Picturing Quantum Processes Aleks Kissinger and Bob Coecke Radboud - - PowerPoint PPT Presentation

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Picturing Quantum Processes Aleks Kissinger and Bob Coecke Radboud - - PowerPoint PPT Presentation

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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

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SLIDE 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

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SLIDE 2

www.cambridge.org/pqp 20% discount @ CUP with code: COECKE2017

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 2 / 40

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SLIDE 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

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SLIDE 4

Spiders

m n

... ... :=

i i i i i i

  • i

n m

... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 4 / 40

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SLIDE 5

Spiders

m n

... ... :=

i i i i i i

  • i

n m

... ... ... ...

...

= ... ... ... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 4 / 40

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SLIDE 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

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SLIDE 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

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SLIDE 8

They connect classical to quantum:

... ... ... ...

...giving something new:

... ... ... ... ... ... ... ... =

Bastard spiders!

... ... ... ... := ... ... ... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 7 / 40

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SLIDE 9

Bastard spiders fuse together: ... ...

...

= ... ... ... ... They also absorb other kinds of spider: = ...

...

... ... ... ... ... ... ... ...

...

= ... ... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 8 / 40

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SLIDE 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

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SLIDE 11

Useful bastards (cont’d)

  • Non-demolition measurement:

= = = =

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 10 / 40

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SLIDE 12

Multicoloured spiders

... ... =

i i i i

  • i

... ... ... ... =

i i i i

  • i

... ... ... ...

...

= ... ... ... ... ... ...

...

= ... ... ... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 11 / 40

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SLIDE 13

Complementary bases

1 1

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 12 / 40

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SLIDE 14

Complementarity

  • Theorem. Two bases are complementary (a.k.a. mutually unbiased):

∀i, j :

i j

=

1 D

if and only if: =

1 D

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 13 / 40

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SLIDE 15

Complementarity

  • Theorem. Two bases are complementary (a.k.a. mutually unbiased):

∀i, j :

i j

=

1 D

if and only if: =

1 D

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 14 / 40

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SLIDE 16

Complementarity

=

1 D

(encode in ) THEN (measure in ) = (no data flow)

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SLIDE 17

Stern-Gerlach

S N

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 16 / 40

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SLIDE 18

Stern-Gerlach

ւ ց

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 17 / 40

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SLIDE 19

Stern-Gerlach

:= :=

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 18 / 40

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SLIDE 20

Stern-Gerlach

:= :=

S N

Z-measurement

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SLIDE 21

Stern-Gerlach

:= :=

S N

Z-measurement

S N

X-measurement

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SLIDE 22

Stern-Gerlach

N S

S N

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 19 / 40

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SLIDE 23

Stern-Gerlach

N S

S N

Z-measurement X-measurement

=

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 19 / 40

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SLIDE 24

Stern-Gerlach

N S

S N

S N

blocked!

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 20 / 40

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SLIDE 25

Stern-Gerlach

N S

S N

S N

blocked!

=

X-measurement 1st Z-measurement 2nd Z-measurement

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 20 / 40

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SLIDE 26

Teleportation with spiders

ρ Aleks Bob

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 21 / 40

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SLIDE 27

Teleportation with spiders

= = = = =

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 31

Universality Q: Can we write any linear map as a diagram of:

... ...

and

... ...

?

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

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SLIDE 32

Universality Q: Can we write any linear map as a diagram of:

... ...

and

... ...

? A: Clearly not! e.g.

1 i

  • Kissinger & Coecke

Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

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SLIDE 33

Universality Q: Can we write any linear map as a diagram of:

... ...

and

... ...

? A: Clearly not! e.g.

1 i

  • Q2: How much more do we need?

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 24 / 40

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SLIDE 34

Phases

1 1

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SLIDE 35

Phases

1

α

α

1 D

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SLIDE 36

Phase spiders

β α

=

α + β

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SLIDE 37

Phase spiders

β α

=

α + β α

... ... :=

α

... ...

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 27 / 40

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SLIDE 38

Phase spiders

β α

=

α + β α

... ... :=

α

... ... ... ...

α

... =

β

... ... ... ...

α+β

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 27 / 40

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SLIDE 39

Universality

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SLIDE 40

Universality

  • Theorem. Any linear map with 2D inputs and outputs can be expressed

as a diagram of: ...

α

... and ...

α

...

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

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SLIDE 41

Universality

  • Theorem. Any linear map with 2D inputs and outputs can be expressed

as a diagram of: ...

α

... and ...

α

...

  • Corollary. Any quantum map from qubits to qubits can be expressed as

a diagram of: ...

α

... , ...

α

... , and

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

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SLIDE 42

Universality

  • Proof. (idea)

Any single-qubit unitary:

  • U

=

β γ α

+ CNOT: ...gives any unitary, which gives any state:

  • ψ

... =

  • U

... ... ...which gives any map by process-state duality.

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 29 / 40

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SLIDE 43

Completeness

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

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SLIDE 44

Completeness

  • Q. Can we find a complete set of equations to describe

the behaviour of: ...

α

... and ...

α

... ?

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

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SLIDE 45

Completeness

  • Q. Can we find a complete set of equations to describe

the behaviour of: ...

α

... and ...

α

... ?

  • A. Yes! The ZX-calculus.

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

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SLIDE 46

ZX-calculus

α

... =

β

... ... ... ...

α+β

... ... ...

β

... ... ...

α+β

... ...

α

= ... = ... ... .... .... =

  • π

2 π 2 π 2

  • π

2 π 2

  • π

2

  • π

2 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 31 / 40

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SLIDE 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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SLIDE 48

ZX-calculus

  • Theorem. The ZX-calculus is complete for Clifford diagrams, i.e.

diagrams where α ∈ {0, π, ± π

2 }.

  • Theorem. (Hot off the press! 1,2) A bigger version of the ZX-calculus is

complete for all diagrams.

1Jeandel, Perdrix, Vilmart. 31 May, 2017. arXiv:1705.11151 2Wang & Ng. 29 June, 2017. arXiv:1706.09877 Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 32 / 40

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SLIDE 49

Application: Quantum Computing

The quantum circuit model:

π 4

prepare a state quantum gates measure some outputs

  • π

4

  • π

4 π 4

  • π

2 π 4

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 33 / 40

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SLIDE 50

Quantum algorithms

f

N bits bit

Classical computation

⇒

  • f

:=

  • Uf

Quantum process

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SLIDE 51

Quantum algorithms

f

N bits bit

Classical computation

⇒

  • f

:=

  • Uf

Quantum process

⇒ Deutsch-Jozsa, Bernstein-Vazirani, quantum seach, and hidden subgroup (e.g. factoring) algorithms.

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 34 / 40

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SLIDE 52

Automation

Quantomatic:

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SLIDE 53

Spekkens’ toy model

:: ∗ → {(0, 0)} :: ∗ → {(0, 1)} :: ∗ → {(1, 0)} :: ∗ → {(1, 1)}

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 36 / 40

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SLIDE 54

Spekkens’ toy model

Z :: (a, b) → (a, b ⊕ 1) Z :: (a, b) → (a, a ⊕ b) H :: (a, b) → (b, a) CZ :: ((a, b), (c, d)) → ((a, b ⊕ c), (c, a ⊕ d)) MEASURE(a, b) := reveal a, randomize b

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 37 / 40

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SLIDE 55

Spekkens’ toy model

a , b...

=

c , d

... ... ... ...

a+c , b+d

...

c , d

... ... ...

a , b

= ...

a+c , b+d

... ... ... ... = ... ... .... .... =

a, a a, a a, a b, b b, b b, b b, b Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 38 / 40

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SLIDE 56

GHZ/Mermin non-locality

quantum theory any local theory

π 2 π 2 π 2 π 2 π 2 π 2

π π π π

= =

yA

i

yC

i

zA

i

yB

i

zC

i

zB

i

yA

i

yB

i

yC

i

zC

i

zB

i

zA

i

= =

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

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SLIDE 57

Thanks!

...

Coecke, Abramsky, Backens, Duncan, Edwards, Gogioso, Hadzihasanovic, Heunen, Lal, Dixon, Merry, Pavlovic, Perdrix, Quick, Selinger, Zamdzhiev, ...

Special thanks ⇒

Jamie Vicary and David Reutter - qubit.zone simulators

www.cambridge.org/pqp quantomatic.github.io

Kissinger & Coecke Picturing Quantum Processes ESSLLI Toulouse 2017 40 / 40