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The method

Quantum in Pictures

Quantum theory, done with diagrams

Quantum in Pictures replaces pages of algebra with diagrams: wires, and green and red spiders. It is built on the ZX-calculus, introduced at the University of Oxford in 2008, and it gives up nothing in rigour: the pictures are the proofs. Today it is taught from textbooks, used inside industry’s quantum compilers, and has been tested in the classroom with school students.

See the method The course
The method

Pages of algebra, or four pictures.

Here is quantum teleportation proved twice. On the left, the way most textbooks do it, with vectors and a change of basis. On the right, the way Quantum in Pictures does it, with nothing lost in rigour. Both arrive at the same answer.

In algebra

  1. Alice’s qubit|ψ⟩ = α|0⟩ + β|1⟩
  2. The Bell pair|Φ⁺⟩ = (|00⟩ + |11⟩) / √2
  3. All three qubits|ψ⟩ ⊗ |Φ⁺⟩ = (α|000⟩ + α|011⟩
  4. + β|100⟩ + β|111⟩) / √2
  5. The Bell basis|Φ±⟩ = (|00⟩ ± |11⟩) / √2
  6. |Ψ±⟩ = (|01⟩ ± |10⟩) / √2
  7. Turned around|00⟩ = (|Φ⁺⟩ + |Φ⁻⟩) / √2
  8. |11⟩ = (|Φ⁺⟩ − |Φ⁻⟩) / √2
  9. |01⟩ = (|Ψ⁺⟩ + |Ψ⁻⟩) / √2
  10. |10⟩ = (|Ψ⁺⟩ − |Ψ⁻⟩) / √2
  11. Regrouped|ψ⟩ ⊗ |Φ⁺⟩ = ½ [ |Φ⁺⟩ ⊗ (α|0⟩ + β|1⟩)
  12. + |Φ⁻⟩ ⊗ (α|0⟩ − β|1⟩)
  13. + |Ψ⁺⟩ ⊗ (β|0⟩ + α|1⟩)
  14. + |Ψ⁻⟩ ⊗ (α|1⟩ − β|0⟩) ]
  15. Alice measuresfour outcomes, each with probability ¼;
  16. she sends Bob two bits, b₀b₁
  17. Bob corrects00 : α|0⟩ + β|1⟩, nothing to do
  18. 10 : Z (α|0⟩ − β|1⟩) = α|0⟩ + β|1⟩
  19. 01 : X (β|0⟩ + α|1⟩) = α|0⟩ + β|1⟩
  20. 11 : Z X (α|1⟩ − β|0⟩) = Z (α|0⟩ − β|1⟩)
  21. = α|0⟩ + β|1⟩
  22. UsingX = 0110Z = 100−1
  23. SoBob holds α|0⟩ + β|1⟩ = |ψ⟩  ∎

In pictures

  1. AliceBobψb₀πb₁πb₁πb₀π

    1Draw the protocol

    Wires carry qubits; Alice’s measurement and Bob’s corrections are spiders.

  2. AliceBobψb₀πb₁πb₁πb₀π

    2Pull the wire straight

    Only the connections matter, so the wire can be pulled taut.

  3. AliceBobψb₀π2b₁πb₀π

    3Let the spiders fuse

    Same-coloured neighbours fuse: 2b₁π, then 2b₀π, each a full turn or none.

  4. AliceBobψ

    4Read the result

    A plain wire from Alice to Bob: Bob has her qubit.

Same result: Bob has Alice’s qubit.

Your turn

Prove it with your own hands.

Two rules of the ZX-calculus are enough to prove teleportation. Learn them on a small diagram, then use them on the real one.

αβ

Drag one green spider onto the other.

Track record

Twenty years from a research idea to the classroom.

The method was built in a university research group, written up in textbooks, put inside industry’s compilers, and then tested with school students.

  1. 2005

    Research

    Quantum mechanics, in pictures

    In “Kindergarten Quantum Mechanics”, Bob Coecke sets out how quantum mechanics can be done with pictures alone, from joint work with Samson Abramsky at Oxford.

    Source

  2. 2008

    Research

    The ZX-calculus

    Bob Coecke and Ross Duncan introduce the ZX-calculus, the language of green and red spiders, at the Oxford University Computing Laboratory.

    Source

  3. 2017

    Book

    Picturing Quantum Processes

    Bob Coecke and Aleks Kissinger publish the full diagrammatic course with Cambridge University Press. It becomes a core text at Oxford.

    Source

  4. 2023

    Book

    Quantum in Pictures

    Bob Coecke and Stefano Gogioso bring the same ideas to readers with no maths background. Published by Quantinuum in February.

  5. 2023

    Experiment

    The Oxford experiment

    Over the summer, 54 UK school students learn quantum theory in pictures, online, two hours a week for eight weeks.

    The results

  6. 2025

    Results

    The study is published

    The experiment is written up in full, with the exam results, and covered by Physics World.

    Source

  7. 2026

    Today

    QPic, the course

    NEUROssance launches Quantum in Pictures as its flagship course. The method has now been taught in 12 countries.

The Oxford experiment, 2023

School students, a postgraduate exam.

In 2023 the University of Oxford, Quantinuum and IBM led an experiment: 54 UK school students, mostly from state schools, learned quantum theory in pictures, online, two hours a week for eight weeks. The final exam used questions from past Oxford postgraduate papers.

students
54
weeks
8
hours
16
Read the study

82%

passed the exam

48%

reached distinction level

…UK schoolchildren – who then beat the average exam scores of Oxford University’s postgraduate physics students.

In industry

Inside quantum compilers.

The ZX-calculus is at the heart of Quantinuum’s TKET compiler, and is used across the field, at PsiQuantum, Google, Microsoft and IBM among others.

ZX-calculus at Quantinuum
  • Quantum circuit optimisation
  • Quantum error correction
  • Measurement-based quantum computing
  • Quantum simulation
  • Quantum foundations
  • Quantum natural language processing

In the press

Reading

The books, and the papers behind them.

Papers

  1. 2005Kindergarten Quantum MechanicsB. Coecke · arXiv
  2. 2010Quantum PicturalismB. Coecke · Contemporary Physics
  3. 2023Basic ZX-calculus for students and professionalsB. Coecke · arXiv
  4. 2023Quantum Picturalism: Learning Quantum Theory in High SchoolS. Dündar-Coecke, L. Yeh, C. Puca and others · IEEE QCE
  5. 2025Making the quantum world accessible to young learners through Quantum Picturalism: an experimental studyS. Dündar-Coecke, C. Puca, L. Yeh and others · arXiv
  6. 2025High schoolers excel at Oxford quantum course using pictorial mathematicsB. Coecke, A. Kissinger, S. Gogioso and others · arXiv

More on the ZX-calculus: zxcalculus.com

About the course

QPic8 weeks16 hoursLive onlineFrom age 14CPD certificate£400

New dates are being set

Quantum physics is often seen as strange, abstract and open only to scientists with advanced mathematical training. Quantum in Pictures explores the quantum world with an intuitive, diagram-based form of mathematics instead: the ZX-calculus.

The ZX-calculus was developed by University of Oxford professors to describe tensors, a framework used across engineering, computing and quantum technologies. It has been proven complete for pure qubit quantum mechanics: any valid equality can be derived with its graphical rules, with no heavy algebra required.

The course opens up ideas that are normally taught only in postgraduate university courses. You learn how diagrams can work as a scientific language that reveals the logic of quantum theory and quantum computing in a way that feels natural, graspable and even playful. The focus is not on memorising, but on seeing how quantum ideas work.

Course content

Eight sessions, one idea at a time.

Each session starts with a one-hour recorded lecture, then a live tutorial of one to two hours where you solve problems with diagrams together. A short quiz follows every other tutorial.

  1. Session 1

    Introduction to quantum thinking

  2. Session 2

    Foundations of diagrammatic mathematics

  3. Session 3

    Quantum teleportation, part I

    The proof you tried above

  4. Session 4

    Errors, corrections and flip-undoability

  5. Session 5

    Spiders and their mathematics: the building blocks of the ZX-calculus

  6. Session 6

    Bits, qubits and quantum gate computing

  7. Session 7

    Quantum computing

  8. Session 8

    The power of ZX: quantum versus ordinary

  1. Weeks 9 and 10

    Take-home exam

    Optional, with conceptual and diagrammatic problems. It is needed for the certificate.

  2. On completion

    CPD certificate

    A Continuing Professional Development certificate, for taking part in the sessions and quizzes and completing the exam.

Faculty

Who leads the course.

Led by Prof Bob Coecke and the QPic teaching team.

“It’s easier to convince kids than adults about quantum mechanics.”
Bob Coecke, The Guardian, December 2023
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Portrait of Prof Bob Coecke

Distinguished Visiting Research Chair, Perimeter Institute. Formerly Professor at Oxford and Chief Scientist at Quantinuum.

A pioneer of diagrammatic reasoning in quantum theory. Bob Coecke spent twenty years at the University of Oxford, where he co-founded and led the Quantum Group, before moving to industry as Chief Scientist at Quantinuum. He continues to supervise students at Oxford and elsewhere, and still teaches at Oxford’s Mathematical Institute.

Career

  1. Relational IntelligenceFounder and CEONow
  2. Perimeter Institute for Theoretical PhysicsDistinguished Visiting Research ChairNow
  3. University of OxfordFellow of Wolfson College; Visiting Fellow, Department of Computer Science and Mathematical InstituteNow
  4. QuantinuumChief Scientist, where he invented Quantum NLP and Compositional Intelligence
  5. University of OxfordProfessor of Quantum Foundations, Logics and Structures, Department of Computer Science20 years

Highlights

  • Pioneered categorical quantum mechanics, the ZX-calculus, DisCoCat and quantum natural language processing
  • Co-founded and led Oxford’s Quantum Group, which grew to 50 members; supervised 70 PhD students
  • Co-author of Picturing Quantum Processes and Quantum in Pictures, and of some 200 research papers
  • A founding father of the QPL and Applied Category Theory communities; founder of the journal Compositionality

Also teaches