SZX-calculus, Scalable Graphical Quantum Reasoning A GTIQ2019 talk - - PowerPoint PPT Presentation

▶
szx calculus scalable graphical quantum reasoning
SMART_READER_LITE
LIVE PREVIEW

SZX-calculus, Scalable Graphical Quantum Reasoning A GTIQ2019 talk - - PowerPoint PPT Presentation

Mar 21, 2024 42 likes •794 views

t rtt rst rr rs rst r Prr

slide-1
SLIDE 1

❚✐t♦✉❛♥ ❈❛r❡tt❡✱ ▲❖❘■❆✱ ❯♥✐✈❡rs✐té ❞❡ ▲♦rr❛✐♥❡ ❉♦♠✐♥✐❝ ❍♦rs♠❛♥✱ ❯♥✐✈❡rs✐té ❞❡ ❣r❡♥♦❜❧❡ ❙✐♠♦♥ P❡r❞r✐①✱ ▲❖❘■❆✱ ❈◆❘❙ presents

SZX-calculus, Scalable Graphical Quantum Reasoning

A GTIQ2019 talk

slide-2
SLIDE 2

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-3
SLIDE 3

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-4
SLIDE 4

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-5
SLIDE 5

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-6
SLIDE 6

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-7
SLIDE 7

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-8
SLIDE 8

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-9
SLIDE 9

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-10
SLIDE 10

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g f g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-11
SLIDE 11

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g f g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ σ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-12
SLIDE 12

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g f g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ σ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-13
SLIDE 13

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g f g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ σ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t I ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-14
SLIDE 14

Picturing processes

◆❛♠❡ ❙②♠❜♦❧ ❉✐❛❣r❛♠ ❍✐❧❜❡rt s♣❛❝❡ ❚r❛♥s❢♦r♠❛t✐♦♥ f f ▼❛tr✐① ❈♦♠♣♦s✐t✐♦♥ g ◦ f f g ♠❛tr✐① ♣r♦❞✉❝t ■❞❡♥t✐t② id ■❞❡♥t✐t② ♠❛tr✐① ❚❡♥s♦r f ⊗ g f g ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t ❙✇❛♣ σ ❇❛s✐s ♣❡r♠✉t❛t✐♦♥ ❯♥✐t I ❙❝❛❧❛rs

Chapitre 1: Graphical Reasoning ✶ ✵✵✵✶ ⑤ ✶✶✵✶

slide-15
SLIDE 15

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

✳ ✳ ✳ ✳ ✳ ✳

✷ ✷

❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-16
SLIDE 16

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

✳ ✳ ✳ ✳ ✳ ✳

✷ ✷

❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-17
SLIDE 17

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

✳ ✳ ✳ ✳ ✳ ✳

✷ ✷

❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-18
SLIDE 18

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

✳ ✳ ✳ ✳ ✳ ✳

✷ ✷

❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-19
SLIDE 19

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

✳ ✳ ✳ ✳ ✳ ✳

✷ ✷

❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-20
SLIDE 20

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

D

✳ ✳ ✳ ✳ ✳ ✳

∈ M✷m,✷n(C) ❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-21
SLIDE 21

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

D

✳ ✳ ✳ ✳ ✳ ✳

∈ M✷m,✷n(C) ❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-22
SLIDE 22

Diagrammatic languages and categories

❉✐❛❣r❛♠s ❛r❡ ♠♦♥♦✐❞❛❧ s②♠♠❡tr✐❝ ❝❛t❡❣♦r✐❡s✳ ❚❤❡② ❛r❡ ♣r♦♣s✿ ❖♥❡ ✇✐r❡ t②♣❡ s♣❛♥s ❛❧❧ t❤❡ ♦t❤❡rs✳ ❋♦r ✉s t❤✐s ✐s t❤❡ ◗✉❜✐t t②♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ❙②♥t❛①✿ ❞✐❛❣r❛♠s ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ❜✉✐❧t ❢r♦♠ ❛ s❡t ♦❢ ❣❡♥❡r❛t♦rs✳ ❙❡♠❛♥t✐❝✿

D

✳ ✳ ✳ ✳ ✳ ✳

∈ M✷m,✷n(C) ❍✐❧❜❡rt s♣❛❝❡ ✐s ❛ ❜✐❣ ♣❧❛❝❡ ❈❛r❧t♦♥ ❈❛✈❡s ❇✉t ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ ♣❛rt ♦❢ ✐t✦

Chapitre 1: Graphical Reasoning ✷ ✵✵✶✵ ⑤ ✶✶✵✶

slide-23
SLIDE 23

ZX-calculus

  • π
  • ✶

✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-24
SLIDE 24

ZX-calculus

  • π
  • ✶

✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-25
SLIDE 25

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

  • ✶

✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-26
SLIDE 26

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-27
SLIDE 27

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-28
SLIDE 28

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵

  • ⊗ I
  • ✶

✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-29
SLIDE 29

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵

  • ⊗ I

  I ⊗     ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶         ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-30
SLIDE 30

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵

  • ⊗ I

  I ⊗     ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶         ◦ [I ⊗ H ⊗ I] ✶ ✵ ✵ ✶

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-31
SLIDE 31

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵

  • ⊗ I

  I ⊗     ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶         ◦ [I ⊗ H ⊗ I] ◦         ✶ ✵ ✵ ✶     ⊗ I    

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-32
SLIDE 32

ZX-calculus

  • π
  • H ⊗
  • ✶

✵ ✵ ✶

       ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶     ⊗ I     ◦

  • H ⊗

✶ ✵ ✵ eiπ

  • ✶

✵ ✵ ✶ ✵ ✶ ✶ ✵

  • ⊗ I

  I ⊗     ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶         ◦ [I ⊗ H ⊗ I] ◦         ✶ ✵ ✵ ✶     ⊗ I     = ?

Chapitre 1: Graphical Reasoning ✸ ✵✵✶✶ ⑤ ✶✶✵✶

slide-33
SLIDE 33

Rewriting rules

π

→ ✵ ✶ ✶ ✵

Chapitre 1: Graphical Reasoning ✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-34
SLIDE 34

Rewriting rules

π

→

π

✵ ✶ ✶ ✵

Chapitre 1: Graphical Reasoning ✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-35
SLIDE 35

Rewriting rules

π

→

π π

→

π

✵ ✶ ✶ ✵

Chapitre 1: Graphical Reasoning ✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-36
SLIDE 36

Rewriting rules

π

→

π π

→

π π

→

π

✵ ✶ ✶ ✵

Chapitre 1: Graphical Reasoning ✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-37
SLIDE 37

Rewriting rules

π

→

π π

→

π π

→

π π

→

π

✵ ✶ ✶ ✵

Chapitre 1: Graphical Reasoning ✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-38
SLIDE 38

Rewriting rules

π

→

π π

→

π π

→

π π

→

π

  • π
  • =

✵ ✶ ✶ ✵

  • Chapitre 1: Graphical Reasoning

✹ ✵✶✵✵ ⑤ ✶✶✵✶

slide-39
SLIDE 39

Check point

π

❚❤❡ ❩❳✲❝❛❧❝✉❧✉s ♣r♦✈✐❞❡s✿ ■♥t✉✐t✐✈❡ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s✳ ❯♥✐✈❡rs❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ q✉❜✐ts✳ ❬❋❲✶✽❪ ❛♥❞ ❬❏P❱✶✽❪✳ ❈♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t ✐♥t❡r❡st✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❲❡ ❝❛♥ ❜❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❛❝t ✇❤✐❧❡ s❝❛❧✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡r ♦❢ q✉❜✐ts✦

Chapitre 1: Graphical Reasoning ✺ ✵✶✵✶ ⑤ ✶✶✵✶

slide-40
SLIDE 40

Check point

π

❚❤❡ ❩❳✲❝❛❧❝✉❧✉s ♣r♦✈✐❞❡s✿ ■♥t✉✐t✐✈❡ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s✳ ❯♥✐✈❡rs❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ q✉❜✐ts✳ ❬❋❲✶✽❪ ❛♥❞ ❬❏P❱✶✽❪✳ ❈♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t ✐♥t❡r❡st✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❲❡ ❝❛♥ ❜❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❛❝t ✇❤✐❧❡ s❝❛❧✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡r ♦❢ q✉❜✐ts✦

Chapitre 1: Graphical Reasoning ✺ ✵✶✵✶ ⑤ ✶✶✵✶

slide-41
SLIDE 41

Check point

π

❚❤❡ ❩❳✲❝❛❧❝✉❧✉s ♣r♦✈✐❞❡s✿ ■♥t✉✐t✐✈❡ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s✳ ❯♥✐✈❡rs❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ q✉❜✐ts✳ ❬❋❲✶✽❪ ❛♥❞ ❬❏P❱✶✽❪✳ ❈♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t ✐♥t❡r❡st✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❲❡ ❝❛♥ ❜❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❛❝t ✇❤✐❧❡ s❝❛❧✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡r ♦❢ q✉❜✐ts✦

Chapitre 1: Graphical Reasoning ✺ ✵✶✵✶ ⑤ ✶✶✵✶

slide-42
SLIDE 42

Check point

π

❚❤❡ ❩❳✲❝❛❧❝✉❧✉s ♣r♦✈✐❞❡s✿ ■♥t✉✐t✐✈❡ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s✳ ❯♥✐✈❡rs❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ q✉❜✐ts✳ ❬❋❲✶✽❪ ❛♥❞ ❬❏P❱✶✽❪✳ ❈♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t ✐♥t❡r❡st✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❲❡ ❝❛♥ ❜❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❛❝t ✇❤✐❧❡ s❝❛❧✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡r ♦❢ q✉❜✐ts✦

Chapitre 1: Graphical Reasoning ✺ ✵✶✵✶ ⑤ ✶✶✵✶

slide-43
SLIDE 43

Check point

π

❚❤❡ ❩❳✲❝❛❧❝✉❧✉s ♣r♦✈✐❞❡s✿ ■♥t✉✐t✐✈❡ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s✳ ❯♥✐✈❡rs❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ q✉❜✐ts✳ ❬❋❲✶✽❪ ❛♥❞ ❬❏P❱✶✽❪✳ ❈♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t ✐♥t❡r❡st✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❲❡ ❝❛♥ ❜❡ ❡✈❡♥ ♠♦r❡ ❝♦♠♣❛❝t ✇❤✐❧❡ s❝❛❧✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡r ♦❢ q✉❜✐ts✦

Chapitre 1: Graphical Reasoning ✺ ✵✶✵✶ ⑤ ✶✶✵✶

slide-44
SLIDE 44

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-45
SLIDE 45

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-46
SLIDE 46

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-47
SLIDE 47

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-48
SLIDE 48

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-49
SLIDE 49

Yes... but why?

❙❝❛❧❛❜❧❡ ❞✐❛❣r❛♠❛t✐❝ r❡❛s♦♥✐♥❣✳ ◗✉❛♥t✉♠ ❛❧❣♦r✐t❤♠s✳ ◗✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♦♥✳ ◗✉❛♥t✉♠ s✐♠✉❧❛t✐♦♥✳ ▲❛r❣❡ s❝❛❧❡ q✉❛♥t✉♠ ❝♦♠♣✐❧❡r❄

Chapitre 2: Scaling ✻ ✵✶✶✵ ⑤ ✶✶✵✶

slide-50
SLIDE 50

SZX-calculus I, Divide and Gather

✷ = ✶ + ✶ ✷ ✷ ✶ ✶ ✶ ✶

✶ ✶ ✶ ✶

Chapitre 2: Scaling ✼ ✵✶✶✶ ⑤ ✶✶✵✶

slide-51
SLIDE 51

SZX-calculus I, Divide and Gather

✷ = ✶ + ✶ ✷ ✷ = ✶ ✶ ✶ ✶

✶ ✶ ✶ ✶

Chapitre 2: Scaling ✼ ✵✶✶✶ ⑤ ✶✶✵✶

slide-52
SLIDE 52

SZX-calculus I, Divide and Gather

✷ = ✶ + ✶ ✷ ✷ = ✶ ✶ ✶ ✶

n + ✶ ✶ n n + ✶ ✶ n

Chapitre 2: Scaling ✼ ✵✶✶✶ ⑤ ✶✶✵✶

slide-53
SLIDE 53

SZX-calculus I, Divide and Gather

✷ = ✶ + ✶ ✷ ✷ = ✶ ✶ ✶ ✶

n + ✶ ✶ n n + ✶ ✶ n

= =

Chapitre 2: Scaling ✼ ✵✶✶✶ ⑤ ✶✶✵✶

slide-54
SLIDE 54

SZX-calculus II, The rewiring theorem

❚❤❡♦r❡♠✿

▲❡t ω ∈ W[a, b] ❛♥❞ ω′ ∈ W[c, d]✿ W ⊢ ω = ω′ ⇔ a = c and b = d

Chapitre 2: Scaling ✽ ✶✵✵✵ ⑤ ✶✶✵✶

slide-55
SLIDE 55

SZX-calculus II, The rewiring theorem

❚❤❡♦r❡♠✿

▲❡t ω ∈ W[a, b] ❛♥❞ ω′ ∈ W[c, d]✿ W ⊢ ω = ω′ ⇔ a = c and b = d a a + b b

Chapitre 2: Scaling ✽ ✶✵✵✵ ⑤ ✶✶✵✶

slide-56
SLIDE 56

SZX-calculus II, The rewiring theorem

❚❤❡♦r❡♠✿

▲❡t ω ∈ W[a, b] ❛♥❞ ω′ ∈ W[c, d]✿ W ⊢ ω = ω′ ⇔ a = c and b = d a a + b b := =

Chapitre 2: Scaling ✽ ✶✵✵✵ ⑤ ✶✶✵✶

slide-57
SLIDE 57

SZX-calculus, The big generators

= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

Chapitre 2: Scaling ✾ ✶✵✵✶ ⑤ ✶✶✵✶

slide-58
SLIDE 58

SZX-calculus, The big generators

=

α::β

✳ ✳ ✳ ✳ ✳ ✳ =

β α

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

α::β

✳ ✳ ✳ ✳ ✳ ✳ =

β α

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

Chapitre 2: Scaling ✾ ✶✵✵✶ ⑤ ✶✶✵✶

slide-59
SLIDE 59

SZX-calculus, Completeness

  • s := (idn, n, n)

✶ ✷ ✵ ✶

✶

✶

✶ ✶ ✳ ✳ ✳ ✳ ✳ ✳

✵ ✶

Chapitre 2: Scaling ✶✵ ✶✵✶✵ ⑤ ✶✶✵✶

slide-60
SLIDE 60

SZX-calculus, Completeness

  • s := (idn, n, n)
  • s := (

✶ √ ✷

n

  • x,y∈{✵,✶}n(−✶)x•y|y

x|, n, n)

✶

✶ ✶ ✳ ✳ ✳ ✳ ✳ ✳

✵ ✶

Chapitre 2: Scaling ✶✵ ✶✵✶✵ ⑤ ✶✶✵✶

slide-61
SLIDE 61

SZX-calculus, Completeness

  • s := (idn, n, n)
  • s := (

✶ √ ✷

n

  • x,y∈{✵,✶}n(−✶)x•y|y

x|, n, n)

  • s := (idn+✶, n + ✶, ✶ ⊗ n)

✳ ✳ ✳ ✳ ✳ ✳

✵ ✶

Chapitre 2: Scaling ✶✵ ✶✵✶✵ ⑤ ✶✶✵✶

slide-62
SLIDE 62

SZX-calculus, Completeness

  • s := (idn, n, n)
  • s := (

✶ √ ✷

n

  • x,y∈{✵,✶}n(−✶)x•y|y

x|, n, n)

  • s := (idn+✶, n + ✶, ✶ ⊗ n)
  • α ✳

✳ ✳ ✳ ✳ ✳

  • s := (
  • x∈{✵,✶}n eix•α|xk

xℓ|, n⊗k, n⊗ℓ)

Chapitre 2: Scaling ✶✵ ✶✵✶✵ ⑤ ✶✶✵✶

slide-63
SLIDE 63

SZX-calculus, Completeness

  • s := (idn, n, n)
  • s := (

✶ √ ✷

n

  • x,y∈{✵,✶}n(−✶)x•y|y

x|, n, n)

  • s := (idn+✶, n + ✶, ✶ ⊗ n)
  • α ✳

✳ ✳ ✳ ✳ ✳

  • s := (
  • x∈{✵,✶}n eix•α|xk

xℓ|, n⊗k, n⊗ℓ)

  • ;

 

π

  ;

  • Chapitre 2: Scaling

✶✵ ✶✵✶✵ ⑤ ✶✶✵✶

slide-64
SLIDE 64

New large scale structures, Bipartite graph matrices.

A

=

A

✳ ✳ ✳ ✳ ✳ ✳

|A|−m

✷

✵ ✶

Chapitre 3: Spicing up with matrices ✶✶ ✶✵✶✶ ⑤ ✶✶✵✶

slide-65
SLIDE 65

New large scale structures, Bipartite graph matrices.

A

=

A

✳ ✳ ✳ ✳ ✳ ✳

|A|−m

∀A ∈ Fm×n

✷

,

  • A

s = (|x → |Ax, n, m)

✵ ✶

Chapitre 3: Spicing up with matrices ✶✶ ✶✵✶✶ ⑤ ✶✶✵✶

slide-66
SLIDE 66

New large scale structures, Bipartite graph matrices.

A

=

A

✳ ✳ ✳ ✳ ✳ ✳

|A|−m

∀A ∈ Fm×n

✷

,

  • A

s = (|x → |Ax, n, m)

✵

=

✶

=

  • A

B

  • =

A B

[CD] =

C D m

Chapitre 3: Spicing up with matrices ✶✶ ✶✵✶✶ ⑤ ✶✶✵✶

slide-67
SLIDE 67

Properties of matrices

A

=

A A A

=

πv A

=

πAv A A n

=

A A m A n

=

m πu A

=

πAtu A A m

❍

=

At n

A B

m

♣

=

A+B A C

♠

=

CA

Chapitre 3: Spicing up with matrices ✶✷ ✶✶✵✵ ⑤ ✶✶✵✶

slide-68
SLIDE 68

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-69
SLIDE 69

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-70
SLIDE 70

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-71
SLIDE 71

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-72
SLIDE 72

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-73
SLIDE 73

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-74
SLIDE 74

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶

slide-75
SLIDE 75

This is the last slide of this talk

❲❡ ❦❡❡♣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❞❛t❛ ♦r❣❛♥✐s❛t✐♦♥✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s✳

  • ✐✈✐♥❣ ❛ ♣♦✇❡r❢✉❧❧ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s t♦ ✈❡r✐❢② ❛♥❞ ❞❡s✐❣♥ ❧❛r❣❡

s❝❛❧❡ q✉❛♥t✉♠ ♣r♦❝❡ss❡s✳ ❚❤❡ ❜✐❣ ✇✐r❡ ❝♦♥str✉❝t✐♦♥ ❞♦ ♥♦t r❡❧✐❡s ♦♥ q✉❛♥t✉♠✳ ■♥t❡r❡st✐♥❣ ❧❛r❣❡ s❝❛❧❡ str✉❝t✉r❡s ✐♥ ♦t❤❡r ✜❡❧❞s❄ ❚❤❡ ❡♥❞✳

Chapitre 3: Spicing up with matrices ✶✸ ✶✶✵✶ ⑤ ✶✶✵✶