quantum information flow Bob Coecke – Quantum Group - Computer Science - Oxford University = f f = f f f ALICE BOB = ALICE BOB f Samson Abramsky & BC (2004) A categorical semantics for quantum protocols. Logic in Computer Science ’04. arXiv:quant- ph/0402130 BC & Ross Duncan (2008, 2011) Interacting quan- tum observables: categorical algebra and diagrammatics. New Journal of Physics 13, 043016. arXiv:0906. 4725 FORTHCOMING: textbook with Aleks Kissinger on a purely diagrammatic presentation of basic quantum information
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quantum information flow · 2013-04-24 · quantum information flow Bob Coecke – Quantum Group - Computer Science - Oxford University f = f = f f f ALICE BOB = f Samson Abramsky
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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:
Overview / general idea:
• Quantum informatic systems and operations and com-putations thereon admit diagrammatic calculus.
Overview / general idea:
• Quantum informatic systems and operations and com-putations thereon admit diagrammatic calculus.
• Pictures expose ‘information flows’.
Overview / general idea:
• Quantum informatic systems and operations and com-putations thereon admit diagrammatic calculus.
• Pictures expose ‘information flows’.
• Important fragments are ‘complete’:
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)
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)– The qubit stabiliser fragment (Backens’12)
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)– The qubit stabiliser fragment (Backens’12)
• The mathematical backbone is category theory.
Category Theory, . . .
Category Theory, . . .
. . . isn’t it just tedious abstract nonsense?
Category Theory, . . .
. . . isn’t it just tedious abstract nonsense? No!
Category Theory, . . .
. . . 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, ...
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, ...
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, ... Let
Af
-B Af ′
-B Af ′′
-B
be boiling, frying, baking.
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, ... 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
4. Let A⊗D be potato A and carrot D and let
A⊗D f⊗h-B ⊗ E
be boiling potato while frying carrot.
4. Let A⊗D be potato A and carrot D and let
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:
A⊗D f⊗h-B⊗E g⊗k
-C⊗F x-M=A⊗D x◦(g⊗k)◦(f⊗h)
-M.
6. Recipe = composition structure on processes.
5. Total process:
A⊗D f⊗h-B⊗E g⊗k
-C⊗F x-M=A⊗D x◦(g⊗k)◦(f⊗h)
-M.
6. Recipe = composition structure on processes.
7. Laws governing recipes:
(1B ⊗ g) ◦ (f ⊗ 1C) = (f ⊗ 1D) ◦ (1A ⊗ g)
5. Total process:
A⊗D f⊗h-B⊗E g⊗k
-C⊗F x-M=A⊗D x◦(g⊗k)◦(f⊗h)
-M.
6. Recipe = composition structure on processes.
7. Laws governing recipes:
(1B ⊗ g) ◦ (f ⊗ 1C) = (f ⊗ 1D) ◦ (1A ⊗ g)i.e.
boil potato then fry carrot = fry carrot then boil potato
5. Total process:
A⊗D f⊗h-B⊗E g⊗k
-C⊗F x-M=A⊗D x◦(g⊗k)◦(f⊗h)
-M.
6. Recipe = composition structure on processes.
7. Laws governing recipes:
(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 =:
— wire and box language —
foutput wire(s)
input wire(s)Box =:
Interpretation: wire := system ; box := process
— wire and box language —
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 —
— wire and box games —
sequential or causal or connected composition:
g ◦ f ≡g
f
— wire and box games —
sequential or causal or connected composition:
g ◦ f ≡g
f
parallel or acausal or disconnected composition:
f ⊗ g ≡ f fg
— merely a new notation? —
— merely a new notation? —
(g ◦ f )⊗ (k ◦ h) = (g ⊗ k) ◦ (f ⊗ h)
=f h
g k
f h
g k
— merely a new notation? —
(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 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[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 —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[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)
[Birkhoff and von Neumann 1936] The Logic ofQuantum Mechanics in Annals of Mathematics.
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[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)
[Birkhoff and von Neumann 1936] The Logic ofQuantum Mechanics in Annals of Mathematics.
[1936 – 2000] many followed them, ... and FAILED.
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[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)
[Birkhoff and von Neumann 1936] The Logic ofQuantum Mechanics in Annals of Mathematics.
[1936 – 2000] many followed them, ... and FAILED.
— the mathematics of it —
— the mathematics of it —
Hilber space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
— the mathematics of it —
Hilber space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
WHY?
— the mathematics of it —
Hilber space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
WHY?
von Neumann: only used it since it was ‘available’.
— the physics of it —
— the physics of it —
von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.
— 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.
— 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.
Quantum Computer Scientists: Schrodinger is right!
— the game plan —
— the game plan —
Task 0. Solve:tensor product structure
the other stuff= ???
— the game plan —
Task 0. Solve:tensor product structure
the other stuff= ???
i.e. axiomatize “⊗” without reference to spaces.
— 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.
— 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 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 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 1a: “Sheer ratio of results to assumptions”confirms that we are probing something very essential.
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 1a: “Sheer ratio of results to assumptions”confirms that we are probing something very essential.
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 1a: “Sheer ratio of results to assumptions”confirms that we are probing something very essential.
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) :
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.
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 —
— the from-words-to-a-sentence process —
Consider meanings of words, e.g. as vectors (cf. Google):
word 1 word 2 word n...
— the from-words-to-a-sentence process —
What is the meaning the sentence made up of these?
word 1 word 2 word n...
— the from-words-to-a-sentence process —
I.e. how do we/machines produce meanings of sentences?
word 1 word 2 word n...?
— the from-words-to-a-sentence process —
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.
— the from-words-to-a-sentence process —
Information flow within a verb:
verb
object subject
— the from-words-to-a-sentence process —
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
meaning vectors of words
not
grammar
does
—−−−→Alice⊗−−→does⊗−→not⊗−−→like⊗−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not
—−−−→Alice⊗−−→does⊗−→not⊗−−→like⊗−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not
—−−−→Alice⊗−−→does⊗−→not⊗−−→like⊗−−→Bob —
Alice like Bob
meaning vectors of words
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.
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
— MBQC is universal for qubit gates —
H
H
H
H
— MBQC is universal for qubit gates —
H H H H
— MBQC is universal for qubit gates —
⇒ Arbitrary one-qubit unitary
— universality —
Thm. (BC & Ross Duncan) The above described graph-ical language, is universal for computing with qubits.
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