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Picturing Quantum Processes Aleks Kissinger and Bob Coecke Radboud University and Oxford University ESSLLI Toulouse 2017 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 1 / 40 www.cambridge.org/pqp 20% discount @ CUP

  1. Picturing Quantum Processes Aleks Kissinger and Bob Coecke Radboud University and Oxford University ESSLLI Toulouse 2017 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 1 / 40

  2. www.cambridge.org/pqp 20% discount @ CUP with code: COECKE2017 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 2 / 40

  3. Chapters 11-13: Quantum Foundations, Computation, and Resources Mermin once summarized a popular attitude towards quantum theory as “Shut up and calculate”. We suggest a different slogan: “Shut up and contemplate”! — Lucien Hardy and Rob Spekkens, 2010. Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 3 / 40

  4. Spiders n n ... ... i i i � := ... i ... i i i m m Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 4 / 40

  5. Spiders n n ... ... i i i � := ... i ... i i i m m ... ... ... ... = ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 4 / 40

  6. (Classical) spiders are linear maps: ... ... ...so we can double them to get quantum spiders : ... � � ... ... := double = ... ... ... ...which also fuse together: ... ... ... ... = ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 5 / 40

  7. Measuring is a spider: i = � = � i i i i i i ...and so is encoding: := Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 6 / 40

  8. They connect classical to quantum: ... ... ... ... ...giving something new: ... ... ... ... = ... ... ... ... Bastard spiders ! ... ... ... ... := ... ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 7 / 40

  9. Bastard spiders fuse together: ... ... ... = ... ... ... ... They also absorb other kinds of spider: ... ... ... ... ... ... ... ... = = ... ... ... ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 8 / 40

  10. Useful bastards Bastard spiders arise naturally in the interaction between classical and quantum data, e.g. • Discarding: = = = • Decoherence: = = = Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 9 / 40

  11. Useful bastards (cont’d) • Non-demolition measurement: = = = = Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 10 / 40

  12. Multicoloured spiders ... ... ... ... i i i i � � = = ... ... ... i ... i i i i i ... ... ... ... ... ... ... = ... = ... ... ... ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 11 / 40

  13. Complementary bases 0 1 0 1 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 12 / 40

  14. Complementarity Theorem. Two bases are complementary (a.k.a. mutually unbiased): j 1 ∀ i , j : = D i if and only if: 1 = D Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 13 / 40

  15. Complementarity Theorem. Two bases are complementary (a.k.a. mutually unbiased): j 1 ∀ i , j : = D i if and only if: 1 = D Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 14 / 40

  16. Complementarity 1 = D (encode in ) THEN (measure in ) = (no data flow) Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 15 / 40

  17. Stern-Gerlach N S Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 16 / 40

  18. Stern-Gerlach ւ ց Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 17 / 40

  19. Stern-Gerlach := := Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 18 / 40

  20. Stern-Gerlach := := Z -measurement N S Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 18 / 40

  21. Stern-Gerlach := := Z -measurement X -measurement N S N S Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 18 / 40

  22. Stern-Gerlach S N N S Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 19 / 40

  23. Stern-Gerlach S N N S X -measurement = Z -measurement Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 19 / 40

  24. Stern-Gerlach N S S N N blocked! S Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 20 / 40

  25. Stern-Gerlach N S S N N blocked! S 0 0 2nd Z -measurement = X -measurement 1st Z -measurement Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 20 / 40

  26. Teleportation with spiders Aleks Bob ρ Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 21 / 40

  27. Teleportation with spiders = = = = = Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 22 / 40

  28. Power of the Graphical Language We now have a fairly powerful language, it is natural to ask: Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 23 / 40

  29. Power of the Graphical Language We now have a fairly powerful language, it is natural to ask: Q: Is it universal ? Can we express any map in terms of spiders? Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 23 / 40

  30. Power of the Graphical Language We now have a fairly powerful language, it is natural to ask: Q: Is it universal ? Can we express any map in terms of spiders? Q: Is it complete ? Can we prove every equation between maps using some set of spider-equations? Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 23 / 40

  31. Universality Q: Can we write any linear map as a diagram of: ... ... and ? ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

  32. Universality Q: Can we write any linear map as a diagram of: ... ... and ? ... ... � 1 � 0 A: Clearly not! e.g. 0 i Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

  33. Universality Q: Can we write any linear map as a diagram of: ... ... and ? ... ... � 1 � 0 A: Clearly not! e.g. 0 i Q2: How much more do we need? Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

  34. Phases 0 1 0 1 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 25 / 40

  35. Phases 0 α 1 D α 1 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 26 / 40

  36. Phase spiders = α + β α β Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 27 / 40

  37. Phase spiders = α + β α β ... ... := α ... α ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 27 / 40

  38. Phase spiders = α + β α β ... ... := α ... α ... ... ... ... α ... = α + β β ... ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 27 / 40

  39. Universality Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 28 / 40

  40. Universality Theorem. Any linear map with 2D inputs and outputs can be expressed as a diagram of: ... ... and α α ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 28 / 40

  41. Universality Theorem. Any linear map with 2D inputs and outputs can be expressed as a diagram of: ... ... and α α ... ... Corollary. Any quantum map from qubits to qubits can be expressed as a diagram of: ... ... and α , α , ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 28 / 40

  42. Universality Proof. (idea) γ Any single-qubit unitary: � = + CNOT: β U α ...gives any unitary, which gives any state: ... ... = � � U ψ ... 0 0 ...which gives any map by process-state duality. Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 29 / 40

  43. Completeness Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 30 / 40

  44. Completeness Q. Can we find a complete set of equations to describe the behaviour of: ... ... and ? α α ... ... Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 30 / 40

  45. Completeness Q. Can we find a complete set of equations to describe the behaviour of: ... ... and ? α α ... ... A. Yes! The ZX-calculus . Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 30 / 40

  46. ZX-calculus ... ... ... ... ... ... α α ... ... = = α + β α + β ... ... β β ... ... ... ... .... .... π π - π - π 2 2 2 2 = = - π 2 ... ... - π π 2 2 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 31 / 40

  47. ZX-calculus Theorem. The ZX-calculus is complete for Clifford diagrams , i.e. diagrams where α ∈ { 0 , π, ± π 2 } . Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 32 / 40

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