Categorical Quantum Mechanicspersonal.strath.ac.uk/ross.duncan/talks/2012/catQM-pt1.pdf ·...

Ross Duncan ● Brussels 2012

Categorical Quantum Mechanics

Ross DuncanLaboratoire d’Information Quantique

Université Libre de Bruxelles

Friday 17 February 2012


Background and Motivations

"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more."

--John von Neumann, letter to G. Birkhoff, 1935



What is the structure?

Mathematically, QM is formalised over complex Hilbert spaces, an 80 year old formalism that was thought flawed by its creator John von Neumann.

Categorical QM aims to reconstruct quantum mechanics in terms of minimal algebraic structures:

• discover the mathematical heart of the theory

• suggest constraints on successor theories e.g. quantum gravity

• clarify the relationship between quantum and classical computation

• suggest new primitives for the design and analysis of quantum algorithms



Motivations



Motivations

λx.λy.λz.xz(yz)



Motivations

Hilbert space, unitary transforms, self-adjoint operators.... ?

λx.λy.λz.xz(yz)



Motivations

Hilbert space, unitary transforms, self-adjoint operators....

λx.λy.λz.xz(yz)



The need for abstraction:

D ~ 2^1764

This is an 8-bit adder



Motivation & Context

New structural axiomatisation of QM:

• Hilbert space formalism not a perfect fit for QM

• Can also study alternative quantum-like theories

• emphasis on composition

We use Monoidal Categories:

• Very general framework, including Hilbert spaces

• Well studied in literature

• Beautiful diagrammatic presentation



Motivations

1. Better Theory

• Usual formulation of QM has lots of “junk”

• Which axioms make quantum mechanics quantum?

• Are there other models which have some of these properties?

2. Better Tools

• Reason more easily about quantum processes

- formal tools!

• Synthesise quantum programs more easily

• Type systems for quantum programs?



According to wikipedia, category theory:

• “deals in an abstract way with mathematical structures and relationships between them.”

Study QM using the tools of category theory to find which structures are responsible for quantum phenomena.

• find other models of those structures

• use these structures as a translation between different quantum settings

• manipulate these structures directly for easier calculations

- diagrammatic notation!

Categorical approach



Monoidal CategoriesCategorical view point:

• don’t think about states

• do think about processes

Monoidal categories allow us to

• combine processes sequentially

• ... and in parallel

Examples:

• Sets and functions

• Sets and relations

• Finite dimensional Hilbert spaces and linear maps

• ... and many many more

− ◦ −−⊗−



What needs to be added?

†-Monoidal Categories

Hilbert Space QM Too much stuffToo little structure

Too few objectsToo few equations







Add more algebraic gadgetry








Unbiasedness(Phase groups)

Observables(copying and deleting)

Complementarity(Bialgebras)

.

.

.








Expressive enough to do real work

Unbiasedness(Phase groups)

Observables(copying and deleting)

Complementarity(Bialgebras)

.

.

.



DISCLAIMER

There are going to be a LOT of definitions in this talk.

sorry.



Monoidal Categories & Diagrams

Pessimistic Diagram Convention

PAST / HEAVEN

FUTURE / HELL



Why Diagrams?



Why Diagrams?

12

�11

�⊗

1 0 0 00 1 0 00 0 1 00 0 0 eiγ

⊗�

0 11 0

�

◦

��1 0 0 00 0 0 1

�⊗

�1 11 −1

��◦

�0 11 0

�⊗

1 0 0 00 0 1 00 1 0 00 0 0 1

◦

1 0 0 00 1 0 00 0 0 10 0 1 0

⊗�

1 00 1

�

⊗�

1 00 1

�

◦

1 0 0 00 1 0 00 0 0 10 0 1 0

◦��

cos π6

i sin π6

�⊗

�1 00 eiβ

��

⊗

1 0 0 00 1 0 00 0 1 00 0 0 eiα

= ?



Historical Examples

Circuits (classical and quantum) :



Historical Examples

Feynmann Diagrams:



Historical Examples

“Applications of Negative Dimensional Tensors”

(Penrose)



Historical Examples

Proofs-Nets in Linear Logic

(Girard, Danos-Regnier, Tortoro de Falco, many others)



Historical Examples

Proofs-Nets in Linear Logic

(Blute-Cocket-Seely-Trimble, 1996)



Historical Examples

Interaction Combinators

(Lafont, 1997)



Historical Examples

“Coherence for Compact Closed Categories”

(Kelly-Laplaza, 1980)



Historical Examples

“Coherence for Compact Closed Categories”

(Kelly-Laplaza, 1980)

?Friday 17 February 2012


Historical Examples

“Geometry of Tensor Calculus I”

(Joyal-Street, 1991)



Historical Examples

“Traced Monoidal Categories”

(Joyal-Street-Verity, 1996)



Diagrams

D

A B

E

j

C

a.k.a. Monoidal categories

j : A⊗B → C ⊗D ⊗ E



Input Systems

Diagrams

D

A B

E

j

C


j : A⊗B → C ⊗D ⊗ E



Input Systems

Diagrams

D

A B

E

j

C

Interaction, or process, or state change, or ...


j : A⊗B → C ⊗D ⊗ E



Input Systems

Diagrams

D

A B

E

j

C

Output Systems



j : A⊗B → C ⊗D ⊗ E



Input Systems

Diagrams

D

A B

E

j

C

Output Systems


“Time” a.k.a. Monoidal categories

j : A⊗B → C ⊗D ⊗ E



A hierarchy of categoriesCategories




Monoidal Categories




Monoidal Categories

Symmetric Monoidal Categories




Monoidal Categories


Compact Categories




Monoidal Categories


Compact Categories

†-Categories




Monoidal Categories


Compact Categories

†-Categories

†-Compact Categories




Monoidal Categories


Compact Categories

†-Categories

†-Compact Categories

F.D. Hilbert spacesSets and RelationsStrongly self-dual convex cones



A category consists of objects etc, and arrows between them:

f : A→ B g : B → C h : C → D

CategoriesA,B, C,

f gA

B

B

C

C

h

D

f : A→ B g : B → C h : C → D




CategoriesA,B, C,

f

g

A

B

C

C

h

D

f : A→ B g : B → C h : C → D

g ◦ f : A → C




CategoriesA,B, C,

f

g

A

B

C

h

D

h ◦ g ◦ f : A → D



idA : A→ A

Categories

idA : A→ A



Categories

idA : A→ A

B

f

f ◦ idA : A → B



Categories

idA : A→ A

B

f

idB ◦ f : A → B



Categories

idA : A→ A

B

f

f : A→ B



The tensor is associative:

A strict monoidal category is a category equipped with a tensor product on both objects and arrows:

(f ⊗ g)⊗ h = f ⊗ (g ⊗ h)

f ⊗ h : A⊗ C → B ⊗D

Monoidal Categories

f h

D

A

B

C

gf

A

B

h

D

C

C

B



and identities:

The tensor product is bifunctorial, meaning that it preserves composition:

idA⊗B = idA ⊗ idB

(g ◦ f)⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)

Monoidal Categories

f h

D

A

B

C

g k

C E

A B



No lines are drawn for I in the graphical notation:

Monoidal categories have a special unit object called I which is a left and right identity for the tensor:

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

I ⊗A = A = A⊗ I

idI ⊗ f = f = f ⊗ idI

Monoidal Categories

ψ : I → A φ† : A → I φ† ◦ ψ : I → Iψ : I → A φ† : A → I φ† ◦ ψ : I → I

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

ψ : I → A φ† : A → I φ† ◦ ψ : I → I



A monoidal category is called †-monoidal if it is equipped with an involutive functor, which reverses the arrows while leaving the objects unchanged, which preserves the tensor structure.

f : A→ B f† : B → A


(·)†

f

A

B A

Bf



An arrow is called unitary when:


f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

BA

f

A

f

f

f

B





f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

BA

A

f

f

B





f : A→ B f† : B → A

(f ⊗ g)† = f† ⊗ g† σ†A,B = σB,A

BA

A B



d : I → A∗ ⊗A e : A⊗A∗ → I

Compact Closure

cut elimination in MLL proof nets



d : I → A∗ ⊗A e : A⊗A∗ → I

Compact Closure

=

cut elimination in MLL proof nets



Thm: one diagram can be deformed to another if and only if their denotations are equal by the structural equations of the category.

Graphical Calculus Theorem

fhD

A

B

C

g

k

C

E

c



Thm: one diagram can be deformed to another if and only if their denotations are equal by the structural equations of the category.

Graphical Calculus Theorem

fhD

A

B

C

g

k

C

E

c

= c

f

A

B

C

D

g

C

k

E

h



The Category FDHilb

FDHilb is the category of finite dimensional complex Hilbert spaces. It is †-monoidal with the following structure.

• Objects: finite dimensional Hilbert spaces, etc

• Arrows: all linear maps

• Tensor: usual (Kronecker) tensor product;

• is the usual adjoint (conjugate transpose)

A linear map picks out exactly one vector. It is a ket and is the corresponding bra.

Hence is the inner product .

A,B, C,

I = Cf†

ψ : I → Aψ† : A→ I

ψ† ◦ φ : I → I �ψ | φ�



The Category FRel

FRel is the category of finite relations. It is †-compact with the following structure.

• Objects: finite sets, etc

• Arrows: Relations

• Tensor: Cartesian product;

• is relational converse:

A relation is simply a subset of X ; its converse is also a subset.

Hence is non-empty iff .

f†

R ⊆ X × Y

X, Y, Z,

X ⊗ Y := X × Y, I = {∗}(x, y) ∈ f ⇔ (y, x) ∈ f†

R ⊆ {∗}×X

S† ◦R : I → I R ∩ S �= ∅



Compact Structure of FDHilbIn FDHilb the compact structure is given by the maps:

whenever is a basis for and is the corresponding basis for the dual space

{ai}i {ai}iA

A∗





d : 1 �→�

i

ai ⊗ ai e : ai ⊗ ai �→ 1

{ai}i {ai}iA

A∗





|00�+ |11�√2

In the case of the map picks out the Bell state

which is the simplest example of quantum entanglement.

2 d

d : 1 �→�

i

ai ⊗ ai e : ai ⊗ ai �→ 1

{ai}i {ai}iA

A∗



Example: Quantum Teleportation

ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks

Bennett et al 1993; Abramsky & Coecke 2004




ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks|00�+ |11�√

2





ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks|00�+ |11�√

2

�00| + �11|√2





ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks





ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks

f





ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks

f†

f





ψ : I → A φ† : A → I φ† ◦ ψ : I → I

Bob

Aleks




Example: Preparing a Bell state

H

∧X



|0� : I → Q


H

∧X



|0� : I → Q|0� : I → Q


H

∧X



|0� : I → Q|0� : I → Q

H : Q→ Q


H

∧X



|0� : I → Q|0� : I → Q

H : Q→ Q

∧X : Q⊗Q→ Q⊗Q


H

∧X



Classical Structure

Copying, deleting, and all that



unless and are orthogonal [Wooters & Zurek 1982]

Theorem: There are no unitary operations D such that

D : |ψ� �→ |ψ� ⊗ |ψ�D : |φ� �→ |φ� ⊗ |φ�

No-Cloning and No-Deleting

unless and are orthogonal [Pati & Braunstein 2000]

Theorem: There are no unitary operations E such that

E : |ψ� �→ |0�E : |φ� �→ |0�

|ψ�

|ψ�

|φ�

|φ�



then the category collapses [Abramsky 2007]

Theorem: if a †-compact category has natural transformations

δ : − ⇒ −⊗−� : − ⇒ I

No-Cloning and No-Deleting, abstractly

(Translation: in our abstract setting there are no universal cloning or deleting operations, just like in quantum mechanics)



“Classical” Quantum States

When can a quantum state be treated as if classical?

• no-go theorems allow copying and deleting of orthogonal states;

In other words:

• A quantum state may be copied and deleted if it is an eigenstate of some known observable.

We’ll use this property to formalise observables in terms of copying and deleting operations.



Classical Structuresδ = δ† = �† =

=

=

Comonoid Laws

= =

� =




=

=

= =

Monoid Laws

� =




Frobenius LawIsometry Law

= = =

� =



Classical Structuresδ = δ† = �† =� =

In other words: a classical structure is a special commutative †-Frobenius algebra



Classical Structures

Given any finite dimensional Hilbert space we can define a classical structure by

δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

i

ai �→ 1

Example:

define a classical structure over qubits; the standard basis is copied and erased. Note however that:

δ : |0� �→ |00�|1� �→ |11� � : |0�+ |1� �→ 1

δ(|+�) = |00�+|11�√2

showing that not every state can be cloned.





δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

i

ai �→ 1

Example:

define a classical structure over qubits; the standard basis is copied and erased. Note however that:

δ : |0� �→ |00�|1� �→ |11� � : |0�+ |1� �→ 1

showing that not every state can be cloned.

=





δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

i

ai �→ 1

Theorem: in FDHilb, classical structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]



Each (well behaved) observable defines a basis, hence each observable defines a classical structure!



δ : A→ A⊗A :: ai �→ ai ⊗ ai

� : A→ I ::�

i

ai �→ 1

Theorem: in FDHilb, classical structures are in bijective correspondence to bases. [Coecke, Pavlovic, Vicary]



Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.

Spider Theorem

Coecke & Paquette 2006



Classical Structure begets Compact Structure



Frobenius




Unit Law

Unit Law





==

Each classical structure induces a self-dual compact structure.

Self duality is addressed in Coecke, Paquette, & Perdrix 2008



Quantum Observables

fundamental : incompatible observables* pos vs mom* X vs Z spins in fd

What is “classical” anyway?



Quantum Observables

http://dresdencodak.com

fundamental : incompatible observables* pos vs mom* X vs Z spins in fd

What is “classical” anyway?


http://desdencodak.com

http://desdencodak.com


Quantum Observables

Each observable quantity O is represented as self-adjoint operator :

•The possible values of O are the eigenvalues of

•When we observe O for a system in state , there is probability of observing .

•If is the outcome of the measurement, the system is then in state .

In general, measuring then will give a different answer than measuring first!

Not every quantum observable is well defined at the same time.

O =�

i λi |ei� �ei|λi O

|ψ��ei | ψ�2 λi

λi|ei�

O1 O2

O2



Z =�

1 00 −1

�X =

�0 11 0

�

X and Z Spins

α |0�+ β |1�

|0�✛

p=α2

|1�

✛p=

β2

|+�

p=

(α+

β)/

2 2✲

|−�

p=(α−

β)/2 2

✲

We can measure the spin of qubit |ψ� = α |0� + β |1�



Z =�

1 00 −1

�X =

�0 11 0

�

X and Z SpinsWe can measure the spin of qubit |ψ� = α |0� + β |1�

|0�

|0�✛

p=1

|1�

✛p=

0

|+�

p=

1/2

✲

|−�

p=1/2

✲



Z =�

1 00 −1

�X =

�0 11 0

�

X and Z SpinsWe can measure the spin of qubit |ψ� = α |0� + β |1�

(|0�+ |1�)/√

2

|0�✛

p=1/

2

|1�

✛p=

1/2

|+�

p=

1✲

|−�

p=0

✲



Complementary Observables

a0

a1




a0

a1

b1 b0




a0

a1

b1 b0

**




a0

a1

b1 b0

**

|�ai | bj�| =1√D



Phase Maps

Spinning around an observable

α



Let be points of ; we can combine them using the monoid operation

Using the monoid operationψ, φ : I → A A

δ† : A⊗A→ A

ψ ⊙ φ := δ† ◦ (ψ ⊗ φ)



Let be points of ; we can combine them using the monoid operation

Using the monoid operationψ, φ : I → A A

δ† : A⊗A→ A

ψ ⊙ φ := δ† ◦ (ψ ⊗ φ)

θ

θ



Example: qubits

|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�

|ψ� =�

ψ1

ψ2

�

|φ� =�

φ1

φ2

�|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�

|ψ� ⊗ |φ� =

ψ1φ1

ψ1φ2

ψ2φ1

ψ2φ2

|ψ� : I → Q|φ� : I → Qδ†Z : Q⊗Q → Q



Example: qubits

|ψ� : I → Q|φ� : I → Q

|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�

ψ1φ1

ψ2φ2

�

δ†Z : Q⊗Q → Q



Example: qubits

|ψ� : I → Q|φ� : I → Q

|ψ� ⊙ |φ� := δ†Z ◦ (|ψ� ⊗ |φ�) =�

ψ1φ1

ψ2φ2

�

a.k.a. the Schur product or convolution product

δ†Z : Q⊗Q → Q



Moreover, each point can be lifted to an endomorphism

Using the monoid operationψ : I → A

Λ(ψ) : A→ A

Λ(ψ) := δ† ◦ (ψ ⊗ idA)

This yields a homomorphism of monoids so we have:



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

ψ2

�



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

ψ2

�

id⊗ |ψ� =

ψ1 0ψ1 00 ψ2

0 ψ2



Example: qubits

δ†Z =�

1 0 0 00 0 0 1

�|ψ� =

�ψ1

ψ2

�

id⊗ |ψ� =

ψ1 0ψ1 00 ψ2

0 ψ2

ΛZ(ψ) := δ†Z ◦ (id⊗ |ψ�) =�

ψ1 00 ψ2

�



Theorem: any maps constructed from δ, ε, some points and their adjoints, whose graph is connected, is determined by the number of inputs and outputs and the product .

Generalised Spider Theoremψi : I → A

�i ψi





�i ψi

φ

φ





�i ψi



A: In Hilbert spaces, is unitary iff is unbiased w.r.t. the basis copied by .

Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ




Q: When is unitary?

α

Unbiased Points

|ψ�Λ(ψ)δ

α=




Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

α=




Q: When is unitary?

Unbiased Points

|ψ�Λ(ψ)δ

= -α




Q: When is unitary?

Prop: 1. the unbiased points for form an abelian group w.r.t. 2. the arrows generated by the unbiased points form an abelian group w.r.t. composition.

⊙(δ, �)

Unbiased Points

|ψ�Λ(ψ)δ



Example: qubits



Example: qubits

|0�

|1�



Example: qubits

π



Example: qubits

Unbiased pointsπ

|0� + eiα |1�



Example: qubits

Unbiased pointsπ

α



Example: qubits

Unbiased pointsπ

α

�1 00 eiα

�=



Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�

1 0 00 eiα 00 0 eiβ



Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22� |0� + eiα |1� + eiβ |2�

Unbiased points

1 0 00 eiα 00 0 eiβ



Example: FRel

D† : (x, x) ∼ x,∀x ∈ X

D† ◦ (id⊗ ψ) is unitary iff ψ = X

Hence the phase group is trivial.



Example: FRelSuppose that we are working with a two element set = {0,1}. Then:

D : 0 ∼ {(0, 0), (1, 1)}1 ∼ {(0, 1), (1, 0)}

E : 0 ∼ ∗Defines a classical structure whose phase group is Z_2



Example: FRel

Theorem (Pavlovic) : The classical structures of FRel are exactly biproducts of Abelian groups.

It’s easy to check that in this gives a phase group which is just the product group




A very general theory of interference



Two kinds of pointsδ = δ† = �† =� =



Two kinds of pointsδ = δ† = �† =

Classical Points

� =

i

Those points which can be copied by

δ




Classical Points

� =

i= i i


δ



* to keep the pictures

tidy, a scalar factor

has been omitted (and everywhere

from here on)


Classical Points

� =

i= i i


δ




Unbiased PointsClassical Points

� =

i= i i


δ

α





� =

i= i i


δ

=α

α





� =

i= i i


δ

=



Complementary Classical Structures

δX = �X =δZ = �Z =

|i� �→ |ii� |±� �→ |±±�|+� �→ 1 |0� �→ 1



i= i i ⇒

i

i=


δX = �X =δZ = �Z =



i= ⇒

i

i=i i


δX = �X =δZ = �Z =




δX = �X =δZ = �Z =

= =




Classical points

Unbiased pointsπ

π




Classical points

Unbiased pointsπ

π!

|0� =

|1� =

�1 00 eiα

�=!




Classical points

Unbiased pointsπ

π!

|0� =

|1� =

�1 00 eiα

�=!

=�

cos α2 i sin α

2i sin α

2 cos α2

�! !

|+� =

|−� =



Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�

1 0 00 eiα 00 0 eiβ

∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�

1 + eiα + eiβ 1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ

1 + ωeiα + ωeiβ 1 + eiα + eiβ 1 + ωeiα + ωeiβ

1 + ωeiα + ωeiβ 1 + ωeiα + ωeiβ 1 + eiα + eiβ



Classical points are eigenvectors

αi




α i




α ii



Complementary Classical Structures form a Hopf Algebra

Theorem:

(* if there are enough classical points)



Complementary Classical Structures form a Hopf Algebra

= =



Closedness Property

δX = �X =δZ = �Z =

Defn: complentary classical structures are called closed when...



Closedness Property

δX = �X =δZ = �Z =

⇒i

= i i

j= j j

=i⊙ j i⊙ j i⊙ j



Closedness Property

δX = �X =δZ = �Z =

In fdHilb it is always possible to construct a pair of mutually unbiased bases with this property...

... but in big enough dimension, it is possible to construct MUBs which are not closed.



Classical Points form a Subgroup

By the defn of complementarity, we have that:

Prop: If the observable structure is closed, and there are finitely many classical points, then they form a subgroup of the unbiased points, i.e.

In particular, each is a permutation on .

CX ⊆ UZ

CZ ⊆ UX

(CZ ,⊙X) ≤ (UX ,⊙X)(CX ,⊙Z) ≤ (UZ ,⊙Z)

ΛX(zi) CZ




Classical points

Unbiased pointsπ

π!

|0� =

|1� ==

�1 00 eiα

�!

=�

cos α2 i sin α

2i sin α

2 cos α2

�! !

|+� =

|−� =




Classical points

Unbiased pointsπ

π!

|0� =

|1� ==

= !

|+� =

|−� =

�0 11 0

�!

�1 00 −1

�!



The Following are Equivalent:

δX = �X =δZ = �Z =

⇒i

=i i

j

=j j =

i⊙ j i⊙ j i⊙ j

⇒i

=i i

j

=j j =

i⊙ j i⊙ j i⊙ j

1. The classical structures are closed




ii

δX = �X =δZ = �Z =

2. The classical maps are comonoid homomorphisms




i i

δX = �X =δZ = �Z =

2. The classical maps are comonoid homomorphisms




δX = �X =δZ = �Z =

3. The classical maps satisfy canonical commutation relations

ij i

j ij




δX = �X =δZ = �Z =

4. The classical structures form a bialgebra



Automorphism Action

Theorem: The classical maps are group automorphisms of the unbiased points:

αii



Automorphism Action


-α

α

-i-i

ii



Automorphism Action


-α

α

-i-i

i

i



Automorphism Action


-α

α

-i

i



Automorphism Action


-i

i



Automorphism Action




Automorphism Action


!! !!

! !!

!!

! ! !! !



Automorphism Action


Classical points are symmetries of the phase group.



Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�

1 0 00 eiα 00 0 eiβ

∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�






Example: qutrits

∆Z :|0� �→ |00�|1� �→ |11�|2� �→ |22�

1 0 00 eiα 00 0 eiβ

∆X :|+� �→ |++�|ω� �→ |ωω�|ω� �→ |ωω�




Phase maps are classical when:

α ∈ {0,π

3,2π

3} β = 2π − α



Example: qutrits

CZ = {

1 0 00 1 00 0 1

,

0 1 00 0 11 0 0

,

0 0 11 0 00 1 0

}

CX = {

1 0 00 1 00 0 1

,

1 0 00 ω 00 0 ω

,

1 0 00 ω 00 0 ω

}



Example: qutrits

0 1 00 0 11 0 0

·

1

eiα

eiβ

=

eiα

eiβ

1

�

1

ei(β−α)

e−iβ



Example: qutrits

0 1 00 0 11 0 0

·

1

eiα

eiβ

=

eiα

eiβ

1

�

1

ei(β−α)

e−iβ

(α, β) �→ (β − α,−α)



Why care about phase groups?

The phase group seems like a rather random structure. Why do we care? The phase group:

• ... determines the underlying arithmetic

• ... fixes the possibility for interference

• ... determines the possible results of CHSH-type experiments, and hence the possibility for non-locality.(I will discuss this more in Part 3 of this series.)



The ZX-calculus

Quantum processes, diagrammatically



Z and X ObservablesThe Pauli Z and X observables are strongly complementary, with some additional features:

• The phase group is [0,2π)

• Dimension 2 implies two classical points

• The action of the classical group is α �→ −α

π

π

π

π



Z and X Observables

• Both observables generate the same compact structure

- can just treat the diagram as an undirected graph

• the Z and X are related by a definable unitary- gives rise to colour change rule

= =



Generators for diagrams

Defn: A diagram is an undirected open graph generated by the above vertices.

Defn: let be the dagger compact category of diagrams s.t.

(·)† : α �→ −α

D



Equations



Equations

· · ·

· · ·

π2

α+ nπ2

π2

π2

π2

· · ·

· · ·

−π2

α

−π2 −π

2

−π2

=

(colour change)



Semantics for diagramsDefn: Let be the traced monoidal functor such and that and define its action on generators by:



Representing Qubits



Representing Paulis



Representing CNot



∧X =

1 0 0 00 1 0 00 0 0 10 0 1 0

Representing Logic Gates



∧X =

1 0 0 00 1 0 00 0 0 10 0 1 0


ππ



Representing Hadamard

π2

π2

π2

H = 1√2

�1 11 −1

�= � �



More on the Hadamard

π2

π2

π2

H :=




π2

π2

π2

H :=

− π2

π2

π2

π2

π2

=




π2

π2

π2

H :=

− π2

π2

π2

π2

π2

=

− π2

π

π2

π2

=




π2

π2

π2

H :=

− π2

π2

π2

π2

π2

=

− π2

π

π2

π2

=

− π2

− π2

π

π2

=




π2

π2

π2

H =

− π2

− π2

− π2

=




π2

π2

π2

H =

− π2

− π2

− π2

=

− π2

− π2

− π2

=

colour change




π2

π2

π2

H =

− π2

− π2

− π2

=

− π2

− π2

− π2

= =

π2

π2

π2

colour change colour change




Corollary 1:

H

H

=H

†H =




Corollary 2:

H H

H H

· · ·

· · ·

α




Corollary 2:

H H

H H

· · ·

· · ·

α

· · ·

· · ·

α+ nπ2

π2

π2

π2

π2

π2

π2

π2

π2

=




Corollary 2:

H H

H H

· · ·

· · ·

α

· · ·

· · ·

α+ nπ2

π2

π2

π2

π2

π2

π2

π2

π2

=

· · ·

· · ·

α

π2

π2

π2

π2

−π2 −π

2

−π2 −π

2

=




Corollary 2:

H H

H H

· · ·

· · ·

α

· · ·

· · ·

α+ nπ2

π2

π2

π2

π2

π2

π2

π2

π2

=

· · ·

· · ·

α

π2

π2

π2

π2

−π2 −π

2

−π2 −π

2

=

· · ·

· · ·

α=

Corollary 2.1 :total symmetry between red and green




Corollary 3:

π2




Corollary 3:

π2

π2

−π2

π2=




Corollary 3:

π2

π2

−π2

π2=

−π2

−π2=




Corollary 3:

π2

π2

−π2

π2=

−π2

−π2= −π

2=




Corollary 3:

π2

π2

−π2

π2=

−π2

−π2= −π

2=

−π2

=




Corollary 3:π2 −π

2

=−π

2π2

=




Corollary 4: Van den Nest’s theorem

Van den Nest, M., Dehaene, J. and De Moor, B. “Graphical description of the action of local clifford transformations on graph states”, Phys. Rev. A 69 (2004) 022316. arXiv:quant-ph/0411115

Duncan, R. and Perdrix, S. “Graph States and the necessity of Euler Decomposition”, Proc. CiE09 (2009), LNCS 5635. arXiv:0902.0500



∧Z =

1 0 0 00 1 0 00 0 1 00 0 0 −1




Example: 2 CZs

H

H



Example: 2 CZs

HHH

H



Example: 2 CZs

H

H



Example: 2 CZs




∧X

H




H



controlledgate

Zπ/2

j1 = |1�

j0 = |0�

input qubits

Example: 2-Qubit Quantum Fourier Transform

π

H−π/4

H

π/4




π

H−π/4

H

π/4




π

H−π/4

ππH

π/4




π

H−π/4

ππ

H

π/4




H−π/4

π

π

ππ/4




−π/4π

π

ππ/4




−π/4π

π

π/4




π

π/4 π/4




π

π/2



output qubits

j0 = |0� + eiπ2 |1�

j1 = |0� + eiπ |1�


π

π/2



Does the ZX-calculus decide the word problem for the Clifford group?

Ross DuncanLaboratoire d’Information Quantique

Université Libre de Bruxelles



The Pauli Group

Pauli Group:

We don’t care about phase so Y = XZ

X =

�0 11 0

�Y =

�0 −ii 0

�Z =

�1 00 −1

�

Pi = 1⊗ · · ·⊗ 1⊗ P ⊗ 1⊗ · · ·⊗ 1



The Clifford Group

The Clifford group is the stabilizer of the Pauli group

CP = P �C



The Clifford Group

The Clifford group is generated by the following matrices:

Zπ/2 =

�1 00 i

�H =

1

2

�1 11 −1

�

CNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0



The Clifford Group

The Clifford group is generated by the following matrices:

CNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

Zπ/2 =

�1 00 i

�Xπ/2 =

�ω ωω ω

�



The Clifford Group

• The Clifford group preserves the the eigenstates of the Pauli operators -- the stabilizer formalism.

• The Clifford group is the basis of quantum error correcting codes

• Clifford circuits can be classically simulated in linear time

• Measurement-based quantum computations with only Pauli measurements can be performed in one layer (i.e. no feed-forward)

• Add almost any gate to the Clifford group to get (approximate) universal QC



The Clifford Group

The Clifford group gets big fast...

|C1| = 24

|C2| = 11520

|C3| = 92827280



The Word Problem

Given two Clifford circuits, are they equal?

• or: given one circuit, does it compute the identity?

Obviously this is decidable by just working out the matrix of the circuit, but this may be very large.

• No presentation by generators and relations is known.



The Completeness Question

How powerful is the ZX-calculus?




Conjecture: the ZX-calculus decides the word problem for Cn






Conjecture: the ZX-calculus decides the word problem for SU(2)






Conjecture: the ZX-calculus decides the word problem for SU(2)

Conjecture: the ZX-calculus decides the word problem for SU(2n)




The Clifford Group C1

The Clifford group on 1 qubit (aka C1) is generated by the unitaries:

Z π2

=�

1 00 i

�H = 1√

2

�1 11 −1

�

π2

π2

π2

π2




Or equivalently:

π2

π2

Z π2

=�

1 00 i

�X π

2= 1√

2

�ω ωω ω

�




0 vertices:

1 vertex: π2 π − π

2π2 π − π

2




2 vertices: π

π= π

π

ππ2

=− π

2

π

π

− π2

=π2

π

π

− π2

=π2

π

ππ2

=− π

2

π

π2π2

− π2

π2

− π2

− π2

π2

− π2

π2

− π2

− π2

− π2

− π2

π2

π2π2




3 vertices:

π2π2π2

=π2π2π2

=− π

2

− π2

− π2

=− π

2

− π2

− π2

π2

− π2

π2

=π2

− π2

π2

=− π

2π2

− π2

=− π

2π2

− π2

− π2

− π2

π2

=− π

2

− π2

π2

=π2π2

− π2

=π2π2

− π2

− π2

π2π2

=− π

2π2π2

=π2

− π2

− π2

=π2

− π2

− π2




Theorem: the ZX-calculus decides the word problem for C1

Proof: the 1 + 6 + 13 + 4 = 24 elements shown are all distinct; any combination of n > 3 vertices can be reduced to smaller number via the spider rule and commutation with Paulis.



CNOT and TONC

CNOT = TONC =



CNOT and TONC



TONC and Swap




Several classes of 2-qubit cliffords:

This gives 24 * 24 * ( 1 + 1 + 9 + 9) = 11520 diagrams

Check mechanically: they are all different. Nice!

A ∈ {1, , } B ∈ {1, , }



C2

IDEA: compose an elements of C2 with a generator of the group, show that you get another one of the diagrams already shown



Theorem: the ZX-calculus decides the word problem for C2


Categorical Quantum Mechanicspersonal.strath.ac.uk/ross.duncan/talks/2012/catQM-pt1.pdf ·...

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