david reutter department of computer science university of ... · higher algebra in quantum...

Higher algebra in quantum information theory

David Reutter

Department of Computer ScienceUniversity of Oxford

March 9, 2018

David Reutter Higher algebra in quantum information March 9, 2018 1 / 21

What is this talk about?

Part I: Shaded tensor networks & biunitaries

I shaded tensor networksI ‘biunitary’ tensors in themI composing these tensors

Part II: Untangling quantum circuitsI a shaded tangle language for quantum circuitsI biunitaries and error correction

Based on joint work with Jamie Vicary:

Biunitary constructions in quantum information

Shaded tangles for the design and verification of quantum programs



Part I: Shaded tensor networks & biunitariesI shaded tensor networks

I ‘biunitary’ tensors in themI composing these tensors







Part I: Shaded tensor networks & biunitariesI shaded tensor networksI ‘biunitary’ tensors in them

I composing these tensors







Part I: Shaded tensor networks & biunitariesI shaded tensor networksI ‘biunitary’ tensors in themI composing these tensors








Part II: Untangling quantum circuits

I a shaded tangle language for quantum circuitsI biunitaries and error correction







Part II: Untangling quantum circuitsI a shaded tangle language for quantum circuits

I biunitaries and error correction





Part 1Shaded tensor networks &

biunitaries


Quantum structures

Let’s start with a very concrete problem.

Hadamard matrices H unitary error bases (UEB) {Ui}1≤i≤n2

|Hi ,j |2 = 1 H†H = n1 Ui unitary Tr(U†i Uj) = nδi ,j

( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )

Important in quantum information ... but hard to construct.Only a handful of known constructions, for example:

Hadamard + Hadamard + Hadamard UEB

(Uab)c,d =1√nAa,dBb,cCc,d

Why do they work? Where do they come from? How can we find them?

An algebraic problem?


Quantum structures


Hadamard matrices H

unitary error bases (UEB) {Ui}1≤i≤n2

|Hi ,j |2 = 1 H†H = n1

Ui unitary Tr(U†i Uj) = nδi ,j

( 1 11 -1 )

( 1 00 1 ), ( 0 1

1 0 ),(0 -ii 0

), ( 1 0

0 -1 )







Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )







Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )

Important in quantum information ...

but hard to construct.Only a handful of known constructions, for example:






Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )

Important in quantum information ... but hard to construct.

Only a handful of known constructions, for example:






Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )







Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )




Why do they work?

Where do they come from? How can we find them?



Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )




Why do they work? Where do they come from?

How can we find them?



Quantum structures




( 1 11 -1 ) ( 1 0

0 1 ), ( 0 11 0 ),

(0 -ii 0

), ( 1 0

0 -1 )







A higher algebraic problem!


What is higher algebra?

Ordinary algebra lets us compose along a line:

xy2zyx3

Higher algebra lets us compose in higher dimensions:

L

M

N

ε

η


Planar algebra = 2-category theory

The language describing algebra in the plane is 2-category theory :

A A Bf−→ A B

g

f

⇑η

objects 1-morphism 2-morphism

We can compose 2-morphisms like this:

A B⇑η

⇑εA B C⇑η ⇑ε

vertical composition horizontal composition

These are pasting diagrams.The dual diagrams are the graphical calculus.




A A Bf−→ A B

g

f

⇑η



A B⇑η

⇑εA B C⇑η ⇑ε


These are pasting diagrams.

The dual diagrams are the graphical calculus.




A

f

A B η

g

f

A Bη



A Bη

ε

A B Cη ε


These are pasting diagrams.The dual diagrams are the graphical calculus.


Monoidal dagger pivotal 2-categories

We use monoidal dagger pivotal 2-categories:

Dagger pivotal 2-categories have a very flexible graphical calculus.

In a monoidal 2-category, we can layer surfaces on top of each other.

η =

η

µν

⇒ surfaces, wires and vertices in three-dimensional space


Monoidal dagger pivotal 2-categories

We use monoidal dagger pivotal 2-categories:

Dagger pivotal 2-categories have a very flexible graphical calculus.

In a monoidal 2-category, we can layer surfaces on top of each other.

η =η

µν

⇒ surfaces, wires and vertices in three-dimensional space


A model for quantum computation: 2Hilb

We work in the 2-category 2Hilb, a categorification of Hilb.

Objects are natural numbers n,m, ...

1-morphisms nH−→ m are matrices of Hilbert spaces

2-morphisms Hφ

=⇒ H ′ are matrices of linear maps H11 · · · H1n...

. . ....

Hm1 · · · Hmn

H11φ11−−→ H ′11 . . . H1n

φ1n−−→ H ′1n...

. . ....

Hm1φm1−−→ H ′m1 . . . Hmn

φmn−−→ H ′mn

This well-studied structure plays a key role in higher representation theory.






2-morphisms Hφ

=⇒ H ′ are matrices of linear maps

H11 · · · H1n...

. . ....

Hm1 · · · Hmn

H11

φ11−−→ H ′11 . . . H1nφ1n−−→ H ′1n

.... . .

...

Hm1φm1−−→ H ′m1 . . . Hmn








2-morphisms Hφ

=⇒ H ′ are matrices of linear maps H11 · · · H1n...

. . ....

Hm1 · · · Hmn

H11φ11−−→ H ′11 . . . H1n

φ1n−−→ H ′1n...

. . ....

Hm1φm1−−→ H ′m1 . . . Hmn




A direct perspective: tensor networks

indexing seti ∈ S

vector spaceV

family of vectorspaces Vi ,j

linear mapF : V −→W

family of linear mapsFi ,j : Vi ,j −→Wi ,j

F

A

C

E F

B

D

L

M

N

A (composed) linear mapE ⊗ F −→ A


A direct perspective: shaded tensor networks

indexing seti ∈ S

vector spaceV




F

A

C

E F

B

D

L

M

N

A family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai



i

indexing seti ∈ S

vector spaceV




F

j

i

A

C

E F

B

D

L

M

N




i

indexing seti ∈ S

i j

vector spaceV




F

j

i

Ai

C

Ei ,j Fj

Bi

Dj

L

M

N




i

indexing seti ∈ S

i j

vector spaceV


i j



Fi,j

j

i

Ai

C

Ei ,j Fj

Bi

Dj

L

M

N

Li

Mi,j

Nj




i

indexing seti ∈ S

i j

vector spaceV


i j



Fi,j

Ai

C

Ei ,j Fj

Bi

Dj

Li

Mi,j

Nj




i

indexing seti ∈ S

i j

vector spaceV


i j



Fi,j

A

C

E F

B

D

L

M

N

A family of linear maps, indexed by i

and j

Ei ,j ⊗ Fj −→ Ai


Biunitarity

A biunitary is a 2-morphism that is

(vertically) unitary:

U

U†

=U†

U

=

horizontally unitary:

U

U† = λ

U† U = λ

These look just like the second Reidemeister move.


Biunitarity

A biunitary is a 2-morphism that is

(vertically) unitary:

= =

horizontally unitary:

= λ = λ

These look just like the second Reidemeister move.


Quantum structures are biunitaries in 2Hilb

Result 1: Hadamards and UEBs are biunitaries of the following type:

H U

Hadamard UEB

Result 2: We can compose biunitaries diagonally:

U

V


Composing quantum structures

H U H

Had UEB Had⇤

1

A

B

C

(Uab)c,d =1√nAa,dBb,cCc,d X



H U H

Had UEB Had⇤

1

Had

A

B

C




H U H

Had UEB Had⇤

1

Had + Had

A

B

C




H U H

Had UEB Had⇤

1

Had + Had + Had

A

B

C




H U H

Had UEB Had⇤

1

Had + Had + Had UEB

A

B

C




H U H

Had UEB Had⇤

1

Had + Had + Had UEB

A

B

C


X



H U H

Had UEB Had⇤

1

Had + Had + Had UEB

A

B

C



Composing biunitaries

H U HQ

Had UEB Had⇤ QLS [4]

1

P

H

Q

1

2

3

4

6

5

7 8

1011

9

V

W

Q

Uabc,de,fg=Hb,ca,egP

c,ge,b,f Qc,g,d

X

Uabc,def ,gh:=∑

r Vb,ca,rf ,gQ

cb,r,dWrc,e,h

X

Q

VH

P

P

CK

D

Q

H

A

B

Uabc,de,fg=∑

r Hb,ca,r Pc,r,dQr,b,f Vr,e,g

X

Uabcd,ef ,gh=1n

∑r,s Af ,hBs,f Cr,hDs,rHd

a,sKcb,rQd,s,ePr,c,g

X


Composing biunitaries

H U HQ

Had UEB Had⇤ QLS [4]

1

P

H

Q

1

2

3

4

6

5

7 8

1011

9

V

W

Q

Uabc,de,fg=Hb,ca,egP

c,ge,b,f Qc,g,dX Uabc,def ,gh:=

∑r V

b,ca,rf ,gQ

cb,r,dWrc,e,hX

Q

VH

P

P

CK

D

Q

H

A

B

Uabc,de,fg=∑

r Hb,ca,r Pc,r,dQr,b,f Vr,e,gXUabcd,ef ,gh=

1n

∑r,s Af ,hBs,f Cr,hDs,rHd

a,sKcb,rQd,s,ePr,c,gX


Taking a step back

Tensor networks:see structural properties hidden in conventional matrix notation

Shaded tensor networks:see structural properties hidden in tensor network notation

⇒ harness combinatorial richness of planar geometry

But now enough of linear algebra and let’s have some fun!

Recall:

Hadamard matrix H


Taking a step back





Recall:

Hadamard matrix

H


Taking a step back





Recall:

Hadamard matrix H


Part 2Untangling quantum circuits


Basic states and gates

|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉



|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉

|i〉 7→∑

j Hij |j〉 |i〉 ⊗ |j〉 7→ Hij |i〉 ⊗ |j〉



|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉

Hadamard gate CZ gate


Creating GHZ states

How to create a GHZ state from |+〉 states?

=


Creating GHZ states


=

CZ

H

CZ

H


Creating GHZ states


|GHZ〉

=

|+〉 |+〉 |+〉

Z

H

Z

H


Quantum error correction

A k-local quantum code is an isometry Henc−→ H⊗n, s.t.

Henc−→ H⊗n

E−→ H⊗nenc†−→ H

is proportional to the identity for every k-local error E : H⊗n −→ H⊗n.

phase error

full error


Quantum error correction

A k-local quantum code is an isometry Henc−→ H⊗n, s.t.

Henc−→ H⊗n

E−→ H⊗nenc†−→ H

is proportional to the identity for every k-local error E : H⊗n −→ H⊗n.

phase error full error


The phase code

The following is a 2−local phase error code H −→ H⊗3:

RII∼a

b

RII∼ ∼

RII∼ ∼

New construction of a phase code from unitary error bases.


Future work: The 5-qubit code

A 2−local full error correcting code H −→ H⊗5:


Future work: The 5-qubit code

Caveat: We cannot yet handle two non-adjacent errors.

Thanks for listening!


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