This paper is an introduction to diagrammatic methods for representing quantum processes and quantum computing. We review basic notions for quantum information and quantum computing. We discuss topological diagrams and some issues about using category theory in representing quantum computing and teleportation. We analyze very carefully the diagrammatic meaning of the usual representation of the Mach-Zehnder interferometer, and we show how it can be generalized to associate to each composition of unitary transformations a "laboratory quantum diagram" such that particles moving though the many alternate paths in this diagram will mimic the quantum process represented by the composition of unitary transformations. This is a finite dimensional way to think about the Feynman Path Integral. We call our representation result the Path Theorem. Then we go back to the basics of networks and matrices and show how elements of quantum measurement can be represented with network diagrams.
This paper is an introduction to diagrammatic methods for representing quantum processes and quantum computing. We review basic notions for quantum information and quantum computing. We discuss topological diagrams and some issues about using category theory in representing quantum computing and teleportation. We analyze very carefully the diagrammatic meaning of the usual representation of the Mach-Zehnder interferometer, and we show how it can be generalized to associate to each composition of unitary transformations a "laboratory quantum diagram" such that particles moving though the many alternate paths in this diagram will mimic the quantum process represented by the composition of unitary transformations. This is a finite dimensional way to think about the Feynman Path Integral. We call our representation result the Path Theorem. Then we go back to the basics of networks and matrices and show how elements of quantum measurement can be represented with network diagrams.
We present "Diagrams of States", a way to graphically represent and analyze how quantum information is elaborated during the execution of quantum circuits. This introductory tutorial illustrates the basics, providing useful examples of quantum computations: elementary operations in single-qubit, two-qubit and three-qubit systems, immersions of gates on higher dimensional spaces, generation of single and multi-qubit states, procedures to synthesize unitary, controlled and diagonal matrices. To perform the analysis of quantum processes, we directly derive diagrams of states from physical implementations of quantum circuits associated to the processes. Complete diagrams are then rearranged into simplified diagrams, to visualize the overall effects of computations. Conversely, diagrams of states help to conceive new quantum algorithms, by schematically describing desired manipulations of quantum information with intuitive diagrams and then by guessing the equivalent complete diagrams, from which the corresponding quantum circuit is obtained effortlessly. Related examples and analysis of complex algorithms will be provided in future works, for whose comprehension this first tutorial offers the necessary introduction.
The tutorial introduces quantum computing to a general audience. It begins with the concept of qubit and its representation, followed by 1-qubit quantum gates and qubit measurement. It will then move on to 2-qubit states, entanglement, and 2-qubit gates. Switching gears, the tutorial will discuss quantum Fourier transforms, unitary transforms, and quantum arithmetic. Finally, we will go through a simple quantum algorithm and discuss solid-state implementations of networks of qubits, highlighting the challenges for reading and controlling them. We conclude with future perspectives.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of unitary transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to "unitary circuit models", which are models of computation in which unitary operators are expressed as a product of "elementary" unitary transformations. However, unitary transformations can also be effected as a composition of operations which are not all unitary themselves: the "one-way measurement model" is one such model of quantum computation. In this thesis, we examine the relationship between representations of unitary operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement model can be described as simulating a unitary circuit, by considering the combinatorial structures which are common to unitary circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from unitary transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the unitary circuit model.
In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley--Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley--Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley--Lieb category which is the collection of all the Temperley--Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topological-like features, and quantum information flow is recast in a correct formulation. In other words, we propose the extended Temperley--Lieb category as a mathematical framework to describe quantum information and computation involving maximally entangled states and local unitary transformations.
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.