Quantum Functional Programming Language & Its Denotational Semantics

Transcription

1 Quantum Functional Programming Language & Its Denotational Semantics Ichiro Hasuo Dept. Computer Science University of Tokyo Naohiko Hoshino Research Inst. for Math. Sci. Kyoto University Talk based on: I. Hasuo & N. Hoshino, Semantics of Higher-Order Quantum Computation via Geometry of Interaction, to appear in Proc. Logic in Computer Science (LICS), June 2011.

2 Quantum Functional Programming Language & Its Denotational Semantics

3 Quantum Functional Programming Language & Its Denotational Semantics

4 Quantum Functional Programming Language & Its Denotational Semantics What? Why?

5 Overview Why programming language? Why functional programming language? Why semantics? Why denotational semantics?

6 Overview Why programming language? Why functional programming language? Why semantics? Why denotational semantics? Contribution First denotational semantics for full-featured QFPL

7 Quantum Functional Programming Language & Its Denotational Semantics

8 Quantum Functional Programming Language & Its Denotational Semantics Q1. Why programming language?

9 Formalisms We need one for describing/studying quantum algorithms

10 Formalisms We need one for describing/studying quantum algorithms Classical Quantum (Boolean) circuit Quantum circuit [Null-Lobur] [beachhandball.es] Programming language int i,j; int factorial(int k) { j=1; for (i=1; i<=k; i++) j=j*i; return j; }

11 Formalisms We need one for describing/studying quantum algorithms Classical Quantum (Boolean) circuit Quantum circuit [Null-Lobur] [beachhandball.es] Programming language int i,j; int factorial(int k) { j=1; for (i=1; i<=k; i++) j=j*i; return j; } Quantum programming language [Selinger-Valiron]

12 Quantum Programming Languages Imperative (Mlnarik) Functional (Selinger, Valiron)

13 Quantum Programming Languages Imperative (Mlnarik) Functional (Selinger, Valiron) High-level new algorithms?

14 Quantum Programming Imperative (Mlnarik) Languages Functional (Selinger, Valiron) High-level new algorithms? Well-developed techniques for correctness guarantee (verification) Type system Program model checking etc.

15 Quantum Functional Programming Language & Its Denotational Semantics

16 Quantum Functional Programming Language & Its Denotational Semantics Q2. Why functional programming language?

17 (Classical) Functional Programming Languages Computation as evaluation of mathematical functions Avoids (memory) state or mutable data Scheme, Erlang, ML (SML, OCaml), Haskell, F#, int i,j; int factorial(int k) { j=1; for (i=1; i<=k; i++) j=j*i; return j; } Factorial in C fun factorial x = if x = 0 then 1 else x * factorial (x-1) Factorial in ML

18 (Classical) Functional Programming Languages Higher-order computation twice f = λx.f(fx) as fun twice (f : int -> int) : int -> int = fn (x : int) => f (f x) Modularity, code reusability

19 (Classical) Functional Programming Languages Higher-order computation twice f = λx.f(fx) as fun twice (f : int -> int) : int -> int = fn (x : int) => f (f x) Modularity, code reusability Mathematically clean Programs as functions!

20 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum

21 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum Uniform treatment of quantum data and classical data

22 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum quantum data, classical control Uniform treatment of quantum data and classical data

23 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum quantum data, classical control Uniform treatment of quantum data and classical data Nicely enforced by types 0 : int + : int * int -> int

24 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum quantum data, classical control Uniform treatment of quantum data and classical data Nicely enforced by types 0 : int + : int * int -> int new 0 : qbit tt :!bit meas : qbit! bit

25 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum quantum data, classical control Uniform treatment of quantum data and classical data Nicely enforced by types 0 : int + : int * int -> int new 0 : qbit tt :!bit meas : qbit! bit! : duplicable

26 Quantum Functional Programming Mathematical Mathematical transfer from classical to quantum quantum data, classical control Uniform treatment of quantum data and classical data Nicely enforced by types 0 : int + : int * int -> int new 0 : qbit tt :!bit meas : qbit! bit --o : linear function (input is used only once)! : duplicable

27 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] Types:

28 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] Types: A, B ::= n-qbit bit! A A B A B

29 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state Types: A, B ::= n-qbit bit! A A B A B

30 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B

31 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit A, duplicable! A A B A B

32 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B A, duplicable linear function

33 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B A, duplicable linear function pair of A & B

34 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B Programs or terms: A, duplicable linear function pair of A & B M, N ::= x Var λx A.M MN M, N let x A,y B = M in N letrec f A x = M in N new 0 meas n+1 i U cmp m,n

35 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B Programs or terms: A, duplicable linear function pair of A & B M, N ::= x Var λx A.M MN M, N let x A,y B = M in N letrec f A x = M in N new 0 meas n+1 i U cmp m,n function

36 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B Programs or terms: A, duplicable linear function pair of A & B M, N ::= x Var λx A.M MN M, N let x A,y B = M in N letrec f A x = M in N new 0 meas n+1 i U cmp m,n function pairing

37 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B Programs or terms: A, duplicable linear function pair of A & B M, N ::= x Var λx A.M MN M, N let x A,y B = M in N letrec f A x = M in N new 0 meas n+1 i U cmp m,n function pairing recursive def. of func.

38 Our Language qλ l λ-calculus (prototype FPL) Based on [Selinger-Valiron, 2008] n-qubit state (classical) bit Types: A, B ::= n-qbit bit! A A B A B Programs or terms: A, duplicable linear function pair of A & B M, N ::= x Var λx A.M MN M, N let x A,y B = M in N letrec f A x = M in N new 0 meas n+1 i U cmp m,n function pairing recursive def. of func. quantum primitives

39 Our Language qλ l Typing rules: N.B. Only some are shown. Very much simplified!,x : A x : A (Ax.1)! new 0 : qbit (Ax.2)! meas :!(qbit bit) (Ax.2) x : A, M : B λx A.M : A B (.I 1)!, Γ 1 M : A B!, Γ 2 N : A!, Γ 1, Γ 2 MN : B (.E), ( )!, Γ 1 M 1 : A 1!, Γ 2 M 2 : A 2!, Γ 1, Γ 2 M 1,M 2 : A 1 A 2 (.I), ( )

40 Our Language qλ l Typing rules: N.B. Only some are shown. Very much simplified!,x : A x : A (Ax.1)! new 0 : qbit (Ax.2)! meas :!(qbit bit) (Ax.2) x : A, M : B λx A.M : A B (.I 1)!, Γ 1 M : A B!, Γ 2 N : A!, Γ 1, Γ 2 MN : B (.E), ( )!, Γ 1 M 1 : A 1!, Γ 2 M 2 : A 2!, Γ 1, Γ 2 M 1,M 2 : A 1 A 2 (.I), ( )

41 Type Discipline Typable safe Guarantees minimal correctness f AB x A : B f AB y AA meas(x qbit ):bit meas(x qbit ), meas(hx qbit )

42 Type Discipline Typable safe Guarantees minimal correctness f AB x A : B f AB y AA meas(x qbit ):bit meas(x qbit ), meas(hx qbit ) Faulty program fun isvalue t = case t of Num _ => 1 _ => false compile Type error ex.sml: Error: types of rules don't agree [literal] earlier rule(s): term -> int this rule: term -> bool in rule: _ => false

43 Examples In [Selinger-Valiron]; similar in ours Quantum teleportation (Fair) cointoss, repeated Hadamard Flip coin: head Hadamard and flip again tail done

44 Quantum Functional Programming Language & Its Denotational Semantics

45 Quantum Functional Programming Language & Its Denotational Semantics Q3. Why semantics?

46 Semantics

47 Semantics Meaning of a program

48 Semantics Meaning of a program For reasoning about programs

49 Semantics Meaning of a program For reasoning about programs M N: M and N have the same meaning, i.e. computational content

50 Semantics Meaning of a program For reasoning about programs M N: M and N have the same meaning, i.e. computational content?? = λx. (x x) λx. 0

51 Semantics Meaning of a program For reasoning about programs M N: M and N have the same meaning, i.e. computational content?? = λx. (x x) λx. 0 (stupid) sort quick sort

52 Semantics For functional languages: Operational: how the program is transformed/evaluated/reduced (λx. 1+x) Denotational: meaning as a mathematical function λx. 1+x = (function N N, n 1+n)

53 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic Denotational λx. 1+x = (function N N, n 1+n) mathematical static

54 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.)

55 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion

56 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N

57 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N

58 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N

59 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N hard to show directly

60 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N = M = N hard to show directly

61 Operational vs. Denotational Operational (λx. 1+x) reduction-based dynamic (akin to) machine implementation Denotational λx. 1+x = (function N N, n 1+n) mathematical static comes with mathematical reasoning principles (fixed pt. induction, well-fdd induction, etc.) powerful esp. for recursion Goal: M = opr. N = M = N hard to show directly Adequate denotational semantics: M = N M = opr. N

62 Quantum Functional Programming Language & Its Denotational Semantics

63 Quantum Functional Programming Language & Its Denotational Semantics Q4. Why no denotational semantics before?

64 Challenges H = : C 2 C 2, isn t it?

65 Challenges H = : C 2 C 2, isn t it? Quantum data, classical control not clear how to accommodate duplicable data in Hilbert spaces

66 Technical Contributions

67 Technical Contributions Quantum functional programming language Based on [Selinger-Valiron] w/ recursion, classical data (by!)

68 Technical Contributions Quantum functional programming language Based on [Selinger-Valiron] w/ recursion, classical data (by!) Its denotational semantics First one for fully-featured QFPL

69 Full-fledged Semantical Technologies

70 Full-fledged Semantical Technologies

71 Geometry of Interaction Originally by J.-Y. Girard, 1989: Computation as player of a game cf. Game semantics (Abramsky et al., Hyland-Ong) We use categorical formulation: Abramsky, Haghverdi and Scott, 2002

72 Geometry of Interaction Originally by J.-Y. Girard, 1989: Computation as player of a game cf. Game semantics (Abramsky et al., Hyland-Ong) We use categorical formulation: Abramsky, Haghverdi and Scott, 2002 Axiomatization of what is classical control

73 (Particle-Style) Geometry of Interaction (countably many) M = M

74 (Particle-Style) Geometry of Interaction (countably many) M = M

75 (Particle-Style) Geometry of Interaction (countably many) M = M

76 (Particle-Style) Geometry of Interaction token (chocolate) (countably many) M = M

77 (Particle-Style) Geometry of Interaction (countably many) M = M

78 (Particle-Style) Geometry of Interaction (countably many) M = M

79 (Particle-Style) Geometry of Interaction (countably many) M = M

80 MN = M N

81 MN = M N

82 MN = M N

83 MN = M N

84 MN = M N

85 MN MN = M N

86 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

87 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

88 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

89 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

90 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

91 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

92 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

93 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

94 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

95 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

96 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

97 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

98 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

99 MN MN = M N M = λx. x +1 N =2 M = λx. 1 N =2 M = λf. f1 N = λx. (x +1)

100 Quantum Geometry of Interaction (countably many) M = M

101 Quantum Geometry of Interaction Not just a token/ particle, but quantum state! (countably many) M = M

102 Quantum Geometry of Interaction (countably many) M = M Not just a token/ particle, but quantum state!

103 Quantum Geometry of Interaction (countably many) M = M Quantum Data Not just a token/ particle, but quantum state!

104 Quantum Geometry of Interaction Classical Control (countably many) M = M Quantum Data Not just a token/ particle, but quantum state!

105 Quantum Functional Programming Language & Its Denotational Semantics

106 Quantum Functional Programming Language & Its Denotational Semantics Q5. Why quantum computation?

107 Ans. You know why!

108 Conclusions Structured programming & mathematical semantics Quantum data, classical control Geometry of interaction as the essence of classical control

109 Conclusions Structured programming & mathematical semantics Quantum data, classical control Geometry of interaction as the essence of classical control Thank you for your attention! Ichiro Hasuo (Dept. CS, U Tokyo) Naohiko Hoshino (RIMS, Kyoto U)

110 Quantum Programming Languages Functional (Selinger, Valiron)

111 Quantum Programming Languages Imperative (Mlnarik) Functional (Selinger, Valiron)

112 Quantum Programming Languages Imperative (Mlnarik) Functional (Selinger, Valiron) High-level new algorithms?

113 Quantum Programming Languages Imperative (Mlnarik) Functional (Selinger, Valiron) High-level new algorithms? (Sometimes) good handling of quantum vs. classical data No-Cloning vs. Duplicable

114 Quantum Programming Imperative (Mlnarik) Languages Functional (Selinger, Valiron) High-level new algorithms? (Sometimes) good handling of quantum vs. classical data No-Cloning vs. Duplicable Model quantum communication protocols

115 Quantum Functional Programming Language & Its Denotational Semantics

116 Quantum Functional Programming Language & Its Denotational Semantics Q4. Why denotational semantics?

117 M N

118 M N

119 M N

120 M N

121 M N

122 M N

123 M N

124 M N

125 M N