Foundations of Quantum Programming

Transcription

1 Foundations of Quantum Programming Mingsheng Ying University of Technology Sydney, Australia Institute of Software, Chinese Academy of Sciences Tsinghua University, China

2 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

3 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

4 How to program quantum computers? Quantum algorithms: Deutsch-Josza, Grover, Shor, HHL,... Quantum algorithm zoo (

5 How to program quantum computers? Quantum algorithms: Deutsch-Josza, Grover, Shor, HHL,... Quantum algorithm zoo ( Quantum computers: IBM Q (20 qubits Nov 2017; 50 qubits 2018) Google (49 qubits 2018) Intel (17 qubits Oct 2017)

6 Quantum programming languages

7 Quantum programming languages Open

8 Quantum programming languages Open

9 Quantum programming languages Open

10 Quantum programming languages Open

11 Quantum programming languages Open Project

12 Quantum programming languages Open Project

13 Quantum programming languages Open Project Q

14 This lecture Principles underlying all of the quantum programming languages

15 This lecture Principles underlying all of the quantum programming languages Not the languages themselves.

16 This lecture Principles underlying all of the quantum programming languages Not the languages themselves. Reference: M. S. Ying, Foundations of Quantum Programming, Morgan Kaufmann 2016, Chapters 3 and 4.

17 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics

18 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics Turing-complete?

19 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics Turing-complete? Comiplers

20 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics Turing-complete? Comiplers Program analysis: Termination,...

21 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics Turing-complete? Comiplers Program analysis: Termination,... How to verify correctness of your programs?

22 Programming languages and tools Programming languages are notations used for specifying, organising and reasoning about computations. [R. Sethi, Programming languages: Concepts and Constructs] Semantics Turing-complete? Comiplers Program analysis: Termination,... How to verify correctness of your programs?...

23 Quantum circuit vs Quantum programs Example: Quantum walk on a circle with an absorbing boundary. H d = span{ L, R } the direction space, L and R indicate directions left and right, respectively.

24 Quantum circuit vs Quantum programs Example: Quantum walk on a circle with an absorbing boundary. H d = span{ L, R } the direction space, L and R indicate directions left and right, respectively. H p = span{ 0, 1,..., n 1 } the position space with orthonormal basis states, the vector i denotes position i for each 0 i n 1.

25 Quantum circuit vs Quantum programs Example: Quantum walk on a circle with an absorbing boundary. H d = span{ L, R } the direction space, L and R indicate directions left and right, respectively. H p = span{ 0, 1,..., n 1 } the position space with orthonormal basis states, the vector i denotes position i for each 0 i n 1. The state space of the walk H = H p H d.

26 Quantum circuit vs Quantum programs Example: Quantum walk on a circle with an absorbing boundary. H d = span{ L, R } the direction space, L and R indicate directions left and right, respectively. H p = span{ 0, 1,..., n 1 } the position space with orthonormal basis states, the vector i denotes position i for each 0 i n 1. The state space of the walk H = H p H d. The initial state 0 p L d.

27 Quantum circuit vs Quantum programs Each step of the walk:

28 Quantum circuit vs Quantum programs Each step of the walk: 1. Measure the system to see whether the current position is 1 (absorbing boundary). If yes, terminates; otherwise, continues: M = {M yes = 1 1 I d, M no = I M yes };

29 Quantum circuit vs Quantum programs Each step of the walk: 1. Measure the system to see whether the current position is 1 (absorbing boundary). If yes, terminates; otherwise, continues: M = {M yes = 1 1 I d, M no = I M yes }; ( 2. A coin-tossing operator C = direction space; ) is applied on the

30 Quantum circuit vs Quantum programs Each step of the walk: 1. Measure the system to see whether the current position is 1 (absorbing boundary). If yes, terminates; otherwise, continues: M = {M yes = 1 1 I d, M no = I M yes }; ( 2. A coin-tossing operator C = direction space; 3. A shift operator S = ) is applied on the n 1 n 1 i 1 i L L + i 1 i R R i=0 i=0 is performed on the state space H.

31 Quantum circuit vs Quantum programs Each step of the walk: 1. Measure the system to see whether the current position is 1 (absorbing boundary). If yes, terminates; otherwise, continues: M = {M yes = 1 1 I d, M no = I M yes }; ( 2. A coin-tossing operator C = direction space; 3. A shift operator S = ) is applied on the n 1 n 1 i 1 i L L + i 1 i R R i=0 i=0 is performed on the state space H. Question: How to specify it in the circuit language?

32 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

33 Classical while-language S ::= skip u := t S 1 ; S 2 if b then S 1 else S 2 fi while b do S od. Conditional statement can be generalised to case statement: or more compactly: if G 1 S 1 G 2 S 2... G n S n fi if ( i G i S i ) fi

34 Quantum while-language Alphabet: a countably infinite set Var of quantum variables q, q, q 0, q 1, q 2,...

35 Quantum while-language Alphabet: a countably infinite set Var of quantum variables q, q, q 0, q 1, q 2,... Each quantum variable q Var has a type H q (a Hilbert space), e.g. Boolean = H 2, integer = H.

36 Quantum while-language Alphabet: a countably infinite set Var of quantum variables q, q, q 0, q 1, q 2,... Each quantum variable q Var has a type H q (a Hilbert space), e.g. Boolean = H 2, integer = H. A quantum register is a finite sequence q = q 1,..., q n of distinct quantum variables. State Hilbert space: H q = n H qi. i=1

37 Syntax of Quantum Programs S ::= skip q := 0 q := U[q] S 1 ; S 2 if ( m M[q] = m S m ) fi while M[q] = 1 do S od.

38 Syntax of Quantum Programs S ::= skip q := 0 q := U[q] S 1 ; S 2 if ( m M[q] = m S m ) fi while M[q] = 1 do S od. Exercise 1 Write quantum walk as a program in quantum while-language.

39 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

40 Notations partial density operator: positive operator ρ, tr(ρ) 1.

41 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H.

42 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = H q. q Var

43 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = H q. q Var E empty program; i.e. termination.

44 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = q Var E empty program; i.e. termination. Configuration: pair S, ρ, where: H q.

45 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = q Var H q. E empty program; i.e. termination. Configuration: pair S, ρ, where: 1. S is a quantum program or the empty program E;

46 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = q Var H q. E empty program; i.e. termination. Configuration: pair S, ρ, where: 1. S is a quantum program or the empty program E; 2. ρ D(H all ), denoting the (global) state of quantum variables.

47 Notations partial density operator: positive operator ρ, tr(ρ) 1. D(H) the set of partial density operators in H. State Hilbert space of all quantum variables: H all = q Var H q. E empty program; i.e. termination. Configuration: pair S, ρ, where: 1. S is a quantum program or the empty program E; 2. ρ D(H all ), denoting the (global) state of quantum variables. Transition: S, ρ S, ρ

48 Operational Semantics (SK) skip, ρ E, ρ (IN) q := 0, ρ E, ρ q 0 (UT) where : ρ q 0 = 0 q i ρ i q 0 i q := U[q], ρ E, UρU (SC) S 1, ρ S 1, ρ S 1 ; S 2, ρ S 1 ; S 2, ρ where : E; S 2 = S 2.

49 Operational Semantics (IF) if ( m M[q] = m S m ) fi, ρ S m, M m ρm m for each possible outcome m of measurement M = {M m }. (L0) while M[q] = 1 do S od, ρ E, M 0 ρm 0 (L1) while M[q] = 1 do S od, ρ S; while M[q] = 1 do S od, M1 ρm 1

50 Computation of Programs 1. A (finite or infinite) transition sequence of program S with input ρ D(H all ): S, ρ S 1, ρ 1... S n, ρ n S n+1, ρ n+1... such that ρ n 0 for all n (except the last n in the case of a finite sequence).

51 Computation of Programs 1. A (finite or infinite) transition sequence of program S with input ρ D(H all ): S, ρ S 1, ρ 1... S n, ρ n S n+1, ρ n+1... such that ρ n 0 for all n (except the last n in the case of a finite sequence). 2. If this sequence cannot be extended, it is called a computation.

52 Computation of Programs 1. A (finite or infinite) transition sequence of program S with input ρ D(H all ): S, ρ S 1, ρ 1... S n, ρ n S n+1, ρ n+1... such that ρ n 0 for all n (except the last n in the case of a finite sequence). 2. If this sequence cannot be extended, it is called a computation. If it is finite and its last configuration is E, ρ, we say it terminates in ρ.

53 Computation of Programs 1. A (finite or infinite) transition sequence of program S with input ρ D(H all ): S, ρ S 1, ρ 1... S n, ρ n S n+1, ρ n+1... such that ρ n 0 for all n (except the last n in the case of a finite sequence). 2. If this sequence cannot be extended, it is called a computation. If it is finite and its last configuration is E, ρ, we say it terminates in ρ. If it is infinite, we say it diverges.

54 Notation Write: S, ρ n S, ρ if there are configurations S 1, ρ 1,..., S n 1, ρ n 1 such that S, ρ S 1, ρ 1... S n 1, ρ n 1 S, ρ,

55 Notation Write: S, ρ n S, ρ if there are configurations S 1, ρ 1,..., S n 1, ρ n 1 such that S, ρ S 1, ρ 1... S n 1, ρ n 1 S, ρ, Write for the reflexive and transitive closures of : S, ρ S, ρ if and only if S, ρ n S, ρ for some n 0.

56 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

57 Semantic Function Semantic function of program S: S : D(H all ) D(H all ) S (ρ) = { ρ : S, ρ E, ρ }

58 Semantic Function Semantic function of program S: S : D(H all ) D(H all ) S (ρ) = { ρ : S, ρ E, ρ } Exercise 2 Try to compute semantic function of your quantum walk program.

59 Structural Representation 1. skip (ρ) = ρ.

60 Structural Representation 1. skip (ρ) = ρ. 2. q := 0 (ρ) = i 0 q i ρ i q 0.

61 Structural Representation 1. skip (ρ) = ρ. 2. q := 0 (ρ) = i 0 q i ρ i q q := U[q] (ρ) = UρU.

62 Structural Representation 1. skip (ρ) = ρ. 2. q := 0 (ρ) = i 0 q i ρ i q q := U[q] (ρ) = UρU. 4. S 1 ; S 2 (ρ) = S 2 ( S 1 (ρ)).

63 Structural Representation 1. skip (ρ) = ρ. 2. q := 0 (ρ) = i 0 q i ρ i q q := U[q] (ρ) = UρU. 4. S 1 ; S 2 (ρ) = S 2 ( S 1 (ρ)). 5. if ( m M[q] = m S m ) fi (ρ) = m S m (M m ρm m).

64 Structural Representation 1. skip (ρ) = ρ. 2. q := 0 (ρ) = i 0 q i ρ i q q := U[q] (ρ) = UρU. 4. S 1 ; S 2 (ρ) = S 2 ( S 1 (ρ)). 5. if ( m M[q] = m S m ) fi (ρ) = m S m (M m ρm m). 6. while M[q] = 1 do S od (ρ) =???

65 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying:

66 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L;

67 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L;

68 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L.

69 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L. x L is called the least element when x y for all y L.

70 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L. x L is called the least element when x y for all y L. x L is called an upper bound of a subset X L if y x for all x X.

71 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L. x L is called the least element when x y for all y L. x L is called an upper bound of a subset X L if y x for all x X. x is called the least upper bound of X, written x = X, if

72 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L. x L is called the least element when x y for all y L. x L is called an upper bound of a subset X L if y x for all x X. x is called the least upper bound of X, written x = X, if x is an upper bound of X;

73 Basic Lattice Theory A partial order (L, ): L is a nonempty set, is a binary relation on L satisfying: 1. Reflexivity: x x for all x L; 2. Antisymmetry: x y and y x imply x = y for all x, y L; 3. Transitivity: x y and y z imply x z for all x, y, z L. x L is called the least element when x y for all y L. x L is called an upper bound of a subset X L if y x for all x X. x is called the least upper bound of X, written x = X, if x is an upper bound of X; for any upper bound y of X, x y.

74 Basic Lattice Theory A complete partial order (CPO) is a partial order (L, ):

75 Basic Lattice Theory A complete partial order (CPO) is a partial order (L, ): 1. it has the least element 0;

76 Basic Lattice Theory A complete partial order (CPO) is a partial order (L, ): 1. it has the least element 0; 2. n=0 x n exists for any increasing sequence {x n }: x 0... x n x n+1...

77 Basic Lattice Theory A complete partial order (CPO) is a partial order (L, ): 1. it has the least element 0; 2. n=0 x n exists for any increasing sequence {x n }: x 0... x n x n+1... A function f from L into itself is continuous if ( ) f x n = n n f (x n ) for any increasing sequence {x n } in L.

78 Knaster-Tarski Theorem Let (L, ) be a CPO and function f : L L continuous. Then f has the least fixed point µf = f (n) (0) n=0 where { f (0) (0) = 0, f (n+1) (0) = f (f (n) (0)) for n 0.

79 CPO of Partial Density Operators Löwner order: operators A B B A is positive.

80 CPO of Partial Density Operators Löwner order: operators A B B A is positive. (D(H), ) is a CPO with the zero operator 0 H as its least element. Exercise 3 Prove the above statement for finite-dimensional H.

81 CPO of Super-operators Each super-operator in H is a continuous function from (D(H), ) into itself.

82 CPO of Super-operators Each super-operator in H is a continuous function from (D(H), ) into itself. QO(H) the set of superoperators in Hilbert space H.

83 CPO of Super-operators Each super-operator in H is a continuous function from (D(H), ) into itself. QO(H) the set of superoperators in Hilbert space H. Löwner order between operators can be lifted to a partial order between super-operators: E F E(ρ) F(ρ) for all ρ D(H)

84 CPO of Super-operators Each super-operator in H is a continuous function from (D(H), ) into itself. QO(H) the set of superoperators in Hilbert space H. Löwner order between operators can be lifted to a partial order between super-operators: (QO(H), ) is a CPO. E F E(ρ) F(ρ) for all ρ D(H)

85 Syntactic Approximation abort denotes a program such that abort (ρ) = 0 Hall for all ρ D(H).

86 Syntactic Approximation abort denotes a program such that abort (ρ) = 0 Hall for all ρ D(H). Write: while while M[q] = 1 do S od.

87 Syntactic Approximation abort denotes a program such that abort (ρ) = 0 Hall for all ρ D(H). Write: while while M[q] = 1 do S od. For integer k 0, the kth syntactic approximation while (k) of while: while (0) abort, while (k+1) if M[q] = 0 skip 1 S; while (k) fi

88 Semantic Function of Loops while = while (k), k=0 where stands for the least upper bound in CPO (QO (H all ), ).

89 Semantic Function of Loops while = while (k), k=0 where stands for the least upper bound in CPO (QO (H all ), ). Fixed Point Characterisation For any ρ D(H all ): ( )) while (ρ) = M 0 ρm 0 ( S + while M 1 ρm 1.

90 Semantic Function of Loops while = while (k), k=0 where stands for the least upper bound in CPO (QO (H all ), ). Fixed Point Characterisation For any ρ D(H all ): ( )) while (ρ) = M 0 ρm 0 ( S + while M 1 ρm 1. Exercise 4 Prove the above equality.

91 Termination and Divergence Probabilities For any quantum program S and ρ D(H all ): tr( S (ρ)) tr(ρ).

92 Termination and Divergence Probabilities For any quantum program S and ρ D(H all ): tr( S (ρ)) tr(ρ). tr( S (ρ)) is the probability that program S with input ρ terminates.

93 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

94 Quantum predicates = Quantum effects A quantum effect is a (Hermitian) operator 0 M I; that is, 0 tr(mρ) 1 for all density operators.

95 Quantum predicates = Quantum effects A quantum effect is a (Hermitian) operator 0 M I; that is, 0 tr(mρ) 1 for all density operators. tr(mρ) may be interpreted as the expected degree to which quantum state ρ satisfies quantum predicate M.

96 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where:

97 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where: S is a quantum program; Partial Correctness, Total Correctness

98 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where: S is a quantum program; P, Q are quantum predicates in Hall. Partial Correctness, Total Correctness

99 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where: S is a quantum program; P, Q are quantum predicates in Hall. P is called the precondition, Q the postcondition. Partial Correctness, Total Correctness

100 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where: S is a quantum program; P, Q are quantum predicates in Hall. P is called the precondition, Q the postcondition. Partial Correctness, Total Correctness Partial correctness: If an input to program S satisfies precondition P, then either S does not terminate, or it terminates in a state satisfying postcondition Q.

101 Correctness Formulas A correctness formula (Hoare triple) is a statement of the form: {P}S{Q} where: S is a quantum program; P, Q are quantum predicates in Hall. P is called the precondition, Q the postcondition. Partial Correctness, Total Correctness Partial correctness: If an input to program S satisfies precondition P, then either S does not terminate, or it terminates in a state satisfying postcondition Q. Total correctness: If an input to program S satisfies precondition P, then S must terminate and it terminates in a state satisfying postcondition Q.

102 Partial Correctness, Total Correctness (Continued) The correctness formula {P}S{Q} is true in the sense of total correctness, written = tot {P}S{Q}, if: tr(pρ) tr(q S (ρ)) for all ρ D(H all ), where S is the semantic function of S.

103 Partial Correctness, Total Correctness (Continued) The correctness formula {P}S{Q} is true in the sense of total correctness, written = tot {P}S{Q}, if: tr(pρ) tr(q S (ρ)) for all ρ D(H all ), where S is the semantic function of S. The correctness formula {P}S{Q} is true in the sense of partial correctness, written = par {P}S{Q}, if: for all ρ D(H all ). tr(pρ) tr(q S (ρ)) + [tr(ρ) tr( S (ρ))]

104 Proof System for Partial Correctness (Ax Sk) {P}Skip{P} (Ax In) If type(q) = Boolean, then { 0 q 0 P 0 q q 0 P 0 q 1 }q := 0 {P} If type(q) = integer, then { n q 0 P 0 q n n= } q := 0 {P} (Ax UT) {U PU}q := Uq{P}

105 Proof System for Partial Correctness (Continued) (R SC) {P}S 1 {Q} {Q}S 2 {R} {P}S 1 ; S 2 {R} (R IF) {P m }S m {Q} for all m { m M mp m M m } if ( m M[q] = m Sm ) fi{q} (R LP) {Q}S { M 0 PM 0 + M 1 QM } 1 {M 0 PM 0 + M 1 QM 1}while M[q] = 1 do S od{p} (R Or) P P {P }S{Q } Q Q {P}S{Q}

106 Soundness Theorem For any quantum while-program S and quantum predicates P, Q P(H all ): qpd {P}S{Q} implies = par {P}S{Q}.

107 Soundness Theorem For any quantum while-program S and quantum predicates P, Q P(H all ): qpd {P}S{Q} implies = par {P}S{Q}. Exercise 5 Prove soundness theorem.

108 Soundness Theorem For any quantum while-program S and quantum predicates P, Q P(H all ): qpd {P}S{Q} implies = par {P}S{Q}. Exercise 5 Prove soundness theorem. (Relative) Completeness Theorem For any quantum while-program S and quantum predicates P, Q P(H all ): = par {P}S{Q} implies qpd {P}S{Q}.

109 Bound (Ranking) Functions Let P P(H all ) be a quantum predicate, real number ɛ > 0.

110 Bound (Ranking) Functions Let P P(H all ) be a quantum predicate, real number ɛ > 0. A function t : D(H all ) ω is a (P, ɛ)-bound function of quantum loop while M[q] = 1 do S od if for all ρ D(H all ):

111 Bound (Ranking) Functions Let P P(H all ) be a quantum predicate, real number ɛ > 0. A function t : D(H all ) ω is a (P, ɛ)-bound function of quantum loop if for all ρ D(H all ): 1. t ( S ( M 1 ρm 1)) t(ρ); while M[q] = 1 do S od

112 Bound (Ranking) Functions Let P P(H all ) be a quantum predicate, real number ɛ > 0. A function t : D(H all ) ω is a (P, ɛ)-bound function of quantum loop while M[q] = 1 do S od if for all ρ D(H all ): 1. t ( S ( M 1 ρm 1)) t(ρ); 2. tr(pρ) ɛ implies ( ( )) t S M 1 ρm 1 < t(ρ)

113 Proof System for Total Correctness {Q}S{M 0 PM 0 + M 1 QM 1} (R LT) for any ɛ > 0, t ɛ is a (M 1 QM 1, ɛ) bound function of loop while M[q] = 1 do S od {M 0 PM 0 + M 1 QM 1}while M[q] = 1 do S od{p}

114 Soundness Theorem For any quantum program S and quantum predicates P, Q P(H all ): qtd {P}S{Q} implies = tot {P}S{Q}.

115 Soundness Theorem For any quantum program S and quantum predicates P, Q P(H all ): qtd {P}S{Q} implies = tot {P}S{Q}. (Relative) Completeness Theorem For any quantum program S and quantum predicates P, Q P(H all ): = tot {P}S{Q} implies qtd {P}S{Q}.

116 Outline Introduction Syntax of Quantum Programs Operational Semantics Denotational Semantics (Floyd-)Hoare Logic for Quantum Programs Research Problems

117 Research problems 1. Compute the expected running time of the quantum walk.

118 Research problems 1. Compute the expected running time of the quantum walk. 2. Develop a logic for recursive quantum programs. Further reading [1] M. S. Ying, S. G. Ying and X. D. Wu, Invariants of quantum programs: characterisations and generation, POPL [2] Y. J. Li and M. S. Ying, Algorithmic analysis of termination problem for quantum programs, POPL 2018.

119 Research problems 1. Compute the expected running time of the quantum walk. 2. Develop a logic for recursive quantum programs. 3. Parallel or distributed quantum programs? Further reading [1] M. S. Ying, S. G. Ying and X. D. Wu, Invariants of quantum programs: characterisations and generation, POPL [2] Y. J. Li and M. S. Ying, Algorithmic analysis of termination problem for quantum programs, POPL 2018.

120 Research problems 1. Compute the expected running time of the quantum walk. 2. Develop a logic for recursive quantum programs. 3. Parallel or distributed quantum programs? 4. Improve the invariant generation and termination analysis algorithms for quantum programs. Further reading [1] M. S. Ying, S. G. Ying and X. D. Wu, Invariants of quantum programs: characterisations and generation, POPL [2] Y. J. Li and M. S. Ying, Algorithmic analysis of termination problem for quantum programs, POPL 2018.

121 Thank You!