[PPT] - Optimisation and verification of quantum circuits with string PowerPoint Presentation

SLIDE 1

Optimisation and verification of quantum circuits with string diagrams

Aleks Kissinger

STRING 3; Birmingham 2019

Aleks Kissinger STRING 3, 2019 1 / 43

SLIDE 2

Quantum software

1. := the code the runs on a quantum computer

factoring search

quantum simulation hidden subgroup linear systems & codes pattern matching collision-finding network flows NLP ...

2. := the code that makes that code (better)
quantum programming languages
compilers
optimisation
verification

Aleks Kissinger STRING 3, 2019 2 / 43

SLIDE 3

Problem: no quantum computers

Rigetti Aspen 16Q IBM Q 20 Tokyo Google Bristlecone

(+ Oxford, Vienna, Delft, Sussex, Grenoble...)

Aleks Kissinger STRING 3, 2019 3 / 43

SLIDE 4

Problem: limited quantum computers

Noisy intermediate-scale quantum (NISQ) devices

short coherence times (e.g. µs-scale)
small memory (8-80 qubits)
noisy primitive operations (gates & measurements)
limited connectivity:
...

⇒ small advances in software give big gains on NISQ hardware!

Aleks Kissinger STRING 3, 2019 4 / 43

SLIDE 5

Quantum circuits

:= the ‘assembly language’ of quantum computation, e.g.

INIT 5 CNOT 1 0 H 2 Z 3 H 0 H 1 CNOT 4 2 ...

⇔

⊕

H H

⊕ ⊕

H Z X Z

⊕ ⊕

S T

⊕ ⊕

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SLIDE 6

Built out of fixed set of complex unitary matrices, or gates, e.g.

⊕ :=     1 1 1 1    

H

:=

1 √ 2

1 1 1 −1

S

:= 1 i

...combined by ⊗ and ◦.
Universal sets can gates can be used to construct any 2n × 2n unitary

matrix

Gates describe physical evolutions of a q. state, which take time and

introduce noise

Lower gate count / gate depth is good!

Aleks Kissinger STRING 3, 2019 6 / 43

SLIDE 7

⊕

H H

⊕ ⊕

H Z X Z

⊕

S T

⊕ ⊕

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SLIDE 8

⊕

H H

⊕ ⊕

H Z X Z

⊕

S T

⊕ ⊕

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SLIDE 9

H H

⊕ ⊕

H Z X Z S T

⊕ ⊕

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SLIDE 10

H H

⊕ ⊕

H Z X Z S T

⊕ ⊕

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SLIDE 11

⊕ ⊕ ⊕ = ⊕

H H

= ⊕

H H

⊕ ⊕ = ⊕ ⊕ ⊕

Z

=

Z H

=

S H S H H

. . .

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SLIDE 12

* Selinger 2015

Aleks Kissinger STRING 3, 2019 8 / 43

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* Selinger 2015

Aleks Kissinger STRING 3, 2019 8 / 43

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* Amy, Chen, & Ross 2018

Aleks Kissinger STRING 3, 2019 8 / 43

SLIDE 15

* Nam et al 2018

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SLIDE 16

⊕

H H

⊕ ⊕

H Z X Z

⊕ ⊕

S T

⊕ ⊕

π π π

π 2 π 4 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2

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SLIDE 17

⊕

H H

⊕ ⊕

H Z X Z

⊕ ⊕

S T

⊕ ⊕

π π π

π 2 π 4 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2

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SLIDE 18

π π π

π 2 π 4 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2

Aleks Kissinger STRING 3, 2019 9 / 43

SLIDE 19

π π π

π 2 π 4 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2

ZX-diagram

Aleks Kissinger STRING 3, 2019 9 / 43

SLIDE 20

ZX-diagrams

...are made of spiders:

α

... ...

=

      1 ... ... eiα      

α

... ... := ...

α

...

:=

1 √ 2 1 1 1 −1

Aleks Kissinger

STRING 3, 2019 10 / 43

SLIDE 21

α

π 2 3π 2

β π

Aleks Kissinger STRING 3, 2019 11 / 43

SLIDE 22

α

π 2 3π 2

β π

Aleks Kissinger STRING 3, 2019 11 / 43

SLIDE 23

α

π 2 3π 2

β π

Aleks Kissinger STRING 3, 2019 11 / 43

SLIDE 24

1

ZX calculus

these 8 rules = ⇒ everything before

1Vilmart 2018. arXiv:1812.09114 Aleks Kissinger STRING 3, 2019 12 / 43

SLIDE 25

⊕

H H

⊕ ⊕

H Z X Z

⊕ ⊕

S T

⊕ ⊕