Aleks Kissinger

Picturing quantum processes, rewriting quantum pictures

Aleks Kissinger (Radboud University)

Slides of lecture 1 and lecture 2.

We provide a self-contained introduction to quantum theory using a unique diagrammatic language. Far from simple visual aids, the diagrams we use are mathematical objects in their own right, which allow us to develop from first principles a completely rigorous treatment of textbook quantum theory. The core calculational tool will be diagram rewriting, where simple local identities on diagrams are used to explain and derive the behaviour of many kinds of quantum processes.
Using diagrams and diagram rewriting, we will show how key notions such as quantum and classical processes, measurements, quantum teleportation, and quantum computations can be expressed and reasoned about. We will also briefly cover how such rewrites can be automated to reason about large quantum processes and perform optimisation of quantum circuits.

Outline of the lecture

  • Quantum theory: a story about penguins, polar bears, and pictures
    • The diagrammatic notation, and diagrams-as-formulae
    • Process theories
    • Compound systems and (non-)separability
    • Maximal non-separability and string diagrams
  • Quantum maps, processes, and measurements
    • Constructing the process theory of quantum maps
    • Quantum processes and the causality postulate
    • Quantum teleportation as rewriting
    • Quantum measurements
  • Spiders and their rewrite theory
    • Classical data operations and spiders
    • Complementarity and strong complementarity
    • Phase spiders
    • The ZX-calculus
  • Applications of a complete graphical calculus
    • Quantum algorithms
    • Quantum security protocols
    • Quantum non-locality
    • Automated rewriting, circuit optimisation

Exercises or experiments

Quantum (non)locality will be demonstrated using “” devices, courtesy of Jamie Vicary and Oxford Hackspace.

Bibliographical references:

  1. Coecke & Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

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