Post on 30-Apr-2020
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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI shaded tensor networks
I ‘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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI shaded tensor networksI ‘biunitary’ tensors in them
I 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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI 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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
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
Based on joint work with Jamie Vicary:
Biunitary constructions in quantum information
Shaded tangles for the design and verification of quantum programs
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI shaded tensor networksI ‘biunitary’ tensors in themI composing these tensors
Part II: Untangling quantum circuitsI a shaded tangle language for quantum circuits
I 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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI 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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
What is this talk about?
Part I: Shaded tensor networks & biunitariesI 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
David Reutter Higher algebra in quantum information March 9, 2018 2 / 21
Part 1Shaded tensor networks &
biunitaries
David Reutter Higher algebra in quantum information March 9, 2018 3 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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 00 1 ), ( 0 1
1 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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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?
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
A higher algebraic problem!
David Reutter Higher algebra in quantum information March 9, 2018 4 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 5 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 5 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 6 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 6 / 21
Planar algebra = 2-category theory
The language describing algebra in the plane is 2-category theory :
A
f
A B η
g
f
A Bη
objects 1-morphism 2-morphism
We can compose 2-morphisms like this:
A Bη
ε
A B Cη ε
vertical composition horizontal composition
These are pasting diagrams.The dual diagrams are the graphical calculus.
David Reutter Higher algebra in quantum information March 9, 2018 6 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 7 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 7 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 7 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 7 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 8 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 8 / 21
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
H11
φ11−−→ H ′11 . . . H1nφ1n−−→ H ′1n
.... . .
...
Hm1φm1−−→ H ′m1 . . . Hmn
φmn−−→ H ′mn
This well-studied structure plays a key role in higher representation theory.
David Reutter Higher algebra in quantum information March 9, 2018 8 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 8 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 8 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded 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 family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded tensor networks
i
indexing seti ∈ S
vector spaceV
family of vectorspaces Vi ,j
linear mapF : V −→W
family of linear mapsFi ,j : Vi ,j −→Wi ,j
F
j
i
A
C
E F
B
D
L
M
N
A family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded tensor networks
i
indexing seti ∈ S
i j
vector spaceV
family of vectorspaces Vi ,j
linear mapF : V −→W
family of linear mapsFi ,j : Vi ,j −→Wi ,j
F
j
i
Ai
C
Ei ,j Fj
Bi
Dj
L
M
N
A family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded tensor networks
i
indexing seti ∈ S
i j
vector spaceV
family of vectorspaces Vi ,j
i j
linear mapF : V −→W
family of linear mapsFi ,j : Vi ,j −→Wi ,j
Fi,j
j
i
Ai
C
Ei ,j Fj
Bi
Dj
L
M
N
Li
Mi,j
Nj
A family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded tensor networks
i
indexing seti ∈ S
i j
vector spaceV
family of vectorspaces Vi ,j
i j
linear mapF : V −→W
family of linear mapsFi ,j : Vi ,j −→Wi ,j
Fi,j
Ai
C
Ei ,j Fj
Bi
Dj
Li
Mi,j
Nj
A family of linear maps, indexed by i and jEi ,j ⊗ Fj −→ Ai
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
A direct perspective: shaded tensor networks
i
indexing seti ∈ S
i j
vector spaceV
family of vectorspaces Vi ,j
i j
linear mapF : V −→W
family of linear mapsFi ,j : Vi ,j −→Wi ,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
David Reutter Higher algebra in quantum information March 9, 2018 9 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 10 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 10 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 10 / 21
Biunitarity
A biunitary is a 2-morphism that is
(vertically) unitary:
= =
horizontally unitary:
= λ = λ
These look just like the second Reidemeister move.
David Reutter Higher algebra in quantum information March 9, 2018 10 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 11 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 11 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 11 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had + Had
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had + Had + Had
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had + Had + Had UEB
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had + Had + Had UEB
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d
X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
Composing quantum structures
H U H
Had UEB Had⇤
1
Had + Had + Had UEB
A
B
C
(Uab)c,d =1√nAa,dBb,cCc,d X
David Reutter Higher algebra in quantum information March 9, 2018 12 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 13 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 13 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 14 / 21
Part 2Untangling quantum circuits
David Reutter Higher algebra in quantum information March 9, 2018 15 / 21
Basic states and gates
|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉
David Reutter Higher algebra in quantum information March 9, 2018 16 / 21
Basic states and gates
|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉
|i〉 7→∑
j Hij |j〉 |i〉 ⊗ |j〉 7→ Hij |i〉 ⊗ |j〉
David Reutter Higher algebra in quantum information March 9, 2018 16 / 21
Basic states and gates
|+〉 = |0〉+ |1〉 |Bell〉 = |00〉+ |11〉 |GHZ〉 = |000〉+ |111〉
Hadamard gate CZ gate
David Reutter Higher algebra in quantum information March 9, 2018 16 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
=
CZ
H
CZ
H
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
Creating GHZ states
How to create a GHZ state from |+〉 states?
|GHZ〉
=
|+〉 |+〉 |+〉
Z
H
Z
H
David Reutter Higher algebra in quantum information March 9, 2018 17 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 18 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 18 / 21
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
David Reutter Higher algebra in quantum information March 9, 2018 18 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
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.
David Reutter Higher algebra in quantum information March 9, 2018 19 / 21
Future work: The 5-qubit code
A 2−local full error correcting code H −→ H⊗5:
David Reutter Higher algebra in quantum information March 9, 2018 20 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21
Future work: The 5-qubit code
Caveat: We cannot yet handle two non-adjacent errors.
Thanks for listening!
David Reutter Higher algebra in quantum information March 9, 2018 21 / 21