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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Diagrammatic Reasoning and Quantum Computation Aleks Kissinger ACA, Kalamata November 4, 2015 Q UANTUM G ROUP Introduction Quantum circuits Spiders ZX-calculus MBQC

  1. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey As an equational theory • The good: • complete for Clifford circuits: � C 1 � = � C 2 � = ⇒ C 1 = E C 2 • unique normal forms • relatively compact (3 generators, 15 rules) • The bad: • rules are large, and don’t carry any intuition or algebraic structure • rewrite strategy is complicated (17 derived gates, 100 derived rules) • The ugly: • proof of completeness is extremely complicated ( > 100 pages long! though mostly machine-generated)

  2. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Can we do better? • Yes!

  3. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Can we do better? • Yes! • We can capture underlying algebraic structure by decomposing gates into smaller pieces ⊕ H ⊕ H H ⊕ Z α ⊕ H

  4. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Can we do better? • Yes! • We can capture underlying algebraic structure by decomposing gates into smaller pieces ⊕ H ⊕ H H ⊕ Z α ⊕ H

  5. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Decomposing CNOT ⊕

  6. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Decomposing CNOT ⊕

  7. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Decomposing CNOT ⊕ | i � | j �

  8. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Decomposing CNOT | i � ⊕ copy | i � | i � | j �

  9. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Decomposing CNOT | i � | i ⊕ j � ⊕ copy xor | i � | j �

  10. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps  | 00 � �→ | 0 �   �   | 0 � �→ | 00 � | 01 � �→ | 1 �  ⊕ | 1 � �→ | 11 � | 10 � �→ | 1 �     | 11 � �→ | 0 � 

  11. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � � | 0 � �→ | 00 � | ++ � �→ | + � ⊕ | 1 � �→ | 11 � |−−� �→ |−�

  12. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � � | 0 � �→ | 00 � | + � �→ | ++ � ⊕ | 1 � �→ | 11 � |−� �→ |−−�

  13. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � � | 0 � �→ | 00 � | + � �→ | ++ � | 1 � �→ | 11 � |−� �→ |−−�

  14. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � | 0 � �→ | 00 � | 1 � �→ | 11 �

  15. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � � | 0 � �→ | 00 � | 0 � �→ 1 | 1 � �→ | 11 � | 1 � �→ 1

  16. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � | 0 � �→ | 00 � � � 0 | + � 1 | | 1 � �→ | 11 �

  17. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey ‘Copy’ maps � | 0 � �→ | 00 � � � 0 | + � 1 | | 1 � �→ | 11 � � | 00 � �→ | 0 � � | 0 � + | 1 � | 11 � �→ | 1 �

  18. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Algebraic identities... These satisfy 8 identities: = = = = = = = ...making them a commutative Frobenius algebra .

  19. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey But luckily... ...you don’t need to remember all that! The only thing to remember is, for: ... � | 0 .. 0 � �→ | 0 ... 0 � := | 1 .. 1 � �→ | 1 ... 1 � ...

  20. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey But luckily... ...you don’t need to remember all that! The only thing to remember is, for: ... � | 0 .. 0 � �→ | 0 ... 0 � := | 1 .. 1 � �→ | 1 ... 1 � ... we have: ... ... ... = ... ... ... ...

  21. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey But luckily... ...you don’t need to remember all that! The only thing to remember is, for: ... � | 0 .. 0 � �→ | 0 ... 0 � := | 1 .. 1 � �→ | 1 ... 1 � ... we have: ... ... ... = ... ... ... ... or equivalently: =

  22. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey But luckily... ...you don’t need to remember all that! The only thing to remember is, for: ... � | + .. + � �→ | + ... + � := |− .. −� �→ |− ... −� ... we have: ... ... ... = ... ... ... ... or equivalently: =

  23. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey What about 2-colour diagrams? Direction of edges doesn’t matter: = =:

  24. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey What about 2-colour diagrams? Direction of edges doesn’t matter: = =: ...in fact, only topology matters : = =

  25. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Interaction: Hopf algebra Red + green spiders also satisfy: = = =

  26. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Interaction: Hopf algebra Red + green spiders also satisfy: = = = ...from which we can derive: = make the overall structure into a Hopf algebra

  27. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Circuit calculation ⊕ = ⊕ ⊕

  28. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Circuit calculation =

  29. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Circuit calculation = � � ⇐

  30. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Making spiders universal ... � | 0 .. 0 � �→ | 0 ... 0 � := | 1 .. 1 � �→ | 1 ... 1 � ... ... � | + .. + � �→ | + ... + � := |− .. −� �→ |− ... −� ...

  31. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Making spiders universal ... � | 0 .. 0 � �→ | 0 ... 0 � := α | 1 .. 1 � �→ e i α | 1 ... 1 � ... ... � | + .. + � �→ | + ... + � := α |− .. −� �→ e i α |− ... −� ...

  32. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Making spiders universal ... ... ... ... � α | 0 .. 0 � �→ | 0 ... 0 � ... = := α α + β | 1 .. 1 � �→ e i α | 1 ... 1 � ... ... β ... ... ... ... ... ... � α | + .. + � �→ | + ... + � ... = := α α + β |− .. −� �→ e i α |− ... −� ... ... β ... ...

  33. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Making spiders universal Theorem Phased spiders are universal for qubit quantum computation. Proof. Let: γ ⊕ := := U β α

  34. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey The ZX-calculus The ZX-calculus consists of the two spider-fusion rules: ... ... ... ... ... ... α α ... ... = = α + β α + β ... ... β β ... ... ... ...

  35. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey The ZX-calculus The ZX-calculus consists of the two spider-fusion rules: ... ... ... ... ... ... α α ... ... = = α + β α + β ... ... β β ... ... ... ... four Interaction rules: α π = = = = π π π - α π

  36. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey The ZX-calculus The ZX-calculus consists of the two spider-fusion rules: ... ... ... ... ... ... α α ... ... = = α + β α + β ... ... β β ... ... ... ... four Interaction rules: α π = = = = π π π - α π and the Colour Change rule: · · · · · · · · · · α α = · · · · · ·

  37. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey The ZX-calculus The ZX-calculus consists of the two spider-fusion rules: ... ... ... ... ... ... α α ... ... = = α + β α + β ... ... β β ... ... ... ... four Interaction rules: α π = = = = π π π - α π and the Colour Change rule: · · · · · π · · · · · 2 where := α α π = 2 · · · π 2 · · ·

  38. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Completeness Theorem (Backens 2013) The ZX-calculus is complete for Clifford ZX-diagrams: � D 1 � = � D 2 � = ⇒ D 1 = zx D 2 . . . . . . · · · · · · · · · · · · D 1 := D 2 := π π π π 2 2 2 2 · · · · · · · · · · · · . . . . . .

  39. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing

  40. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model

  41. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  42. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  43. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  44. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  45. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  46. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  47. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  48. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  49. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  50. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  51. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurement-based quantum computing • Measurement-based quantum computing is an alternative (and equivalent) paradigm to the circuit model • Rather than repeatedly applying operations to a small number of systems, start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ... • But crucially, the choices of measurements can depend on past measurement outcomes . This is called feed-forward , and it’s where all the magic happens.

  52. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Graph states and cluster states • Graph states are prepared by starting with many qubits in the | + � state and creating entanglement with controlled-Z operations: =

  53. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Graph states and cluster states • Graph states are prepared by starting with many qubits in the | + � state and creating entanglement with controlled-Z operations: = • Since controlled-Z’s commute, the only relevant part is the graph: ... ... ... ... ← → ... ... ... ... ... ... ... ... ... ... ...

  54. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurements and feed-forward • Compute with single qubit ONB measurements of this form: � � � � π α α + π , ,

  55. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurements and feed-forward • Compute with single qubit ONB measurements of this form: � � � � π α α + π , , • We want to get the first outcome and treat the second outcome as an error: α error π ...

  56. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurements and feed-forward • We can propagate the error out using the ZX-rules: α α α π = = π ... ... ... π π α α = = π π π ... ... π

  57. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurements and feed-forward • We can propagate the error out using the ZX-rules: α α α π = = π ... ... ... π π α α = = π π π ... ... π • If we know an error occurred, we can modify our later measurement choices to account for it: γ + π α − β γ α β = π π ... ...

  58. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Measurements and feed-forward • We can propagate the error out using the ZX-rules: α α α π = = π ... ... ... π π α α = = π π π ... ... π • If we know an error occurred, we can modify our later measurement choices to account for it: γ + π α − β γ α β = π π ... ... � � • ⇐

  59. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Notable results

  60. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Notable results: MBQC • Duncan & Perdrix used the ZX-calculus to offer a new technique for transforming MBQC patterns to circuits, which has some advantages over other known methods, e.g. not requiring ancillas. 1 • For more details, Ducan has written a self-contained introduction to MBQC from the diagrammatic/ZX point of view, which is available on the arXiv. 2 1 Rewriting measurement-based quantum computations with generalised flow. R. Duncan, S. Perdrix, ICALP 2010. personal.strath.ac.uk/ross.duncan/papers/gflow.pdf 2 A graphical approach to measurement-based quantum computing. R. Duncan. arXiv:1203.6242

  61. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Notable results: quantum algorithms • Vicary gave graphical characterisations of standard quantum algorithms 3 • ...a framework since used by Vicary & Zeng to develop new algorithms as generalisations 4 3 The Topology of Quantum Algorithms. LICS 2013, J. Vicary. arXiv:1209.3917 4 Abstract structure of unitary oracles for quantum algorithms. J.Vicary, W. Zeng. arXiv:1406.1278

  62. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Notable results: quantum protocols • Coecke, along with 3 Wangs and a Zhang give graphical proof of QKD 5 • Hillebrand gave rewriting proofs of many ( ∼ 25) quantum protocols. 6 • Zamdzhiev used ZX-calculus to verify 3 kinds of quantum secret sharing. 7 5 Graphical Calculus for Quantum Key Distribution. B. Coecke, Q. Wang, B. Wang, Y. Wang, and Q. Zhang. QPL 2011. 6 Quantum Protocols involving Multiparticle Entanglement and their Representations in the zx-calculus. A. Hillebrand. Masters thesis, Oxford 2011. www.cs.ox.ac.uk/people/bob.coecke/Anne.pdf 7 An Abstract Approach towards Quantum Secret Sharing. Masters thesis, Oxford 2012. www.cs.ox.ac.uk/people/bob.coecke/VladimirZamdzhievThesis.pdf

  63. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Notable results: quantum non-locality • AK, Coecke, Duncan, and Wang gave diagrammatic presentation of GHZ/Mermin non-locality argument 8 α 1 α 2 α 3 = = = � α i � α i � α i • ...which has since been generalised to arbitrary dimensions and quantum-like theories 9 8 Strong Complementarity and Non-locality in Categorical Quantum Mechanics. B. Coecke, R. Duncan, A. Kissinger, Q. Wang. LICS 2012. 9 Mermin Non-Locality in Abstract Process Theories. QPL 2015

  64. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Where do we go from here? • Completeness (Clifford + T, full)

  65. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Where do we go from here? • Completeness (Clifford + T, full) • Automation: implementation of Clifford decision procedure, theory synthesis

  66. Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Where do we go from here? • Completeness (Clifford + T, full) • Automation: implementation of Clifford decision procedure, theory synthesis • Bigger algorithms , more sophisticated protocols , and generally more expressiveness of the diagrammatic language

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