Simplification by Rotation for Frobenius/Hopf algebras Aleks - - PowerPoint PPT Presentation

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Simplification by Rotation for Frobenius/Hopf algebras Aleks Kissinger September 9, 2017 The goal Simplification for special commutative Frobenius algebras: = = = = = = = = = = = The goal Simplification for commutative Hopf

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Simplification by Rotation for Frobenius/Hopf algebras

Aleks Kissinger September 9, 2017

SLIDE 2

The goal

Simplification for special commutative Frobenius algebras: = = = = = = = = = = =

SLIDE 3

The goal

Simplification for commutative Hopf algebras: = = = = = = = = = = = =

SLIDE 4

The goal

Simplification for the system IB:

Frobenius Frobenius Hopf Hopf

SLIDE 5

The goal

Simplification for the system IB:

Frobenius Frobenius Hopf Hopf

:= = := =

SLIDE 6

The goal

Simplification for the system IB:

Frobenius Frobenius Hopf Hopf

:= = := = (a.k.a. the phase-free fragment of the ZX-calculus)

SLIDE 7

The (first) problem

(Biased) AC rules are not terminating:

= =

SLIDE 8

The (first) problem

(Biased) AC rules are not terminating:

= =

Solution: use unbiased simplifications:

⇒ ⇐

SLIDE 9

The (first) problem

(Biased) AC rules are not terminating:

= =

Solution: use unbiased simplifications:

⇒ ⇐

⇒ need infinitely many rules, or rule schemas

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!-boxes: simple diagram schemas

⇒ ... = · · · , , , ,

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!-boxes: simple diagram rule schemas

... = ... ... ⇒ =

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!-boxes

=

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!-boxes

= = ⇒ =

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Unbiased Frobenius algebras

... = ... ... ... ... ... ⇒ =

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Unbiased bialgebras

... ... = ... ... ... ... ... ... ⇒ =

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To quanto!

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Interacting bialgebras are linear relations IB ∼

= LinRelZ2

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Interacting bialgebras are linear relations IB ∼

= LinRelZ2

LinRelZ2 has:
objects: N
morphisms: R : m → n is a subspace R ⊆ Zm

2 × Zn 2

tensor is ⊕, composition is relation-style

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Interacting bialgebras are linear relations IB ∼

= LinRelZ2

LinRelZ2 has:
objects: N
morphisms: R : m → n is a subspace R ⊆ Zm

2 × Zn 2

tensor is ⊕, composition is relation-style
Pseudo-normal forms can be interpreted as:
white spiders := place-holders
grey spiders := vectors spanning the subspace

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Lets see how this works...

Subspaces can be represented as:

↔

     1 1 1       ,       1 1      

The 1’s indicate where edges appear for each vector.

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Lets see how this works...

Subspaces can be represented as:

↔

     1 1 1       ,       1 1      

The 1’s indicate where edges appear for each vector.

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Lets see how this works...

Subspaces can be represented as:

↔

     1 1 1       ,       1 1      

The 1’s indicate where edges appear for each vector.

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Lets see how this works...

Not unique! We can always add or remove a vector that is the sum of

two other spanning vectors and get the same space: ↔

     1 1 1       ,       1 1       ,       1 1 1      

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Addition is a !-box rule

‘Addition’ operation can be written as a !-box rule:

=

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Addition is a !-box rule

‘Addition’ operation can be written as a !-box rule:

=

We can also apply this forward then backward to get a ‘rotation’ rule:

=

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Addition is a !-box rule

‘Addition’ operation can be written as a !-box rule:

=

We can also apply this forward then backward to get a ‘rotation’ rule:

=

Note this rule decreases the arity of the white dot on the left by 1.

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Thanks!

Joint work with Lucas Dixon, Alex Merry, Ross Duncan, Vladimir

Zamdzhiev, David Quick, Hector Miller-Bakewell and others

See: quantomatic.github.io