in quantum computation


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1 in computation Kestrel Institute and Oxford University  Oxford, August 2008
2 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements 
3 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements Quantum programming λ 
4 What do programmers do? Quantum programming λ f x Z m 2 f(x) Z n 2 
5 What do programmers do? x f x Z m 2 f(x) Z n 2 x Quantum programming λ  y f(x) y
6 What do programmers do? Quantum programming λ x x B m = C Zm 2 U f f  y f(x) y B n = C Zn 2
7 What do programmers do? Quantum programming λ z H m x z = P x ( 1)z x x f x H m z x = P z ( 1)x z z  U f y f(x) y
8 What do programmers do? Simon s algorithm f : Z m 2 Zn 2 : x f(x) f : Z m+n 2 Z m+n 2 : x, y x, f(x) y U f : C Zm+n 2 C Zm+n 2 : x, y x, f(x) y Simon = (H m id)u f (H m id) 0, 0 = ( 1) x z z, f(x) Quantum programming λ  x,z Z m 2
9 What do programmers do? Simon s algorithm f : Z m 2 Zn 2 : x f(x) f : Z m+n 2 Z m+n 2 : x, y x, f(x) y U f : C Zm+n 2 C Zm+n 2 : x, y x, f(x) y Simon = (H m id)u f (H m id) 0, 0 = ( 1) x z z, f(x) Quantum programming λ  x,z Z m 2... to find a hidden subgroup measurement find c such that f(x + c) = f(x)
10 What do programmers do? Shor s algorithm f : Z m+n q U f : C Zm+n q f : Z m q Z n q : x a x mod q Z m+n q : x, y x, a x + y mod q C Zm+n q : x, y x, a x + y mod q Quantum programming λ  Shor = (FT m id)u f (FT m id) 0, 0 = ( 1) x z z, f(x) x,z Z m q... to find a hidden subgroup measurement find c such that a x+c = a x mod q
11 What do programmers do? Hallgren s algorithm h : Z m Z n : x I x (fraction ideal) h : Z m+n Z m+n : x, y x, y h(x) U h : C Zm+n C Zm+n : x, y x, y h(x) Hallgren = (FT m id)u h (FT m id) d, d = x,z Z m ( 1) x z z, h(x) Quantum programming λ to find a hidden subgroup measurement find R such that h(x + R) = h(x)
12 General design pattern of software engineering ;) CLASS QUAN T Quantum programming λ  MEAS T CLASS
13 Quantum prog. = functional prog. + superposition + entanglement CLASS QUAN T superposition entanglement MEAS T Quantum programming λ  CLASS
14 Standard type universes where programmers work CLASS FSet, FSet op, FFMod R... Quantum programming λ QUAN T FHilb, CPM(FHilb)...  MEAS T FHilb, CPM(FHilb)... CLASS
15 Function in programming CLASS[x] QUAN T [x] FSet[x],FSet op [x], FFMod R[x]... FHilb[x], CPM(FHilb)[x]... Quantum programming λ  MEAS T {x} FHilb{x}, CPM(FHilb){x}... CLASS
16 Function in programming CLASS[x] QUAN T [x] = KL (X ) = KL (X ) Quantum programming λ  MEAS T {x} EM(X ) CLASS
17 λ 2x 3 + 3x + 1 in Z[x] λx. 2x 3 + 3x + 1 in Z Z Z[x] x ad x Quantum programming λ  Z f a f S a
18 λ in cartesian closed categories S p(x) : B in S[x : X] λx. p(x) : B X in S S[x] 1 x X ad x F a Quantum programming λ  F C 1 a FX
19 λ in cartesian closed categories S A q(x) B in S[x : X] A λx.q(x) B X in S S[x] 1 x X ad x F a Quantum programming λ  F C 1 a FX
20 λ in cartesian closed categories A ϕ,x B X X ǫ B A ϕ B X S[x](A, B) S(A, B X ) A q(x) B A λx.q(x) B X S[x] 1 x X ad x Quantum programming λ  S F a F C 1 a FX
21 λ in cartesian closed categories Theorem (Lambek, Adv. in Math. 79) Let S be a cartesian closed category, and S[x] the free cartesian closed category generated by S and x : 1 X. Quantum programming λ 
22 λ in cartesian closed categories Theorem (Lambek, Adv. in Math. 79) Let S be a cartesian closed category, and S[x] the free cartesian closed category generated by S and x : 1 X. Then the inclusion ad x : S S[x] has a right adjoint ab x : S[x] S : A A X and the transpositions A ϕ,x B X X ǫ B A ϕ B X S[x](ad x A, B) S(A, ab x B) A q(x) B A λx.q(x) B X Quantum programming λ  model λ and application.
23 λ in cartesian closed categories Theorem (Lambek, Adv. in Math. 79) Let S be a cartesian closed category, and S[x] the free cartesian closed category generated by S and x : 1 X. Then the inclusion ad x : S S[x] has a right adjoint ab x : S[x] S : A A X and the transpositions A ϕ,x B X X ǫ B A ϕ B X S[x](ad x A, B) S(A, ab x B) A q(x) B A λx.q(x) B X Quantum programming λ  model λ and application. S[x] is isomorphic with the Kleisli category for the power monad ( ) X.
24 κ in cartesian Theorem (Lambek, Adv. in Math. 79) categories Let S be a cartesian category, and S[x] the free cartesian category generated by S and x : 1 X. Then the inclusion ad x : S S[x] has a left adjoint ab x : S[x] S : A X A and the transpositions A x,id X A ϕ B X A ϕ B S[x](A, ad x B) S(ab x A, B) A f x B X A κ x.fx B Quantum programming λ  model first order and application. S[x] is isomorphic with the Kleisli category for the product comonad X ( ).
25 κ in monoidal Theorem (DP, MSCS 95) categories Let C be a monoidal category, and C[x] the free monoidal category generated by C and x : 1 X. Then the strong adjunctions ab x ad x : C C[x] are in onetoone correspondence with the internal comonoid structures on X. The transpositions A X A x A B ϕ C[x](A, ad x B) A f x B Quantum programming λ  X A ϕ B C(ab x A, B) X A κ x.fx B model action and application. C[x] is isomorphic with the Kleisli category for the comonad X ( ), induced by any of the comonoid structures.
26 κ in monoidal categories Task Extend this to. Quantum programming λ 
27 κ in monoidal categories Task Extend this to. Problem Lots of complicated diagram chasing. Quantum programming λ 
28 κ in monoidal categories Task Extend this to. Problem Lots of complicated diagram chasing. Solution? What does mean graphically? Quantum programming λ 
29 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements 
30 Objects X A B D X A B D 
31 Identities X A B D X A B D X A B D X A B D id 
32 Morphisms B X B X h h  X A B D X A B D
33 Tensor (parallel composition) B X C B X C h f h f  X A B D X X A B D X
34 Sequential composition B X C B X C h f h f  X A B D X X A B D X g X A B g D D X X A B D D X
35 Elements (vectors) and coelements (functionals) b B X C h f B X B X b B X C h f  X A B D X X A B D X g X A B g a D D X X A B D D X X a D D X X I D D X
36 Symmetry b B X C h f B X c D g B X B X b B X C h f X A D B X X A c X X A B D X X A B g  X A a B D D X X A B D D X X a D D X X I D D X
37 Polynomials X b B X C h f B X c D g A B D D X a B X B X b B X C h f X A D B X id x X A D B I X A c r X A B D X A B g X A B D D X x a D D x I I D D I 
38 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements Abstraction with pictures Consequences 
39 Abstraction with pictures Theorem (again) Let C be a symmetric monoidal category, and C[x] the free symmetric monoidal category generated by C and x : 1 X. Then there is a onetoone correspondence between adjunctions ab x ad x : C C[x] satisfying 1. ab x (A B) = ab x (A) B 2. η(a B) = η(a) B 3. η I = x and commutative comonoids on X. Abstraction with pictures Consequences 
40 Abstraction with pictures Theorem (again) Let C be a symmetric monoidal category, and C[x] the free symmetric monoidal category generated by C and x : 1 X. Then there is a onetoone correspondence between adjunctions ab x ad x : C C[x] satisfying 1. ab x (A B) = ab x (A) B 2. η(a B) = η(a) B 3. η I = x and commutative comonoids on X. Abstraction with pictures Consequences  C[x] is isomorphic with the Kleisli category for the commutative comonad X ( ), induced by any of the comonoid structures.
41 Proof ( ) Given ab x ad x : C C[x], conditions imply ab x (A) = X A η(a) = x A Abstraction with pictures Consequences 
42 Proof ( ) Therefore the correspondence C ( ab x (A), B ) C[x] ( A, ad x (B) ) Abstraction with pictures Consequences 
43 Proof ( )... is actually C(X A, B) C[x](A, B) Abstraction with pictures Consequences 
44 Proof ( )... with C(X A, B) C[x](A, B) B f X A ( ) (x A) B f A Abstraction with pictures Consequences 
45 Proof ( )... and C(X A, B) C[x](A, B) h f κx. h f Abstraction with pictures Consequences  g g X
46 Proof ( ) The bijection corresponds to the conversion: ( ) (x A) C(X A, B) = C[x](A, B) κx. ( ) ( κx. ϕ(x) x A) = ϕ(x) (βrule Abstraction with pictures Consequences  κx. ( f (x A) ) = f (ηrule
47 Proof ( ) The comonoid structure (X,, ) is = κx. = κx. X X id I Abstraction with pictures Consequences 
48 Proof ( ) The conversion rules imply the comonoid laws = = Abstraction with pictures Consequences  = =
49 Proof ( ) Given (X,, ), use its copying and deleting power, and the symmetries, to normalize every C[x]arrow: ϕ(x) = ϕ (x A) h f h f Abstraction with pictures Consequences  g g
50 Proof ( ) Then set κx. ϕ(x) = ϕ to get ( ) (x A) C(X A, B) = C[x](A, B) κx. ( ) ( κx. ϕ(x) x A) = ϕ(x) (βrule Abstraction with pictures Consequences  κx. ( f (x A) ) = f (ηrule
51 Remark C[x] = C X and C[x, y] = C X Y = KL(X Y ), reduce the finite polynomials to the Kleisli morphisms. Abstraction with pictures Consequences 
52 Remark C[x] = C X and C[x, y] = C X Y = KL(X Y ), reduce the finite polynomials to the Kleisli morphisms. But the extensions C[X], where X is large are also of interest. Abstraction with pictures Consequences 
53 Remark C[x] = C X and C[x, y] = C X Y = KL(X Y ), reduce the finite polynomials to the Kleisli morphisms. But the extensions C[X], where X is large are also of interest. Abstraction with pictures Consequences  Cf. N[N], Set[Set], and CPM(C).
54 Interpretation Abstraction with pictures Consequences 
55 Interpretation Upshot In symmetric monoidal categories, applies just to copiable and deletable data. Abstraction with pictures Consequences 
56 Interpretation Upshot In symmetric monoidal categories, applies just to copiable and deletable data. Definition A vector ϕ C(I, X) is a base vector (or a setlike element) with respect to the operation κx if it can be copied and deleted in C[x] (κx.x x) ϕ = ϕ ϕ Abstraction with pictures Consequences  (κx.id I ) ϕ = id I
57 Interpretation Upshot In symmetric monoidal categories, applies just to copiable and deletable data. Definition A vector ϕ C(I, X) is a base vector (or a setlike element) with respect to the operation κx if it can be copied and deleted in C[x] (κx.x x) ϕ = ϕ ϕ Abstraction with pictures Consequences  (κx.id I ) ϕ = id I Proposition ϕ C(I, X) is a base vector with respect to κx if and only if it is a homomorphism for the comonoid structure X X X I corresponding to κx.
58 Interpretation Upshot In other words, only the base vectors can be substituted for variables. Abstraction with pictures Consequences 
59 Interpretation Upshot In other words, only the base vectors can be substituted for variables. Definition Substitution is a structure preserving ioof C[x] C. Abstraction with pictures Consequences 
60 Interpretation Upshot In other words, only the base vectors can be substituted for variables. Definition Substitution is a structure preserving ioof C[x] C. Corollary The substitution functors C[x] C are in onetoone correspondence with the base vectors of type X. Abstraction with pictures Consequences 
61 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in  Abstraction in Category of measurements
62 Definitions A category C is given with an involutive ioof : C op C.  Abstraction in
63 Definitions A category C is given with an involutive ioof : C op C. A morphism f in a category C is called unitary if f = f 1.  Abstraction in
64 Definitions A category C is given with an involutive ioof : C op C. A morphism f in a category C is called unitary if f = f 1. A (symmetric) monoidal category C is monoidal if its monoidal isomorphisms are unitary.  Abstraction in
65 Using the monoidal s for: vectors: C(A) = C(I, A) scalars: I = C(I, I)  Abstraction in
66 Using the monoidal s 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 (ϕ, ψ : I A) I A ψ I  Abstraction in
67 Using the monoidal s 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 (ϕ, ψ : I A) I A ψ I partial inner product AB : C(A B) C(A) C(B) ( ) (ϕ : I A B, ψ : I A) ϕ I A B ψ B B  Abstraction in
68 Using the monoidal s 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 (ϕ, ψ : I A) I A ψ I partial inner product AB : C(A B) C(A) C(B) ( ) (ϕ : I A B, ψ : I A) ϕ I A B ψ B B  Abstraction in entangled vectors η C(A A), such that ϕ C(A) η ϕ AA = ϕ
69 Using entangled vectors ηa : I A A and, ηb : I B Band their adjoints εa = η A : A A I and εb = η B : B B I  Abstraction in
70 Using entangled vectors ηa : I A A and, ηb : I B Band their adjoints εa = η A : A A I and εb = η B : B B I we can define for every f : A B the dual f : B A  Abstraction in f = B Bη BAA BfA BBA εa A
71 Using entangled vectors ηa : I A A and, ηb : I B Band their adjoints εa = η A : A A I and εb = η B : B B I we can define for every f : A B the dual f : B A  Abstraction in f = B Bη BAA BfA BBA εa A the conjugate f : A B f = f = f
72 Proposition For every object A in a monoidal category C holds (a) (b) = (c),  Abstraction in
73 Proposition For every object A in a monoidal category C holds (a) (b) = (c), where (a) η C(A A) is entangled  Abstraction in
74 Proposition For 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  Abstraction in
75 Proposition For 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 selfadjunction A A, in the sense  Abstraction in A η A A A A A ε A = id A A A η A A A ε A A = id A The three conditions are equivalent if I generates C.
76 Proposition in pictures For every object A in a monoidal category C holds (a) (b) = (c), where (a) (b) ψ η η ψ ϕ = = ψ ψ ϕ  Abstraction in (c) η η = = X η η
77 Definition A object in a monoidal category is an object equipped with the structure from the preceding proposition.  Abstraction in
78 Definition A object in a monoidal category is an object equipped with the structure from the preceding proposition.  Remark The subcategory of objects in any monoidal category is compact (strongly compact) with all objects selfadjoint. Abstraction in
79 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x].  Abstraction in
80 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x]. Then the following conditions are equivalent:  Abstraction in
81 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x]. Then the following conditions are equivalent: (a) ad x : C C[x] creates : C[x] op C[x] such that x x = x x = id I.  Abstraction in
82 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x]. Then the following conditions are equivalent: (a) ad x : C C[x] creates : C[x] op C[x] such that x x = x x = id I. (b) η = and ε = η = realize X X.  Abstraction in
83 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x]. Then the following conditions are equivalent: (a) ad x : C C[x] creates : C[x] op C[x] such that x x = x x = id I. (b) η = and ε = η = realize X X. (c) (X ) ( X) = = ( X) (X )  Abstraction in
84 Abstraction in Theorem Let C be a monoidal category, and X X X I a comonoid that induces ab x ad x : C C[x]. Then the following conditions are equivalent: (a) ad x : C C[x] creates : C[x] op C[x] such that x x = x x = id I. (b) η = and ε = η = realize X X. (c) (X ) ( X) = = ( X) (X )  Abstraction in where X X X I is the induced monoid = =
85 Abstraction in Theorem in pictures (b) = = X  Abstraction in
86 Abstraction in Theorem in pictures (c) = =  Abstraction in
87 Proof of (b)= (c) Lemma 1 If (b) holds then  = = Abstraction in
88 Proof of (b)= (c) Then (c) also holds because = = =  Abstraction in =
89 Proof of Lemma 1 Lemma 2 If η ε = = X ε η  Abstraction in then η ε ε = = X X ε ε η η η
90 Proof of Lemma 1 Using Lemma 2, and the fact that (b) implies = =, we get ε ε = η = η =  Abstraction in η η = η
91 The message of the proof  Abstraction in
92 The message of the proof There is more to categories than just diagram chasing.  Abstraction in
93 The message of the proof There is more to categories than just diagram chasing.  There is also picture chasing. Abstraction in
94 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem.  Abstraction in
95 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem. Let C be the category of classical objects and comonoid homomorphisms in C.  Abstraction in
96 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem. Let C be the category of classical objects and comonoid homomorphisms in C. Question: What is classical about classical objects?  Abstraction in
97 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem. Let C be the category of classical objects and comonoid homomorphisms in C. Question: What is classical about classical objects? classical structure structure  Abstraction in
98 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem. Let C be the category of classical objects and comonoid homomorphisms in C. Question: What is classical about classical objects? classical structure structure Answer: classical elements = base vectors  Abstraction in
99 Definition A classical object in a monoidal category C is a comonoid X X X I satisfying the equivalent conditions from the preceding theorem. Let C be the category of classical objects and comonoid homomorphisms in C. Question: What is classical about classical objects? classical structure structure Answer: classical elements = base vectors is neither injective or surjective  Abstraction in
100 Consequences Upshot The Frobenius condition (c) assures the preservation of the operation under.  Abstraction in
101 Consequences Upshot The Frobenius condition (c) assures the preservation of the operation under. This leads to entanglement.  Abstraction in
102 Consequences Proposition The vectors C(X) of any classical object X form a algebra.  Abstraction in
103 Consequences Proposition The vectors C(X) of any classical object X form a algebra.  Abstraction in ϕ ψ = (ϕ ψ) ǫ = ϕ = ϕ = ϕ
104 Definition Two vectors ϕ, ψ C(A) in a monoidal category are orthonormal if their inner product is idempotent: ϕ ψ = ϕ ψ 2  Abstraction in
105 Definition Two vectors ϕ, ψ C(A) in a monoidal category are orthonormal 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 category are orthonormal. Abstraction in
106 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. id I )ϕ = id I }  Abstraction in
107 Definition A classical object X is standard if it is (regularly) generated by its base vectors in the sense B(X) = {ϕ C(X) (κx. x x)ϕ = ϕ ϕ (κx. id I )ϕ = id I } f, g C(X, Y). ( ϕ B(X). fϕ = gϕ) = f = g  Abstraction in A base is regular if C(X, Y) C(Y) B(X) splits.
108 Proposition 1. All standard classical structures, that an object X C may carry, induce the bases with the same number of elements.  Abstraction in
109 Proposition 1. All standard classical structures, that an object X C may carry, induce the bases with the same number of elements.  Abstraction in Proposition 2. Let X C be a classical object with a regular base. Then the equipotent regular bases on any Y C are in onetoone correspondence with the unitaries X Y.
110 Definition A qubit type in an arbitrary monoidal category C is a classical object B with a unitary H of order 2. The induced bases are usually denoted by 0, 1, and +,.  Abstraction in
111 Definition A qubit type in an arbitrary monoidal category C is a classical object B with a unitary H of order 2. The induced bases are usually denoted by 0, 1, and +,.  Computing with qubits A monoidal category with B suffices for the basic algorithms. Abstraction in
112 Definition A qubit type in an arbitrary monoidal category C is a classical object B with a unitary H of order 2. The induced bases are usually denoted by 0, 1, and +,.  Computing with qubits A monoidal category with B suffices for the basic algorithms. A Klein group of unitaries on B suffices for all teleportation and dense coding schemes. Abstraction in
113 Proposition (Coecke, Vicary, P) Every classical object X in FHilb is regular, and X = C n. The classical structure is induced by a base B(X) = { i i n }, with i = ii = 1 n n i i=1  Abstraction in
114 Proposition (Coecke, Vicary, P) Every classical object X in FHilb is regular, and X = C n. The classical structure is induced by a base B(X) = { i i n }, with i = ii = 1 n n i i=1  Abstraction in Moreover, FHilb FSet
115 Proof A algebra in FHilb is a C algebra.  Abstraction in
116 Proof A algebra in FHilb is a C algebra. Thus for a classical X FHilb, : FHilb(X) FHilb(X, X) ( ) ϕ I X ( X ϕ X X X X ) is a representation of a commutative C algebra.  Abstraction in
117 Proof Working through the GelfandNaimark duality, we get X = C n  Abstraction in
118 Proof Working through the GelfandNaimark duality, we get X = C n  Abstraction in because the spectrum of a commutative finitely dimensional C algebra is a discrete set n of minimal central projections, while the representing spaces are the full matrix algebras C(1)
119 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements 
120 Category of measurements ((this was not presented)) 
121 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements 
122 Claim: Task: Simple algorithms have simple categorical semantics. Implement and analyze algorithms in nonstandard models: network computation, data mining. 
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