geometry of abstraction in quantum computation · future work geometry of abstraction in quantum...
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![Page 1: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/1.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Geometry of abstractionin quantum computation
Dusko Pavlovic
Kestrel Instituteand
Oxford University
Oxford, August 2008
![Page 2: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/2.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 3: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/3.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 4: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/4.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
f
x∈Zm2
f (x)∈Zn2
![Page 5: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/5.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
f (x)⊕y
x
y
f
x
x∈Zm2
f (x)∈Zn2
![Page 6: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/6.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
f
Uf
|y〉
|x〉 |x〉∈B⊗m= CZ
m2
|f (x)⊕y〉∈B⊕n= CZ
n2
![Page 7: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/7.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
f
Uf
H⊗m H⊗m
|y〉
|x〉 |x〉
|f (x)⊕y〉
|z〉
|x〉=P
z(−1)x·z |z〉
|z〉
|z〉=P
x(−1)z·x |x〉
![Page 8: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/8.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
Simon’s algorithm
f : Zm2 −→ Z
n2 : x 7→ f (x)
f ′ : Zm+n2 −→ Z
m+n2 : x , y 7→ x , f (x)⊕ y
Uf : CZ
m+n2 −→ C
Zm+n2 : |x , y〉 7→ |x , f (x)⊕ y〉
Simon = (H⊗m ⊗ id)Uf (H⊗m ⊗ id) |0,0〉
=∑
x,z∈Zm2
(−1)x·z |z, f (x)〉
![Page 9: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/9.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
Simon’s algorithm
f : Zm2 −→ Z
n2 : x 7→ f (x)
f ′ : Zm+n2 −→ Z
m+n2 : x , y 7→ x , f (x)⊕ y
Uf : CZ
m+n2 −→ C
Zm+n2 : |x , y〉 7→ |x , f (x)⊕ y〉
Simon = (H⊗m ⊗ id)Uf (H⊗m ⊗ id) |0,0〉
=∑
x,z∈Zm2
(−1)x·z |z, f (x)〉
. . . to find a hidden subgroupmeasurement find c such that f (x + c) = f (x)
![Page 10: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/10.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
Shor’s algorithm
f : Zmq −→ Z
nq : x 7→ ax mod q
f ′ : Zm+nq −→ Z
m+nq : x , y 7→ x ,ax + y mod q
Uf : CZ
m+nq −→ C
Zm+nq : |x , y〉 7→ |x ,ax + y mod q〉
Shor = (FTm ⊗ id)Uf (FTm ⊗ id) |0,0〉=
∑
x,z∈Zmq
(−1)x·z |z, f (x)〉
. . . to find a hidden subgroupmeasurement find c such that ax+c = ax mod q
![Page 11: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/11.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionWhat do quantum programmers do?
Hallgren’s algorithm
h : Zm −→ Z
n : x 7→ Ix (fraction ideal)
h′ : Zm+n −→ Z
m+n : x , y 7→ x , y − h(x)
Uh : CZm+n −→ C
Zm+n: |x , y〉 7→ |x , y − h(x)〉
Hallgren = (FTm ⊗ id)Uh(FTm ⊗ id) |d , d̃〉=
∑
x,z∈Zm
(−1)x·z |z,h(x)〉
. . . to find a hidden subgroupmeasurement find R such that h(x + R) = h(x)
![Page 12: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/12.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionGeneral design pattern of quantum software engineering ;)
CLASS
QUANT
MEAS ′T
|−〉
∡
CLASS
|−|
![Page 13: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/13.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionQuantum prog. = functional prog. + superposition + entanglement
CLASS
QUANT
MEAS ′T
|−〉
∡
CLASS
superposition
entanglement
|−|
![Page 14: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/14.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionStandard type universes where quantum programmers work
CLASS
QUANT
MEAS ′T
|−〉
∡
CLASS
FHilb, CPM(FHilb)...
FHilb, CPM(FHilb)...
FSet, FSetop℘℘, FFModR ...
|−|
![Page 15: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/15.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionFunction abstraction in quantum programming
CLASS[x ]
QUANT [x ]
MEAS ′T {x}
|−〉
∡
CLASS
FHilb[x], CPM(FHilb)[x]...
FHilb{x}, CPM(FHilb){x}...
FSet[x],FSetop℘℘[x], FFModR [x]...
|−|
![Page 16: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/16.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
IntroductionFunction abstraction in quantum programming
CLASS[x ]
QUANT [x ]
MEAS ′T {x}
|−〉
∡
CLASS
∼= KL (X⊗)
≃ EM (X⊗)
∼= KL (X×)
|−|
![Page 17: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/17.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction
Sf
x
a
fa
adxZ[x ]
Z
2x3 + 3x + 1 in Z[x ]
λx . 2x3 + 3x + 1 in Z −→ Z
![Page 18: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/18.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
adx1 x−→ X
1 a−→ FX
S
S[x ]
F
Fa
C
λx . p(x) : BX in S
p(x) : B in S[x : X ]
![Page 19: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/19.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
adx1 x−→ X
1 a−→ FX
S
S[x ]
F
Fa
C
Aλx.q(x)−→ BX in S
Aq(x)−→ B in S[x : X ]
![Page 20: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/20.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
adx1 x−→ X
1 a−→ FX
S
S[x ]
F
Fa
C
S(A,BX )
S[x ](A,B) Aq(x)−→B
Aλx.q(x)−→ BX
A〈ϕ,x〉−→ BX×X ǫ
→B
Aϕ
−→BX
![Page 21: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/21.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
Theorem (Lambek, Adv. in Math. 79)
Let S be a cartesian closed category, and S[x ] the freecartesian closed category generated by S and x : 1→ X.
![Page 22: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/22.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
Theorem (Lambek, Adv. in Math. 79)
Let S be a cartesian closed category, and S[x ] the freecartesian closed category generated by S and x : 1→ X.
Then the inclusion adx : S −→ S[x ] has a right adjointabx : S[x ] −→ S : A 7→ AX and the transpositions
S(A,abxB)
S[x ](adxA,B) Aq(x)−→B
Aλx.q(x)−→ BX
A〈ϕ,x〉−→ BX×X
ǫ→B
Aϕ
−→BX
model λ-abstraction and application.
![Page 23: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/23.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
λ-abstraction in cartesian closed categories
Theorem (Lambek, Adv. in Math. 79)
Let S be a cartesian closed category, and S[x ] the freecartesian closed category generated by S and x : 1→ X.
Then the inclusion adx : S −→ S[x ] has a right adjointabx : S[x ] −→ S : A 7→ AX and the transpositions
S(A,abxB)
S[x ](adxA,B) Aq(x)−→B
Aλx.q(x)−→ BX
A〈ϕ,x〉−→ BX×X
ǫ→B
Aϕ
−→BX
model λ-abstraction and application.
S[x ] is isomorphic with the Kleisli category for the powermonad (−)X .
![Page 24: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/24.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
κ-abstraction in cartesian categories
Theorem (Lambek, Adv. in Math. 79)
Let S be a cartesian category, and S[x ] the freecartesian category generated by S and x : 1→ X.
Then the inclusion adx : S −→ S[x ] has a left adjointabx : S[x ] −→ S : A 7→ X × A and the transpositions
S(abxA,B)
S[x ](A,adxB) Afx−→B
X×Aκx .fx−→B
A〈x,id〉−→ X×A
ϕ→B
X×Aϕ
−→B
model first order abstraction and application.
S[x ] is isomorphic with the Kleisli category for the productcomonad X × (−).
![Page 25: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/25.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
κ-abstraction in monoidal categories
Theorem (DP, MSCS 95)
Let C be a monoidal category, and C[x ] the freemonoidal category generated by C and x : 1→ X.
Then the strong adjunctions abx ⊣ adx : C −→ C[x ] are inone-to-one correspondence with the internal comonoidstructures on X. The transpositions
C(abxA,B)
C[x ](A,adxB) Afx−→B
X⊗Aκx .fx−→B
Ax⊗A−→X⊗A
ϕ→B
X⊗Aϕ
−→B
model action abstraction and application.
C[x ] is isomorphic with the Kleisli category for thecomonad X ⊗ (−), induced by any of the comonoidstructures.
![Page 26: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/26.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
κ-abstraction in monoidal categories
TaskExtend this to ‡-monoidal categories.
![Page 27: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/27.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
κ-abstraction in monoidal categories
TaskExtend this to ‡-monoidal categories.
ProblemLots of complicated diagram chasing.
![Page 28: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/28.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
IntroductionQuantum programming
λ-abstraction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
κ-abstraction in monoidal categories
TaskExtend this to ‡-monoidal categories.
ProblemLots of complicated diagram chasing.
Solution?What does abstraction mean graphically?
![Page 29: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/29.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 30: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/30.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Objects
X A B D X⊗A⊗B⊗D
![Page 31: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/31.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Identities
X A B D X⊗A⊗B⊗D
DX A B X⊗A⊗B⊗D
id
![Page 32: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/32.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Morphisms
h
X A
B X
B D
h
B⊗X
X⊗A⊗B⊗D
![Page 33: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/33.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Tensor (parallel composition)
h
X A
f
X
B X C
B D
B⊗X⊗C
h⊗f
X⊗A⊗B⊗D⊗X
![Page 34: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/34.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Sequential composition
h
X
X
A
g
f
X
B X C
D D
B D
X⊗A⊗B⊗D⊗D⊗X
B⊗X⊗C
X⊗A⊗B⊗g
h⊗f
X⊗A⊗B⊗D⊗X
![Page 35: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/35.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Elements (vectors) and coelements (functionals)
h
X
X
A
g
f
X
B X C
D D
B D
X⊗A⊗B⊗D⊗D⊗X
B⊗X⊗C
b
a
B⊗X
B⊗X⊗b
X⊗A⊗B⊗g
h⊗f
X⊗I⊗D⊗D⊗XX⊗a⊗D⊗D⊗X
X⊗A⊗B⊗D⊗X
![Page 36: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/36.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Symmetry
h
X XA
g
f
X
B X C
c
D D
B
B
D
X⊗A⊗B⊗D⊗D⊗X
B⊗X⊗C
b
a
B⊗X
B⊗X⊗b
X⊗A⊗B⊗g
X⊗A⊗D⊗B⊗X
h⊗f
X⊗I⊗D⊗D⊗XX⊗a⊗D⊗D⊗X
X⊗A⊗B⊗D⊗X
X⊗A⊗c⊗X
![Page 37: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/37.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Polynomials
h
X XA
g
f
X
B X C
c
D D
B
B
D
I⊗I⊗D⊗D⊗I
X⊗A⊗B⊗D⊗D⊗X
X⊗A⊗B⊗D
X⊗A⊗D⊗B⊗I
B⊗X⊗C
X⊗A⊗c⊗r
x⊗a⊗D⊗D⊗x
b
a
B⊗X
B⊗X⊗b
X⊗A⊗B⊗g
X⊗A⊗D⊗B⊗X
id⊗x
h⊗f
![Page 38: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/38.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 39: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/39.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Abstraction with pictures
Theorem (again)
Let C be a symmetric monoidal category, and C[x ] thefree symmetric monoidal category generated by C andx : 1→ X.
Then there is a one-to-one correspondence between◮ adjunctions abx ⊣ adx : C −→ C[x ] satisfying
1. abx (A⊗ B) = abx (A)⊗ B2. η(A⊗ B) = η(A) ⊗ B3. ηI = x
and◮ commutative comonoids on X.
![Page 40: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/40.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Abstraction with pictures
Theorem (again)
Let C be a symmetric monoidal category, and C[x ] thefree symmetric monoidal category generated by C andx : 1→ X.
Then there is a one-to-one correspondence between◮ adjunctions abx ⊣ adx : C −→ C[x ] satisfying
1. abx (A⊗ B) = abx (A)⊗ B2. η(A⊗ B) = η(A) ⊗ B3. ηI = x
and◮ commutative comonoids on X.
C[x ] is isomorphic with the Kleisli category for thecommutative comonad X ⊗ (−), induced by any of thecomonoid structures.
![Page 41: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/41.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
Given abx ⊣ adx : C −→ C[x ],conditions 1.-3. imply
◮ abx (A) = X ⊗ A◮ η(A) = x ⊗ A
![Page 42: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/42.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
Therefore the correspondence
C(abx(A),B
)C[x ]
(A,adx(B)
)
![Page 43: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/43.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
. . . is actually
C(X ⊗ A,B) C[x ](A,B)
![Page 44: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/44.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
. . . with
C(X ⊗ A,B) C[x ](A,B)
f f
X A
B B
A
(−)◦(x⊗A)
![Page 45: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/45.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
. . . and
C(X ⊗ A,B) C[x ](A,B)
h
g
fh
g
f
X
∆
κx.
![Page 46: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/46.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
The bijection corresponds to the conversion:
C(X ⊗ A,B) C[x ](A,B)
(κx . ϕ(x)
)◦
(x ⊗ A) = ϕ(x)
κx .(f ◦ (x ⊗ A)
)= f
κx.
(−)◦(x⊗A)
(η-rule
(β-rule
∼=
![Page 47: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/47.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
The comonoid structure (X ,∆,⊤) is
⊤
∆ =
=
κx .
κx .
X X
idI
![Page 48: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/48.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↓)
The conversion rules imply the comonoid laws
⊤
∆
∆=
∆
∆
∆
=
∆
=
∆=
⊤
![Page 49: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/49.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↑)
Given (X ,∆,⊤), use its copying and deleting power, andthe symmetries, to normalize every C[x ]-arrow:
h
g
f h
g
f
∆
=ϕ(x) ϕ ◦ (x ⊗ A)
![Page 50: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/50.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Proof (↑)
Then set κx . ϕ(x) = ϕ to get
C(X ⊗ A,B) C[x ](A,B)
(κx . ϕ(x)
)◦
(x ⊗ A) = ϕ(x)
κx .(f ◦ (x ⊗ A)
)= f
κx.
(−)◦(x⊗A)
(η-rule
(β-rule
∼=
![Page 51: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/51.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Remark
◮ C[x ] ∼= CX⊗ and C[x , y ] ∼= CX⊗Y⊗ = KL(X ⊗ Y⊗),reduce the finite polynomials to the Kleislimorphisms.
![Page 52: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/52.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Remark
◮ C[x ] ∼= CX⊗ and C[x , y ] ∼= CX⊗Y⊗ = KL(X ⊗ Y⊗),reduce the finite polynomials to the Kleislimorphisms.
◮ But the extensions C[X ], where X is largeare also of interest.
![Page 53: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/53.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Remark
◮ C[x ] ∼= CX⊗ and C[x , y ] ∼= CX⊗Y⊗ = KL(X ⊗ Y⊗),reduce the finite polynomials to the Kleislimorphisms.
◮ But the extensions C[X ], where X is largeare also of interest.
◮ Cf. N[N], Set[Set], and CPM(C).
![Page 54: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/54.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
![Page 55: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/55.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn symmetric monoidal categories,abstraction applies just to copiable and deletable data.
![Page 56: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/56.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn symmetric monoidal categories,abstraction applies just to copiable and deletable data.
DefinitionA vector ϕ ∈ C(I,X ) is a base vector (or a set-likeelement) with respect to the abstraction operation κx if itcan be copied and deleted in C[x ]
(κx .x ⊗ x) ◦ ϕ = ϕ⊗ ϕ(κx .idI) ◦ ϕ = idI
![Page 57: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/57.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn symmetric monoidal categories,abstraction applies just to copiable and deletable data.
DefinitionA vector ϕ ∈ C(I,X ) is a base vector (or a set-likeelement) with respect to the abstraction operation κx if itcan be copied and deleted in C[x ]
(κx .x ⊗ x) ◦ ϕ = ϕ⊗ ϕ(κx .idI) ◦ ϕ = idI
Propositionϕ ∈ C(I,X ) is a base vector with respect to κx if and onlyif it is a homomorphism for the comonoid structure
X ⊗ X ∆←− X ⊤−→ I corresponding to κx .
![Page 58: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/58.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn other words,only the base vectors can be substituted for variables.
![Page 59: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/59.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn other words,only the base vectors can be substituted for variables.
DefinitionSubstitution is a structure preserving ioof C[x ] −→ C.
![Page 60: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/60.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstractionAbstraction with pictures
Consequences
Geometry of‡-abstraction
Measurements
Future work
Interpretation
UpshotIn other words,only the base vectors can be substituted for variables.
DefinitionSubstitution is a structure preserving ioof C[x ] −→ C.
CorollaryThe substitution functors C[x ] −→ C are in one-to-onecorrespondence with the base vectors of type X .
![Page 61: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/61.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 62: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/62.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
Definitions
A ‡-category C is given with an involutive ioof‡ : Cop −→ C.
![Page 63: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/63.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
Definitions
A ‡-category C is given with an involutive ioof‡ : Cop −→ C.
A morphism f in a ‡-category C is called unitary iff ‡ = f−1.
![Page 64: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/64.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
Definitions
A ‡-category C is given with an involutive ioof‡ : Cop −→ C.
A morphism f in a ‡-category C is called unitary iff ‡ = f−1.
A (symmetric) monoidal category C is ‡-monoidal if itsmonoidal isomorphisms are unitary.
![Page 65: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/65.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing the monoidal notations for:◮ vectors: C(A) = C(I,A)◮ scalars: I = C(I, I)
![Page 66: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/66.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing the monoidal notations for:◮ vectors: C(A) = C(I,A)◮ scalars: I = C(I, I)
in every ‡-monoidal category we can define
◮ abstract inner product
〈−|−〉A : C(A)× C(A) −→ I
(ϕ, ψ : I −→ A) 7−→(
Iϕ→ A
ψ‡
→ I)
![Page 67: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/67.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing the monoidal notations for:◮ vectors: C(A) = C(I,A)◮ scalars: I = C(I, I)
in every ‡-monoidal category we can define
◮ abstract inner product
〈−|−〉A : C(A)× C(A) −→ I
(ϕ, ψ : I −→ A) 7−→(
Iϕ→ A
ψ‡
→ I)
◮ partial inner product
〈−|−〉AB : C(A⊗ B)× C(A) −→ C(B)
(ϕ : I → A⊗ B, ψ : I → A) 7−→(
Iϕ→ A⊗ B
ψ‡⊗B−→ B)
![Page 68: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/68.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing the monoidal notations for:◮ vectors: C(A) = C(I,A)◮ scalars: I = C(I, I)
in every ‡-monoidal category we can define
◮ abstract inner product
〈−|−〉A : C(A)× C(A) −→ I
(ϕ, ψ : I −→ A) 7−→(
Iϕ→ A
ψ‡
→ I)
◮ partial inner product
〈−|−〉AB : C(A⊗ B)× C(A) −→ C(B)
(ϕ : I → A⊗ B, ψ : I → A) 7−→(
Iϕ→ A⊗ B
ψ‡⊗B−→ B)
◮ entangled vectors η ∈ C(A⊗ A), such that ∀ϕ ∈ C(A)
〈η|ϕ〉AA = ϕ
![Page 69: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/69.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing
◮ entangled vectors ηA : I −→ A⊗ A and,ηB : I −→ B ⊗ Band
◮ their adjoints εA = η‡A : A⊗ A −→ I andεB = η‡B : B ⊗ B −→ I
![Page 70: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/70.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing
◮ entangled vectors ηA : I −→ A⊗ A and,ηB : I −→ B ⊗ Band
◮ their adjoints εA = η‡A : A⊗ A −→ I andεB = η‡B : B ⊗ B −→ I
we can define for every f : A −→ B
◮ the dual f ∗ : B −→ A
f ∗ = BBη−→ BAA BfA−→ BBA εA−→ A
![Page 71: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/71.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categoriesUsing
◮ entangled vectors ηA : I −→ A⊗ A and,ηB : I −→ B ⊗ Band
◮ their adjoints εA = η‡A : A⊗ A −→ I andεB = η‡B : B ⊗ B −→ I
we can define for every f : A −→ B
◮ the dual f ∗ : B −→ A
f ∗ = BBη−→ BAA BfA−→ BBA εA−→ A
◮ the conjugate f∗ : A −→ B
f∗ = f ∗‡ = f ‡∗
![Page 72: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/72.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
PropositionFor every object A in a ‡-monoidal category C holds(a) ⇐⇒ (b)⇐= (c),
![Page 73: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/73.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
PropositionFor every object A in a ‡-monoidal category C holds(a) ⇐⇒ (b)⇐= (c), where
(a) η ∈ C(A ⊗ A) is entangled
![Page 74: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/74.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
PropositionFor every object A in a ‡-monoidal category C holds(a) ⇐⇒ (b)⇐= (c), where
(a) η ∈ C(A ⊗ A) is entangled
(b) ε = η‡ ∈ C(A⊗ A, I) internalizes the inner product
ε ◦ (ψ∗ ⊗ ϕ) = 〈ϕ|ψ〉
![Page 75: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/75.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
PropositionFor every object A in a ‡-monoidal category C holds(a) ⇐⇒ (b)⇐= (c), where
(a) η ∈ C(A ⊗ A) is entangled
(b) ε = η‡ ∈ C(A⊗ A, I) internalizes the inner product
ε ◦ (ψ∗ ⊗ ϕ) = 〈ϕ|ψ〉
(c) (η, ε) realize the self-adjunction A ⊣ A, in the sense
Aη⊗A−→ A⊗ A⊗ A A⊗ε−→ A = idA
AA⊗η−→ A⊗ A⊗ A ε⊗A−→ A = idA
The three conditions are equivalent if I generates C.
![Page 76: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/76.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
‡-monoidal categories
Proposition in picturesFor every object A in a ‡-monoidal category C holds(a) ⇐⇒ (b)⇐= (c), where
= =(c)η‡
X
η‡
ηη
=η‡
(b)
(a)
ϕψ ϕ
ψ‡
=ψ‡
η ψ
![Page 77: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/77.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Quantum objects
Definition
A quantum object in a ‡-monoidal category is an objectequipped with the structure from the precedingproposition.
![Page 78: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/78.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Quantum objects
Definition
A quantum object in a ‡-monoidal category is an objectequipped with the structure from the precedingproposition.
RemarkThe subcategory of quantum objects in any ‡-monoidalcategory is ‡-compact (strongly compact) — with allobjects self-adjoint.
![Page 79: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/79.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
![Page 80: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/80.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
Then the following conditions are equivalent:
![Page 81: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/81.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
Then the following conditions are equivalent:
(a) adx : C −→ C[x ] creates ‡ : C[x ]op −→ C[x ]such that 〈x |x〉 = x‡ ◦ x = idI .
![Page 82: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/82.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
Then the following conditions are equivalent:
(a) adx : C −→ C[x ] creates ‡ : C[x ]op −→ C[x ]such that 〈x |x〉 = x‡ ◦ x = idI .
(b) η = ∆ ◦ ⊥ and ε = η‡ = ∇ ◦⊤ realize X ⊣ X.
![Page 83: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/83.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
Then the following conditions are equivalent:
(a) adx : C −→ C[x ] creates ‡ : C[x ]op −→ C[x ]such that 〈x |x〉 = x‡ ◦ x = idI .
(b) η = ∆ ◦ ⊥ and ε = η‡ = ∇ ◦⊤ realize X ⊣ X.
(c) (X ⊗∇) ◦ (∆⊗ X ) = ∆ ◦ ∇ = (∇⊗ X ) ◦ (X ⊗∆)
![Page 84: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/84.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem
Let C be a ‡-monoidal category,
and X ⊗ X ∆←− X ⊤−→ I a comonoid that inducesabx ⊣ adx : C −→ C[x ].
Then the following conditions are equivalent:
(a) adx : C −→ C[x ] creates ‡ : C[x ]op −→ C[x ]such that 〈x |x〉 = x‡ ◦ x = idI .
(b) η = ∆ ◦ ⊥ and ε = η‡ = ∇ ◦⊤ realize X ⊣ X.
(c) (X ⊗∇) ◦ (∆⊗ X ) = ∆ ◦ ∇ = (∇⊗ X ) ◦ (X ⊗∆)
where X ⊗ X ∇−→ X ⊥←− I is the induced monoid
∇ = ∆‡
⊥ = ⊤‡
![Page 85: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/85.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem in pictures
∇
∆
=∇
∆
=(b)
⊥ ⊥
⊤⊤
X
![Page 86: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/86.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Abstraction in ‡-monoidal categories
Theorem in pictures
∇
∆ ∇
∆=
∇
∆
=(c)
![Page 87: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/87.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof of (b)=⇒(c)
Lemma 1
If (b) holds then
∇
∆
∆=∇
∆
=
⊥ ⊥
![Page 88: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/88.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof of (b)=⇒(c)
Then (c) also holds because
∇
∆
∇
∆
=
=
∇∆
∇
⊥
= ∇∆
∇
⊥
=
![Page 89: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/89.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof of Lemma 1
Lemma 2
= =ε
X
ε
ηη
then
If
= =ε
X η
ε
η
ε
η
ε
η
X
![Page 90: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/90.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof of Lemma 1
Using Lemma 2, and the fact that (b) implies∇ = ∆‡ = ∆∗, we get
=η
η
∆ =∇
ε
η
η
∇
ε
=
=η
∇
![Page 91: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/91.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
The message of the proof
![Page 92: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/92.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
The message of the proof
There is more to categories than just diagram chasing.
![Page 93: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/93.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
The message of the proof
There is more to categories than just diagram chasing.
There is also picture chasing.
![Page 94: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/94.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
![Page 95: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/95.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
Let C∆ be the category of classical objects and comonoidhomomorphisms in C.
![Page 96: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/96.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
Let C∆ be the category of classical objects and comonoidhomomorphisms in C.
Question: What is classical about classical objects?
![Page 97: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/97.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
Let C∆ be the category of classical objects and comonoidhomomorphisms in C.
Question: What is classical about classical objects?◮ classical structure 99K quantum structure
![Page 98: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/98.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
Let C∆ be the category of classical objects and comonoidhomomorphisms in C.
Question: What is classical about classical objects?◮ classical structure 99K quantum structure
Answer: classical elements = base vectors
![Page 99: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/99.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Classical objects
Definition
A classical object in a ‡-monoidal category C is a
comonoid X ⊗ X ∆←− X ⊤−→ I satisfying the equivalentconditions from the preceding theorem.
Let C∆ be the category of classical objects and comonoidhomomorphisms in C.
Question: What is classical about classical objects?◮ classical structure 99K quantum structure
Answer: classical elements = base vectors◮ 99K is neither injective or surjective
![Page 100: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/100.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Consequences
Upshot
The Frobenius condition (c) assures the preservation ofthe abstraction operation under ‡.
![Page 101: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/101.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Consequences
Upshot
The Frobenius condition (c) assures the preservation ofthe abstraction operation under ‡.This leads to entanglement.
![Page 102: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/102.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Consequences
Proposition
The vectors C(X ) of any classical object X form a⋆-algebra.
![Page 103: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/103.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Consequences
Proposition
The vectors C(X ) of any classical object X form a⋆-algebra.
ϕ · ψ = ∇ ◦ (ϕ⊗ ψ)
ǫ = ⊥ϕ⋆ = ϕ‡∗ = ϕ∗‡
![Page 104: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/104.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Definition
Two vectors ϕ,ψ ∈ C(A) in a ‡-monoidal category areorthonormal if their inner product is idempotent:
〈ϕ | ψ〉 = 〈ϕ | ψ〉2
![Page 105: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/105.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Definition
Two vectors ϕ,ψ ∈ C(A) in a ‡-monoidal category areorthonormal if their inner product is idempotent:
〈ϕ | ψ〉 = 〈ϕ | ψ〉2
Proposition
Any two base vectors are orthonormal.In particular, any two variables in a polynomial categoryare orthonormal.
![Page 106: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/106.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Definition
A classical object X is standard if it is (regularly)generated by its base vectors
B(X ) = {ϕ ∈ C(X )| (κx . x ⊗ x)ϕ = ϕ⊗ ϕ∧ (κx . idI)ϕ = idI}
![Page 107: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/107.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Definition
A classical object X is standard if it is (regularly)generated by its base vectors
B(X ) = {ϕ ∈ C(X )| (κx . x ⊗ x)ϕ = ϕ⊗ ϕ∧ (κx . idI)ϕ = idI}
in the sense
∀f ,g ∈ C(X ,Y ). (∀ϕ ∈ B(X ). fϕ = gϕ) =⇒ f = g
A base is regular if C(X ,Y ) C(Y )B(X) splits.
![Page 108: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/108.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Proposition 1.All standard classical structures, that an object X ∈ Cmay carry, induce the bases with the same number ofelements.
![Page 109: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/109.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Proposition 1.All standard classical structures, that an object X ∈ Cmay carry, induce the bases with the same number ofelements.
Proposition 2.Let X ∈ C be a classical object with a regular base. Thenthe equipotent regular bases on any Y ∈ C are inone-to-one correspondence with the unitaries X −→ Y .
![Page 110: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/110.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
DefinitionA qubit type in an arbitrary ‡-monoidal category C is aclassical object B with a unitary H of order 2. The inducedbases are usually denoted by |0〉, |1〉, and |+〉, |−〉.
![Page 111: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/111.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
DefinitionA qubit type in an arbitrary ‡-monoidal category C is aclassical object B with a unitary H of order 2. The inducedbases are usually denoted by |0〉, |1〉, and |+〉, |−〉.
Computing with qubitsA ‡-monoidal category with B suffices for the basicquantum algorithms.
![Page 112: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/112.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
DefinitionA qubit type in an arbitrary ‡-monoidal category C is aclassical object B with a unitary H of order 2. The inducedbases are usually denoted by |0〉, |1〉, and |+〉, |−〉.
Computing with qubitsA ‡-monoidal category with B suffices for the basicquantum algorithms.A Klein group of unitaries on B suffices for allteleportation and dense coding schemes.
![Page 113: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/113.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Proposition (Coecke, Vicary, P)
Every classical object X in FHilb is regular, and X ∼= Cn.
The classical structure is induced by a baseB(X ) =
{|i〉 | i ≤ n
}, with
∆|i〉 = |ii〉
⊤ =1√n
n∑
i=1
|i〉
![Page 114: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/114.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Base
Proposition (Coecke, Vicary, P)
Every classical object X in FHilb is regular, and X ∼= Cn.
The classical structure is induced by a baseB(X ) =
{|i〉 | i ≤ n
}, with
∆|i〉 = |ii〉
⊤ =1√n
n∑
i=1
|i〉
Moreover,
FHilb∆ ≃ FSet
![Page 115: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/115.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof
A ⋆-algebra in FHilb is a C⋆-algebra.
![Page 116: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/116.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof
A ⋆-algebra in FHilb is a C⋆-algebra.
Thus for a classical X ∈ FHilb,
∇ : FHilb(X ) −→ FHilb(X ,X )(
Iϕ−→ X
)7−→
(X
ϕ⊗X−→ X ⊗ X ∇−→ X)
is a representation of a commutative C⋆-algebra.
![Page 117: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/117.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof
Working through the Gelfand-Naimark duality, we get
X ∼= Cn
![Page 118: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/118.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction‡-monoidal categories
Quantum objects
Abstraction in‡-monoidal categories
Classical objects
Base
Measurements
Future work
Proof
Working through the Gelfand-Naimark duality, we get
X ∼= Cn
— because the spectrum of a commutative finitelydimensional C∗-algebra is a discrete set n of minimalcentral projections, while the representing spaces are thefull matrix algebras C(1)
![Page 119: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/119.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 120: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/120.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Category of measurements((this was not presented))
![Page 121: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/121.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
OutlineIntroduction
Quantum programmingλ-abstraction
Graphical notation
Geometry of abstractionAbstraction with picturesConsequences
Geometry of ‡-abstraction‡-monoidal categoriesQuantum objectsAbstraction in ‡-monoidal categoriesClassical objectsBase
Category of measurements
Future work
![Page 122: Geometry of abstraction in quantum computation · Future work Geometry of abstraction in quantum computation Dusko Pavlovic Kestrel Institute and Oxford University Oxford, August](https://reader033.vdocuments.site/reader033/viewer/2022060403/5f0ebd7a7e708231d440b503/html5/thumbnails/122.jpg)
Geometry ofquantum
abstraction
Dusko Pavlovic
Introduction
Graphicalnotation
Geometry ofabstraction
Geometry of‡-abstraction
Measurements
Future work
Future work
Claim: Simple quantum algorithms have simplecategorical semantics.
Task: Implement and analyze quantum algorithmsin nonstandard models: networkcomputation, data mining.