quantum information ﬂow · 2013-04-24 · quantum information ﬂow bob coecke – quantum group...

quantum information flowBob Coecke – Quantum Group - Computer Science - Oxford University

=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f

Samson Abramsky & BC (2004) A categorical semantics forquantum protocols. Logic in Computer Science ’04. arXiv:quant-ph/0402130 BC & Ross Duncan (2008, 2011) Interacting quan-tum observables: categorical algebra and diagrammatics. NewJournal of Physics 13, 043016. arXiv:0906. 4725

FORTHCOMING: textbook with Aleks Kissinger on a purelydiagrammatic presentation of basic quantum information

Overview / general idea:


• Quantum informatic systems and operations and com-putations thereon admit diagrammatic calculus.



• Pictures expose ‘information flows’.




• Important fragments are ‘complete’:





– Bipartite entanglement generated (Selinger’08)





– Bipartite entanglement generated (Selinger’08)– The qubit stabiliser fragment (Backens’12)





– Bipartite entanglement generated (Selinger’08)– The qubit stabiliser fragment (Backens’12)

• The mathematical backbone is category theory.

Category Theory, . . .


. . . isn’t it just tedious abstract nonsense?


. . . isn’t it just tedious abstract nonsense? No!


. . . isn’t it just tedious abstract nonsense? No!

Symmetric Monoidal Categories are everywhere!

1. Let A be a raw potato.

1. Let A be a raw potato.A admits many states e.g. dirty, clean, skinned, ...


2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ...


2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ... Let

Af

-B Af ′

-B Af ′′

-B

be boiling, frying, baking.


2. We want to process A into cooked potato B.B admits many states e.g. boiled, fried, deep fried,baked with skin, baked without skin, ... Let

Af

-B Af ′

-B Af ′′

-B

be boiling, frying, baking. States are processes

I := unspecifiedψ

-A.

3. LetA

g ◦ f-C

be the composite process of first boiling Af

-B andthen salting B

g-C.

3. LetA

g ◦ f-C

be the composite process of first boiling Af

-B andthen salting B

g-C. Let

X1X -X

be doing nothing. We have 1Y ◦ ξ = ξ ◦ 1X = ξ.

4. Let A⊗D be potato A and carrot D and let


A⊗D f⊗h-B ⊗ E

be boiling potato while frying carrot.


A⊗D f⊗h-B ⊗ E

be boiling potato while frying carrot. Let

C ⊗ F x-M

be mashing spice-cook-potato and spice-cook-carrot.

5. Total process:

A⊗D f⊗h-B⊗E g⊗k

-C⊗F x-M=A⊗D x◦(g⊗k)◦(f⊗h)

-M.

5. Total process:



-M.

6. Recipe = composition structure on processes.

5. Total process:



-M.


7. Laws governing recipes:

(1B ⊗ g) ◦ (f ⊗ 1C) = (f ⊗ 1D) ◦ (1A ⊗ g)

5. Total process:



-M.



(1B ⊗ g) ◦ (f ⊗ 1C) = (f ⊗ 1D) ◦ (1A ⊗ g)i.e.

boil potato then fry carrot = fry carrot then boil potato

5. Total process:



-M.



(1B ⊗ g) ◦ (f ⊗ 1C) = (f ⊗ 1D) ◦ (1A ⊗ g)i.e.

boil potato then fry carrot = fry carrot then boil potato

⇒ Symmetric Monoidal Category

— Why does a tiger have stripes and a lion doesn’t? —

— Why does a tiger have stripes and a lion doesn’t? —

prey ⊗ predator ⊗ environment

dead prey ⊗ eating predatorhunt

?

BOXES AND WIRES

Roger Penrose (1971) Applications of negative dimensional tensors. In: Com-binatorial Mathematics and its Applications, 221–244. Academic Press.

Andre Joyal and Ross Street (1991) The Geometry of tensor calculus I. Ad-vances in Mathematics 88, 55–112.

Bob Coecke and Eric Oliver Paquette (2011) Categories for the practicingphysicist. In: New Structures for Physics, B. Coecke (ed), Springer-Verlag.arXiv:0905.3010

— wire and box language —

foutput wire(s)

input wire(s)Box =:


foutput wire(s)

input wire(s)Box =:

Interpretation: wire := system ; box := process


foutput wire(s)

input wire(s)Box =:

Interpretation: wire := system ; box := process

one system: n subsystems: no system:

︸︷︷︸1

. . .︸︷︷︸

n︸︷︷︸

0

— wire and box games —


sequential or causal or connected composition:

g ◦ f ≡g

f


sequential or causal or connected composition:

g ◦ f ≡g

f

parallel or acausal or disconnected composition:

f ⊗ g ≡ f fg

— merely a new notation? —


(g ◦ f )⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)

=f h

g k

f h

g k


(g ◦ f )⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)

=f h

g k

f h

g k

peel potato and then fry it,while,

clean carrot and then boil it=

peel potato while clean carrot,and then,

fry potato while boil carrot

MINIMAL QUANTUM PROCESS LANGUAGE

Samson Abramsky & BC (2004) A categorical semantics for quantum proto-cols. In: IEEE-LiCS’04. quant-ph/0402130

BC (2005) Kindergarten quantum mechanics. quant-ph/0510032

BC (2010) Quantum picturalism. Contemporary Physics. arXiv:0908.1787

— genesis —

[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”

— genesis —


[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)

— genesis —



[Birkhoff and von Neumann 1936] The Logic ofQuantum Mechanics in Annals of Mathematics.

— genesis —



[Birkhoff and von Neumann 1936] The Logic ofQuantum Mechanics in Annals of Mathematics.

[1936 – 2000] many followed them, ... and FAILED.

— the mathematics of it —


Hilber space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.



WHY?



WHY?

von Neumann: only used it since it was ‘available’.

— the physics of it —


von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.



Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.



Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.

Quantum Computer Scientists: Schrodinger is right!

— the game plan —


Task 0. Solve:tensor product structure

the other stuff= ???




i.e. axiomatize “⊗” without reference to spaces.





Task 1. Investigate which assumptions (i.e. which struc-ture) on ⊗ is needed to deduce physical phenomena.





Task 1. Investigate which assumptions (i.e. which struc-ture) on ⊗ is needed to deduce physical phenomena.

Task 2. Investigate wether such an “interaction struc-ture” appear elsewhere in “our classical reality”.

Outcome 1a: “Sheer ratio of results to assumptions”

Outcome 1a: “Sheer ratio of results to assumptions”confirms that we are probing something very essential.

Hans Halvorson (2010) Editorial to: Deep Beauty: Understanding the Quan-tum World through Mathematical Innovation, Cambridge University Press.


Outcome 1b: Exposing this structure has already helpedto solve open problems elsewhere. (e.g. 2× ICALP’10)

E.g.: Ross Duncan & Simon Perdrix (2010) Rewriting measurement-basedquantum computations with generalised flow. ICALP’10.


Outcome 1b: Exposing this structure has already helpedto solve open problems elsewhere. (e.g. 2× ICALP’10)

Outcome 1c: Framework is a simple intuitive (butrigorous) diagrammatic language, meanwhile adoptedby others e.g. Lucien Hardy in arXiv:1005.5164:

“... we join the quantum picturalism revolution [1]”

[1] BC (2010) Quantum picturalism. Contemporary Physics 51, 59–83.

Outcome 2a:Behaviors of matter (Abramsky-C; LiCS’04, quant-ph/0402130) :

=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f

Meaning in language (Clark-C-Sadrzadeh; Linguistic Analysis, arXiv:1003.4394) :

=

not

likeBobAlice

does

Alice not like

not

Bob

meaning vectors of words

pregroup grammar

Knowledge updating (C-Spekkens; Synthese, arXiv:1102.2368) :

conditionalindependence

=P(C|AB)

A A

=

A

=A

B

A

B=

B(BA) 1-

A

C 1- C 1-

C

P(AB|C) P(A|C) P(B|C) P(C|A) P(C|B)P(C|B) P(C|A)

BOXES AND WIRES II

— quantitative metric —

f : A→ B

f

A

B

— quantitative metric —

f† : B → A

f

B

A

— asserting (pure) entanglement —

quantum

classical=

==

— asserting (pure) entanglement —

quantum

classical=

==

⇒ introduce ‘parallel wire’ between systems:

subject to: only topology matters!

— quantum-like —

E.g.

=

Transpose:

ff

=Conjugate:

ff

=

classical data flow?

f

=

fff


f

=

f


f

ALICE

BOB

=

ALICE

BOB

f

⇒ quantum teleportation

Applying “decorated” normalization 3

=f

f

f f

⇒ Entanglement swapping


f

=

f

g

g

‘

⇒ gate teleportation MBQC

— symbolically: dagger compact categories —

Thm. [Kelly-Laplaza ’80; Selinger ’05] An equa-tional statement between expressions in dagger com-pact categorical language holds if and only if it isderivable in the graphical notation via homotopy.

Thm. [Hasegawa-Hofmann-Plotkin; Selinger ’08]An equational statement between expressions in dag-ger compact categorical language holds if and onlyif it is derivable in the dagger compact category of fi-nite dimensional Hilbert spaces, linear maps, tensorproduct and adjoints.

— symbolically: dagger compact categories —

In words: Any equation involving:

• states, operations, effects

• unitarity, adjoints (e.g. self-adjoint), projections

• Bell-states/effects, transpose, conjugation

• inner-product, trace, Hilbert-Schmidt norm

• positivity, completely positive maps, ...

holds in quantum theory if and only if it can be derivedin the graphical language via homotopy.

What are these diagrams in ordinary QM?

• Step 1: only maps (no vectors, numbers)

• Step 2: cast Dirac in 2D

• Step 3: summations 7→ topological entity

– E.g.∑

i |ii〉 7→

. . . glass board tutorial

A SLIGHTLY DIFFERENT LANGUAGEFOR NATURAL LANGUAGE MEANING

BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundationsfor a compositional distributional model of meaning. arXiv:1003.4394

BC (2012) The logic of quantum mechanics – Take II. arXiv:1204.3458

— the from-words-to-a-sentence process —


Consider meanings of words, e.g. as vectors (cf. Google):

word 1 word 2 word n...


What is the meaning the sentence made up of these?

word 1 word 2 word n...


I.e. how do we/machines produce meanings of sentences?

word 1 word 2 word n...?


I.e. how do we/machines produce meanings of sentences?

word 1 word 2 word n...grammar

Gerald Gazdar (1996) Paradigm merger in natural language processing. In:Computing tomorrow: future research directions in computer science, eds.,I. Wand and R. Milner, Cambridge University Press.


Information flow within a verb:

verb

object subject


Information flow within a verb:

verb

object subject

Again we have:

=

— grammar as pregroups – Lambek ’99 —

A Al A A

A Al

r

A Ar

=

A

A

A

A=A

A A

A

r

r

=A

A

A

A=A

A A

Al l

ll

r

r

—−−−→Alice⊗−−→does⊗−→not⊗−−→like⊗−−→Bob —

Alice not like Bob


not

grammar

does


Alice like Bob


grammar

not


Alice like Bob


grammar

not= not

like

BobAlice

— analogy: quantizing grammar! —

Topological quantum field theory:

F : nCob→ FVectC :: 7→ V

Grammatical quantum field theory:

F : Pregroup→ FVectR+ :: 7→ V

Louis Crane was the first one to notice this analogy.

UNIVERSAL QUANTUM REASONING

WIRES?

not expressive enough

not expressive enough

what is expressive enough?

SPIDERS!

— spiders —

‘spiders’ =

m︷︸︸︷....

....

︸︷︷︸n

such that, for k > 0:

m+m′−k︷︸︸︷

........

....

....

....

︸︷︷︸n+n′−k

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

— spiders —

‘cups/caps’ =

m︷︸︸︷....

....

︸︷︷︸n


m+m′−k︷︸︸︷

........

....

....

....

︸︷︷︸n+n′−k

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

— spiders —

‘(co-)mult.’ =

m︷︸︸︷....

....

︸︷︷︸n


m+m′−k︷︸︸︷

........

....

....

....

︸︷︷︸n+n′−k

=

....

....

Theorem. The following are the same:


• Linear maps {H⊗n → H⊗m}n,m as spiders.



• Orthonormal bases forH.



• Orthonormal bases forH.

• Copying/erasing pairs

∆ : H → H⊗H ε : H → C

that are commutative, dagger Frobenius & special.

To proof:1a) How does an ONB define a copying/deleting pair?1b) ... and conversely? (CPV’08 thm)2a) How does a copying/deleting pair define spiders?2b) ... and conversely? (TQFT thm)

. . . glass board tutorial

... decorated spiders

... allergic spiders

UNBIASEDNESS:

“allergic spiders”

Thm. Unbiasedness⇔

Thm. Unbiasedness⇔

BC & Ross Duncan (2008) Interacting quantum observables. ICALP’08 &New Journal of Physics 13, 043016. arXiv:0906.4725

Z-spin:

δZ : |i〉 7→ |ii〉X-spin:

δX : |±〉 7→ | ± ±〉

MEASUREMENT & CLASSICAL CONTROL

QUANTUM CIRCUITS

i.e.

(δ†Z ⊗ 1) ◦ (1⊗ δX) =

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

= CNOT

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

◦

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

= ?

STRONG UNBIASEDNESS

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

◦σ◦

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

◦σ◦

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

= ?

Def. Strong unbiasedness⇔

+ comults copy units of other colour.

BC & Ross Duncan (2008) Interacting quantum observables. ICALP’08 &New Journal of Physics 13, 043016. arXiv:0906.4725

Strong complementarity⇒ complementarity

PHASES:

“decorated spiders”

Thm. ‘Unbiased states’ always form an Abelian groupfor spider multiplication with conjugate as inverse.

m︷︸︸︷....

....α︸︷︷︸n

∣∣ n,m ∈ N0, α ∈ G

m︷︸︸︷

........

....

....

....α

β︸︷︷︸n

=

m︷︸︸︷

....

....

α+β

︸︷︷︸n

For qubits in FHilb with green ≡ {|0〉, |1〉} ≡ Z:

=

(1

eiα

)Z

= Zα =

(1 0

0 eiα

)Z

These are relative phases for Z, hence in X-Y:

α

For qubits in FHilb with red ≡ {|+〉, |−〉} ≡ X:

=

(1

eiα

)X

= Xα =

(1 0

0 eiα

)X

These are relative phases for X, hence in Z-Y :

α

— colour changer —

.... HH H

....α

H

....α

H H ....=

— one CZ gate —

H

H

H

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

— MBQC is universal for qubit gates —

H

H

H

H


H H H H


⇒ Arbitrary one-qubit unitary

— universality —

Thm. (BC & Ross Duncan) The above described graph-ical language, is universal for computing with qubits.

Proof. CNOT-gate + arbitrary one-qubit unitaries viatheir Euler angle decomposition yield all unitaries:

ΛZ(γ) ◦ ΛX(β) ◦ ΛZ(α) =γ

αβ .

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

= .

— completeness II —

Thm. (Miriam Backens, about a year ago) The abovedescribed graphical calculus with phases restricted toπ2-multiples, is complete for qubit stabiliser fragmentof quantum computing, provided that we add the Eulerangle decomposition of the Hadamard gate.

— applications to QC models —

Example 18. The ubiquitous CNOT operation can be computed by the patternP = X3

4Z24Z2

1M03 M0

2 E13E23E34N3N4 [5]. This yields the diagram,

DP =

H

H

H

!, {3}

!, {2}

!, {2}

!, {3}!, {2}

,

where each qubit is represented by a vertical “path” from top to bottom, withqubit 1 the leftmost, and qubit 4 is the rightmost.

By virtue of the soundness of R and Proposition 10, if DP can be rewrittento a circuit-like diagram without any conditional operations, then the rewritesequence constitutes a proof that the pattern computes the same operation asthe derived circuit.

Example 19. Returning to the CNOT pattern of Example 18, there is a rewritesequence, the key steps of which are shown below, which reduces the DP tothe unconditional circuit-like pattern for CNOT introduced in Example 7. Thisproves two things: firstly that P indeed computes the CNOT unitary, and thatthe pattern P is deterministic.

H

H

H

!, {3}

!, {2}

!, {2}

!, {3}!, {2}

!!H

H

H

!, {3}

!, {2}

!, {2}!, {2} !, {3}

!! H

H

H

!, {3}!, {3}

!, {2}

!, {2}

!, {2}

!!!, {2}

!, {2}!, {2}

!!!, {2}!, {2}

!, {2} !, {2}!!

One can clearly see in this example how the non-determinism introduced bymeasurements is corrected by conditional operations later in the pattern. Thepossibility of performing such corrections depends on the geometry of the pat-tern, the entanglement graph implicitly defined by the pattern.

Definition 20. Let P be a pattern; the geometry of P is an open graph !(P) =(G, I,O) whose vertices are the qubits of P and where i ! j i! Eij occurs in thecommand sequence of P.

Definition 21. Given a geometry " = ((V,E), I, O) we can define a diagramD! = ((VD, ED), ID, OD) as follows:

Ross Duncan & Simon Perdrix (2010) Rewriting measurement-based quantumcomputations with generalised flow. ICALP’10. Survey: arXiv:1203.6242

Clare Horsman (2011) Quantum picturalism for topological cluster-state com-puting. New Journal of Physics 13, 095011. arXiv:1101.4722.

Sergio Boixo & Chris Heunen (2012) Entangled and sequential quantum pro-tocols with dephasing. PRL 108, 120402. arXiv:1108.3569

— applications to quantum foundations —

Toy qubits vs. true quantum theory in one language:

Spekkens’ qubit QMstabilizer qubit QM

=Z2 × Z2

Z4=

localnon-local

Coecke, Bill Edwards & Robert W. Spekkens (2010) Phase groups and theorigin of non-locality for qubits. QPL’10 arXiv:1003.5005

— applications to quantum foundations —

Toy qubits vs. true quantum theory in one language:

Spekkens’ qubit QMstabilizer qubit QM

=Z2 × Z2

Z4=

localnon-local

Coecke, Bill Edwards & Robert W. Spekkens (2010) Phase groups and theorigin of non-locality for qubits. QPL’10 arXiv:1003.5005

Generalized Mermin arg.⇔ strong complementarity

Coecke, Ross Duncan, Aleks Kissinger & Quanlong Wang (2012) Strong com-plementarity and non-locality in categorical quantum mechanics. LiCS’12.arXiv:1203.4988

— multipartite entanglement structure —

Tripartite SLOCC-classes as comm. Frobenius algs:

GHZ = |000〉 + |111〉W = |001〉 + |010〉 + |100〉 =

‘special’ CFAs‘anti-special’ CFAs

=

=

=

=×+

⇒ distributivity

Coecke & Aleks Kissinger (2010) The compositional structure of multipartitequantum entanglement. ICALP’10. arXiv:1002.2540

— GHZ-spiders —

Data: m︷︸︸︷....

....

︸︷︷︸n

∣∣ n,m ∈ N

Rules:

m+m′−k︷︸︸︷........

....

....

....

︸︷︷︸n+n′−k

=

m+m′−k︷︸︸︷....

....

︸︷︷︸n+n′−k

— W-spiders —

Data: m︷︸︸︷....

....

︸︷︷︸n

, ,∣∣ n,m ∈ N

Rules:

m+m′−1︷︸︸︷........

.... ....

︸︷︷︸n+n′−1

=

m+m′−1︷︸︸︷....

....

︸︷︷︸n+n′−1

— W-spiders —

Data: m︷︸︸︷....

....

︸︷︷︸n

, ,∣∣ n,m ∈ N

Rules:

m+m′−2︷︸︸︷........

.... ....

︸︷︷︸n+n′−2

=

m+m′−2︷︸︸︷....

....︸︷︷︸n+n′−2

quantum information ﬂow · 2013-04-24 · quantum information ﬂow bob coecke – quantum group...

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