation for several fragments of quantum mechanics.
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.
This thesis is about the application of graphical languages to quantum computing. By graphical language, we mean the use of diagrams, similar to circuits, representing the evolution of quantum systems. The thesis introduces those languages in the formalism of category theory and focuses mainly on one language: the ZX-calculus, and its close relatives, the ZW-calculus and ZH-calculus. The notion of flexsymmetry is introduced, describing diagrams whose inputs and outputs are all interchangeable. This notion is used to classify all languages similar to the ZX-calculus. It is shown that the only admissible languages are the ZX-calculus, the ZH-calculus, and the ZW-calculus. Then is tackled the question of extending those languages to mixed-state quantum mechanics. A general categorical construction is proposed and is applied to provide extensions of the different languages. Finally, the thesis introduces notations allowing to handle in a compact way quantum algorithms relying on arbitrary large diagrams. To challenge their efficiency, those notations are used to show the correction of various quantum algorithms.
In the models of quantum computing usually considered, some quantum data is manipulated by means of operations which are controlled in an essentially classical way. Controlling these operations in a quantum way is actually possible, but has been much less studied. In particular, quantum control misses a formalism in which one could represent it in a simple way in order to efficiently reason on processes involving it. The first contribution of this thesis is to lay the foundations of a formal framework dedicated to quantum control, in the form of a graphical language. Our main result about this language is the introduction of a complete equational theory, that is, a set of equations that makes it possible, by successive local rewriting, to transform a given diagram into any other diagram representing the same program or physical process. A second contribution is to apply this formalism, on the one hand, to the problem of resource optimisation of processes involving quantum control, and on the other hand, to the characterisation of the observational equivalence of quantum communication channels. A third contribution of this thesis is to introduce a language for linear optical circuits. We equip this language with a complete equational theory, together with a simple normal form, reachable via a strongly normalising and confluent rewriting system. The last contribution of this thesis, maybe the most significant one, is to introduce a complete equational theory for the language of quantum circuits. We obtain this result by exploiting a correspondence between quantum circuits and optical circuits, which allows us to transfer the equational theory already obtained for optical circuits.
Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded languages. This creates obstacles to formally reasoning about graphical languages, since imposing an inductive structure obfuscates the diagrammatic nature of graphical languages, along with their corresponding equational theories. To address this gap, we present Vy ZX, a verified library for reasoning about inductively defined graphical languages. These inductive constructs arise naturally from category-theoretic definitions. We developed Vy ZX to V erify the ZX -calculus, a graphical language for reasoning about quantum computation. The ZX-calculus comes with a collection of diagrammatic rewrite rules that preserve the graph’s semantic interpretation. We show how inductive graphs in Vy ZX are used to prove the soundness of the ZX-calculus rewrite rules and apply them in practice using standard proof assistant techniques. We also provide an IDE-integrated visualizer for proof engineers to directly reason about diagrams in graphical form.
Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not sufficient: the new formalisms also need to have equivalent deductive power. This requirement is captured by the property of completeness, which means that any equality that can be derived using some standard formalism can also be derived graphically. In this thesis, I consider the ZX-calculus, a graphical language for pure state qubit quantum mechanics. I show that it is complete for pure state stabilizer quantum mechanics, so any problem within this fragment of quantum theory can be fully analysed using graphical methods. This includes questions of central importance in areas such as error-correcting codes or measurement-based quantum computation. Furthermore, I show that the ZX-calculus is complete for the single-qubit Clifford+T group, which is approximately universal: any single-qubit unitary can be approximated to arbitrary accuracy using only Clifford gates and the T-gate. [...] Lastly, I extend the use of rigorous graphical languages outside quantum theory to Spekkens' toy theory, a local hidden variable model that nevertheless exhibits some features commonly associated with quantum mechanics. [...] I develop a graphical calculus similar to the ZX-calculus that fully describes Spekkens' toy theory, and show that it is complete. Hence, stabilizer quantum mechanics and Spekkens' toy theory can be fully analysed and compared using graphical formalisms. Intuitive graphical languages can replace conventional formalisms for the analysis of many questions in quantum computation and foundations without loss of mathematical rigour or deductive power.
The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement-based quantum computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The latter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Additionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits.