Quantum Recursion and Second Quantisation

Transcription

1 Quantum Recursion and Second Quantisation Mingsheng Ying University of Technology Sydney, Australia and Tsinghua University, China

2 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

3 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

4 Recursion is one of the central ideas of computer science!

5 Recursion is one of the central ideas of computer science! Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004]. Quantum while-loops [Ying, Feng, Acta Informatica 2010].

6 Recursion is one of the central ideas of computer science! Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004]. Quantum while-loops [Ying, Feng, Acta Informatica 2010]. Quantum data, classical control [Selinger]: The control flow of quantum recursions is classical: branchings are determined by the outcomes of quantum measurements.

7 Recursion is one of the central ideas of computer science! Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004]. Quantum while-loops [Ying, Feng, Acta Informatica 2010]. Quantum data, classical control [Selinger]: The control flow of quantum recursions is classical: branchings are determined by the outcomes of quantum measurements. Example: while M[q] = 1 do S od

8 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

9 How to define quantum control flow? Quantum data, quantum control [Altenkirch and Grattage, LICS 2005]

10 How to define quantum control flow? Quantum data, quantum control [Altenkirch and Grattage, LICS 2005] Coined quantum case statement [Ying, Foundations of Quantum Programming, Morgan Kaufmann 2016] Quantum Case Statement

11 How to define quantum control flow? Quantum data, quantum control [Altenkirch and Grattage, LICS 2005] Coined quantum case statement [Ying, Foundations of Quantum Programming, Morgan Kaufmann 2016] Quantum Case Statement Classical Case Statement: if b then S 1 else S 2 fi

12 How to define quantum control flow? Quantum data, quantum control [Altenkirch and Grattage, LICS 2005] Coined quantum case statement [Ying, Foundations of Quantum Programming, Morgan Kaufmann 2016] Quantum Case Statement Classical Case Statement: if b then S 1 else S 2 fi Classical Case Statement in Quantum Programming: if M[q] = 0 S 1 fi 1 S 2

13 Quantum Case Statement Introduce external quantum coin c: H c = span{ 0, 1 }

14 Quantum Case Statement Introduce external quantum coin c: H c = span{ 0, 1 } U 0, U 1 unitary operators on quantum system q: H q

15 Quantum Case Statement Introduce external quantum coin c: H c = span{ 0, 1 } U 0, U 1 unitary operators on quantum system q: H q Quantum case statement employing coin c: qif [c] 0 U 0 [q] 1 U 1 [q] fiq

16 Quantum Case Statement Introduce external quantum coin c: H c = span{ 0, 1 } U 0, U 1 unitary operators on quantum system q: H q Quantum case statement employing coin c: qif [c] 0 U 0 [q] 1 U 1 [q] fiq Semantics: unitary operator U in H c H q : U 0, ψ = 0 U 0 ψ, U 1, ψ = 1 U 1 ψ

17 Quantum Choice V a unitary operator in H c.

18 Quantum Choice V a unitary operator in H c. Quantum choice of U 0 [q], U 1 [q] with coin-tossing V[c]: U 0 [q] V[c] U 1 [q] def = V[c]; qif [c] 0 U 0 [q] fiq 1 U 1 [q]

19 External Quantum Coin Superpositions of time evolutions of a quantum system [Aharonov, Anandan, Popescu, Vaidman, Plysical Review Letters 1990].

20 External Quantum Coin Superpositions of time evolutions of a quantum system [Aharonov, Anandan, Popescu, Vaidman, Plysical Review Letters 1990]. Quantum walks [Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC 2001; Aharonov, Ambainis, Kempe, Vazirani, STOC 2001]. Quantum walk

21 External Quantum Coin Superpositions of time evolutions of a quantum system [Aharonov, Anandan, Popescu, Vaidman, Plysical Review Letters 1990]. Quantum walks [Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC 2001; Aharonov, Ambainis, Kempe, Vazirani, STOC 2001]. Quantum walk One-dimensional random walk - a particle moves on a line marked by integers Z; at each step it moves one position left or right, depending on the flip of a fair coin.

22 External Quantum Coin Superpositions of time evolutions of a quantum system [Aharonov, Anandan, Popescu, Vaidman, Plysical Review Letters 1990]. Quantum walks [Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC 2001; Aharonov, Ambainis, Kempe, Vazirani, STOC 2001]. Quantum walk One-dimensional random walk - a particle moves on a line marked by integers Z; at each step it moves one position left or right, depending on the flip of a fair coin. Hadamard walk - a quantum variant.

23 Quantum walk Hilbert space H d H p.

24 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right.

25 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right. H p = span{ n : n Z}, n the position marked by integer n.

26 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right. H p = span{ n : n Z}, n the position marked by integer n. One step of Hadamard walk: W = T(H I)

27 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right. H p = span{ n : n Z}, n the position marked by integer n. One step of Hadamard walk: W = T(H I) Translation T: unitary operator in Hd H p T L, n = L, n 1, T R, n = R, n + 1

28 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right. H p = span{ n : n Z}, n the position marked by integer n. One step of Hadamard walk: W = T(H I) Translation T: unitary operator in Hd H p Hadamard transform Hd : T L, n = L, n 1, T R, n = R, n + 1 H = 1 2 ( )

29 Quantum walk Hilbert space H d H p. H d = span{ L, R }, L, R direction Left and Right. H p = span{ n : n Z}, n the position marked by integer n. One step of Hadamard walk: W = T(H I) Translation T: unitary operator in Hd H p Hadamard transform Hd : T L, n = L, n 1, T R, n = R, n + 1 H = 1 2 ( ) Hadamard walk repeated applications of operator W.

30 Look quantum walk in a new way! Define left and right translation T L and T R in space H p : T L n = n 1, T R n = n + 1

31 Look quantum walk in a new way! Define left and right translation T L and T R in space H p : T L n = n 1, T R n = n + 1 Translation operator T is a quantum case statement: T = qif [d] L T L [p] R T R [p] fiq

32 Look quantum walk in a new way! Define left and right translation T L and T R in space H p : T L n = n 1, T R n = n + 1 Translation operator T is a quantum case statement: T = qif [d] L T L [p] R T R [p] fiq Single-step walk operator W is a quantum choice: T L [p] H[d] T R [p]

33 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

34 Quantum recursion with quantum control flow can be defined based on quantum case statement

35 Quantum recursion with quantum control flow can be defined based on quantum case statement A Quantum Programming Language Alphabet: Two sets of quantum variables: 1. principal system variables p, q,...;

36 Quantum recursion with quantum control flow can be defined based on quantum case statement A Quantum Programming Language Alphabet: Two sets of quantum variables: 1. principal system variables p, q,...; 2. coin variables c, d,... Syntax P ::= X abort skip U[q, c] P 1 ; P 2 qif [c]( i i P i ) fiq

37 Quantum recursion with quantum control flow can be defined based on quantum case statement A Quantum Programming Language Alphabet: Two sets of quantum variables: 1. principal system variables p, q,...; 2. coin variables c, d,... A set of procedure identifiers X, X 1, X 2,... Syntax P ::= X abort skip U[q, c] P 1 ; P 2 qif [c]( i i P i ) fiq

38 Quantum choice [P(c)] i ( i P i ) = P; qif [c] ( i i P i ) end.

39 Quantum choice [P(c)] i ( i P i ) = P; qif [c] ( i i P i ) end. Superposition of Programs Quantum choice first runs a coin-tossing subprogram P followed by an alternation of a family of subprograms P 0, P 1,... Coin-tossing subprogram P creates a superposition of the execution paths of P 0, P 1,...,

40 Quantum choice [P(c)] i ( i P i ) = P; qif [c] ( i i P i ) end. Superposition of Programs Quantum choice first runs a coin-tossing subprogram P followed by an alternation of a family of subprograms P 0, P 1,... Coin-tossing subprogram P creates a superposition of the execution paths of P 0, P 1,..., During the execution of the alternation, each P i is running along its own path, but the whole program is executed in a superposition of execution paths of P 0, P 1,...

41 Semantics of quantum programs without recursion H Hilbert space of the principal system.

42 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers:

43 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers: 1. If P = abort, then P = 0 (the zero operator in H);

44 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers: 1. If P = abort, then P = 0 (the zero operator in H); 2. If P = skip, then P = I (the identity operator in H);

45 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers: 1. If P = abort, then P = 0 (the zero operator in H); 2. If P = skip, then P = I (the identity operator in H); 3. If P = U[q, c], then P = U;

46 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers: 1. If P = abort, then P = 0 (the zero operator in H); 2. If P = skip, then P = I (the identity operator in H); 3. If P = U[q, c], then P = U; 4. If P = P 1 ; P 2, then P = P 2 P 1 ;

47 Semantics of quantum programs without recursion H Hilbert space of the principal system. Semantics P of a program P without procedure identifiers: 1. If P = abort, then P = 0 (the zero operator in H); 2. If P = skip, then P = I (the identity operator in H); 3. If P = U[q, c], then P = U; 4. If P = P 1 ; P 2, then P = P 2 P 1 ; 5. If P = qif [c]( i i P i ) fiq, then P = i ( i P i ) = ( i i P i ). i

48 Quantum recursive programs Let X 1,..., X m be different procedure identifiers. A declaration for X 1,..., X m is a system of equations: X 1 P 1, S :... X m P m, where P i = P i [X 1,..., X m ] contains at most procedure identifiers X 1,..., X m.

49 Quantum recursive programs Let X 1,..., X m be different procedure identifiers. A declaration for X 1,..., X m is a system of equations: X 1 P 1, S :... X m P m, where P i = P i [X 1,..., X m ] contains at most procedure identifiers X 1,..., X m. A recursive program consists of a main statement P = P[X 1,..., X m ] and a declaration S for X 1,..., X m. Question: How to define semantics of quantum recursive programs?

50 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

51 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d], and then a quantum case statement:

52 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d], and then a quantum case statement: if coin d is in state L then the walker moves one position left;

53 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d], and then a quantum case statement: if coin d is in state L then the walker moves one position left; if d is in state R then it moves one position right, followed by a procedure behaving as the recursive walk itself.

54 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d], and then a quantum case statement: if coin d is in state L then the walker moves one position left; if d is in state R then it moves one position right, followed by a procedure behaving as the recursive walk itself. Program X declared by recursive equation: X T L [p] H[d] (T R [p]; X)

55 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d] and then a quantum case statement:

56 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d] and then a quantum case statement: if coin d is in state L then the walker moves one position left, followed by a procedure behaving as the recursive walk itself;

57 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d] and then a quantum case statement: if coin d is in state L then the walker moves one position left, followed by a procedure behaving as the recursive walk itself; if d is in state R then it moves one position right, also followed by a procedure behaving as the recursive walk itself.

58 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d] and then a quantum case statement: if coin d is in state L then the walker moves one position left, followed by a procedure behaving as the recursive walk itself; if d is in state R then it moves one position right, also followed by a procedure behaving as the recursive walk itself. Program X declared by the recursive equation: X (T L [p]; X) H[d] (T R [p]; X)

59 Recursive Hadamard Walk The walk first runs coin-tossing operator H[d] and then a quantum case statement: if coin d is in state L then the walker moves one position left, followed by a procedure behaving as the recursive walk itself; if d is in state R then it moves one position right, also followed by a procedure behaving as the recursive walk itself. Program X declared by the recursive equation: X (T L [p]; X) H[d] (T R [p]; X) A variant declared by system of recursive equations: { X T L [p] H[d] (T R [p]; Y), Y (T L [p]; X) H[d] T R [p]

60 More Interesting Recursive Hardamard Walk Let n 2. Program declared by recursive equation: X ((T L [p]; X) H[d] (T R [p]; X)); (T L [p] H[d] T R [p]) n (X, L d 0 p ) [(X, L d 3 p ) + (X, R d 1 p ) + (X, L d 1 p ) (X, R d 1 p ) + (X, L d 1 p ) + (X, R d 1 p ) (X, L d 1 p ) + (X, R d 3 p )]

61 More Interesting Recursive Hardamard Walk Let n 2. Program declared by recursive equation: X ((T L [p]; X) H[d] (T R [p]; X)); (T L [p] H[d] T R [p]) n (X, L d 0 p ) [(X, L d 3 p ) + (X, R d 1 p ) + (X, L d 1 p ) (X, R d 1 p ) + (X, L d 1 p ) + (X, R d 1 p ) (X, L d 1 p ) + (X, R d 3 p )] Configurations (X, R d 1 p ) and (X, R d 1 p ) cancel one another. Question: how to solve quantum recursive equations?

62 Syntactic Approximation Recursive program X declared by equation X F(X)

63 Syntactic Approximation Recursive program X declared by equation X F(X) Syntactic approximations: { X (0) = Abort, X (n+1) = F[X (n) /X] for n 0. X (n) is the nth syntactic approximation of X.

64 Syntactic Approximation Recursive program X declared by equation X F(X) Syntactic approximations: { X (0) = Abort, X (n+1) = F[X (n) /X] for n 0. X (n) is the nth syntactic approximation of X. Semantics X of X is the limit: X = lim n X (n)

65 Example - Recursive Hadamard Walk X (0) = abort, X (1) = T L [p] H[d] (T R [p]; abort), X (2) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; abort)), X (3) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; T L [p] H[d2 ] (T R [p]; abort))),...

66 Example - Recursive Hadamard Walk X (0) = abort, X (1) = T L [p] H[d] (T R [p]; abort), X (2) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; abort)), X (3) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; T L [p] H[d2 ] (T R [p]; abort))),... Observation Continuously introduce new coin to avoid variable conflict. Variables d, d 1, d 2,... denote identical particles.

67 Example - Recursive Hadamard Walk X (0) = abort, X (1) = T L [p] H[d] (T R [p]; abort), X (2) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; abort)), X (3) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; T L [p] H[d2 ] (T R [p]; abort))),... Observation Continuously introduce new coin to avoid variable conflict. Variables d, d 1, d 2,... denote identical particles. The number of coin particles needed in running recursive walk is unknown beforehand.

68 Example - Recursive Hadamard Walk X (0) = abort, X (1) = T L [p] H[d] (T R [p]; abort), X (2) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; abort)), X (3) = T L [p] H[d] (T R [p]; T L [p] H[d1 ] (T R [p]; T L [p] H[d2 ] (T R [p]; abort))),... Observation Continuously introduce new coin to avoid variable conflict. Variables d, d 1, d 2,... denote identical particles. The number of coin particles needed in running recursive walk is unknown beforehand. We need to deal with quantum systems where the number of particles of the same type may vary.

69 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

70 Fock Spaces The principle of symmetrisation: the states of n identical particles are either completely symmetric or completely antisymmetric with respect to the permutations of the particles. [ bosons - symmetric; fermions - antisymmetric]

71 Fock Spaces The principle of symmetrisation: the states of n identical particles are either completely symmetric or completely antisymmetric with respect to the permutations of the particles. [ bosons - symmetric; fermions - antisymmetric] H the Hilbert space of one particle.

72 Fock Spaces The principle of symmetrisation: the states of n identical particles are either completely symmetric or completely antisymmetric with respect to the permutations of the particles. [ bosons - symmetric; fermions - antisymmetric] H the Hilbert space of one particle. For each permutation π of 1,..., n, define operator P π in H n : P π ψ 1... ψ n = ψ π(1)... ψ π(n)

73 Fock Spaces The principle of symmetrisation: the states of n identical particles are either completely symmetric or completely antisymmetric with respect to the permutations of the particles. [ bosons - symmetric; fermions - antisymmetric] H the Hilbert space of one particle. For each permutation π of 1,..., n, define operator P π in H n : P π ψ 1... ψ n = ψ π(1)... ψ π(n) Define symmetrisation and antisymmetrisation operators in H n : S + = 1 n! P π, S = 1 π n! ( 1) π P π π

74 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n.

75 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. State spaces of n bosons and fermions: H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H}

76 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. State spaces of n bosons and fermions: H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Vacuum state 0 H 0 v = H 0 = span{ 0 }

77 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. State spaces of n bosons and fermions: H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Vacuum state 0 H 0 v = H 0 = span{ 0 } Space of the states of variable particle number is Fock space: F v (H) = Hv n n=0

78 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. State spaces of n bosons and fermions: H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Vacuum state 0 H 0 v = H 0 = span{ 0 } Space of the states of variable particle number is Fock space: F v (H) = Hv n n=0 Free Fock space: F(H) = H n n=0

79 Evolution Fock Spaces Evolution of one particle: unitary operator U.

80 Evolution Fock Spaces Evolution of one particle: unitary operator U. Evolution of n particles without mutual interactions: operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n

81 Evolution Fock Spaces Evolution of one particle: unitary operator U. Evolution of n particles without mutual interactions: operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n Symmetrisation: U ψ 1,..., ψ n v = Uψ 1,...Uψ n v.

82 Evolution Fock Spaces Evolution of one particle: unitary operator U. Evolution of n particles without mutual interactions: operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n Symmetrisation: U ψ 1,..., ψ n v = Uψ 1,...Uψ n v. Extend to Fock spaces F v (H) and F(H): ( ) U Ψ(n) = U Ψ(n) n=0 n=0

83 Creation and Annihilation of Particles Transitions between states of different particle numbers.

84 Creation and Annihilation of Particles Transitions between states of different particle numbers. Creation operator a (ψ) in F v (H): a (ψ) ψ 1,..., ψ n v = n + 1 ψ, ψ 1,..., ψ n v Add a particle in individual state ψ to the system of n particles without modifying their respective states.

85 Creation and Annihilation of Particles Transitions between states of different particle numbers. Creation operator a (ψ) in F v (H): a (ψ) ψ 1,..., ψ n v = n + 1 ψ, ψ 1,..., ψ n v Add a particle in individual state ψ to the system of n particles without modifying their respective states. Annihilation operator a(ψ) Hermitian conjugate of a (ψ): a(ψ) 0 = 0, a(ψ) ψ 1,..., ψ n v = 1 n n i=1(v) i 1 ψ ψ i ψ 1,..., ψ i 1, ψ i+1,..., ψ n v Decrease the number of particles by one unit, while preserving the symmetry of state.

86 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

87 Second quantisation provides us with necessary tools for defining semantics of quantum recursions!

88 Second quantisation provides us with necessary tools for defining semantics of quantum recursions! A domain of operators in free Fock space H and K two Hilbert spaces F(H) free Fock space over H.

89 Second quantisation provides us with necessary tools for defining semantics of quantum recursions! A domain of operators in free Fock space H and K two Hilbert spaces F(H) free Fock space over H. O(F(H) K) the set of all operators of the form A = A(n), n=0 where A(n) is an operator in H n K.

90 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n)

91 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n) A B if and only if

92 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n) A B if and only if either for all n 0, A(n) = B(n), or

93 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n) A B if and only if either for all n 0, A(n) = B(n), or there exists an integer n 0 such that A(n) = B(n) for all 0 n n 0 and A(n) = 0 for all n > n 0.

94 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n) Product: A B if and only if either for all n 0, A(n) = B(n), or there exists an integer n 0 such that A(n) = B(n) for all 0 n n 0 and A(n) = 0 for all n > n 0. A B = (A(n) B(n)). n=0

95 Order and Operations on O(F(H) K) Flat order: A = n=0 A(n), B = n=0 B(n) Product: A B if and only if either for all n 0, A(n) = B(n), or there exists an integer n 0 such that A(n) = B(n) for all 0 n n 0 and A(n) = 0 for all n > n 0. Guarded composition: A B = i ( i A i ) = (A(n) B(n)). n=0 ( ) ( i i A i (n)). n=0 i

96 Notation C the set of quantum coins inp = P[X 1,..., X m ].

97 Notation C the set of quantum coins inp = P[X 1,..., X m ]. H C = c C H c, where H c is the Hilbert space of coin c.

98 Notation C the set of quantum coins inp = P[X 1,..., X m ]. H C = c C H c, where H c is the Hilbert space of coin c. The principal system of P is the composition of the systems denoted by principal variables in P.

99 Notation C the set of quantum coins inp = P[X 1,..., X m ]. H C = c C H c, where H c is the Hilbert space of coin c. The principal system of P is the composition of the systems denoted by principal variables in P. H the Hilbert space of the principal system.

100 Semantic functional Semantic functional of program scheme P: P : O(F(H C ) H) m O(F(H C ) H). For any A 1,..., A m O(F(H C ) H), is inductively defined: P (A 1,..., A m ) If P = abort, then P (A 1,..., A m ) is the zero operator A = n=0 A(n) with A(n) = 0 (the zero operator in H n C H);

101 Semantic functional Semantic functional of program scheme P: P : O(F(H C ) H) m O(F(H C ) H). For any A 1,..., A m O(F(H C ) H), is inductively defined: P (A 1,..., A m ) If P = abort, then P (A 1,..., A m ) is the zero operator A = n=0 A(n) with A(n) = 0 (the zero operator in H n C H); If P = skip, then P (A 1,..., A m ) is the identity operator A = n=0 A(n) with A(n) = I (the identity operator in H); H n C

102 Semantic functional Semantic functional of program scheme P: P : O(F(H C ) H) m O(F(H C ) H). For any A 1,..., A m O(F(H C ) H), is inductively defined: P (A 1,..., A m ) If P = abort, then P (A 1,..., A m ) is the zero operator A = n=0 A(n) with A(n) = 0 (the zero operator in H n C H); If P = skip, then P (A 1,..., A m ) is the identity operator A = n=0 A(n) with A(n) = I (the identity operator in H); HC n If P = U[q, c], then P (A 1,..., A m ) is the cylindrical extension of U: A = n=0 A(n) with A(n) = U I 1 I 2 I 3, where I 1 is the identity operator in the Hilbert space of those coins not in c, I 2 is the identity operator in H (n 1) C, and I 3 is the identity operator in the Hilbert space of those principal variables not in q;

103 Semantic functional If P = X j (1 j m), then P (A 1,..., A m ) = A j ; If P = P 1 ; P 2, then P (A 1,..., A m ) = P 2 (A 1,..., A m ) P 1 (A 1,..., A m ); If P = qif [c]( i i P i ) fiq, then P (A 1,..., A m ) = i ( i P i (A 1,..., A m )).

104 Theorem Continuity of Semantic Functionals Semantic functional P : (O(F(H C ) H) m, ) (O(F(H C ) H), ) is continuous.

105 Creation functional Creation functional C : O(F(H C ) H) O(F(H C ) H) is defined: for any A = n=0 A(n), C(A) = n=0 (I A(n)) where I is the identity operator in H C.

106 Creation functional Creation functional C : O(F(H C ) H) O(F(H C ) H) is defined: for any A = n=0 A(n), C(A) = n=0 where I is the identity operator in H C. (I A(n)) Intuition creation functional C moves all coins c 0, c 1, c 2,... one position to the right so that c i becomes c i+1 for all i = 0, 1, 2,... Thus, a new position is created at the left end for a new coin c 0. Lemma Continuity of Creation Functional Creation functional is continuous. C : (O(F(H C ) H), ) (O(F(H C ) H), )

107 Corollary P = P[X 1,..., X m ] a program scheme. Define: (C m P )(A 1,..., A m ) = P (C(A 1 ),..., C(A m )). Then functional C m P : (O(F(H C ) H) m, ) (O(F(H C ) H), ) is continuous.

108 Corollary P = P[X 1,..., X m ] a program scheme. Define: (C m P )(A 1,..., A m ) = P (C(A 1 ),..., C(A m )). Then functional C m P : (O(F(H C ) H) m, ) (O(F(H C ) H), ) is continuous. Notation Consider a recursive program P declared by system of recursive equations: X 1 P 1, S :... X m P m,

109 Notation System S of recursive equations induces semantic functional: S : O(F(H C ) H) m O(F(H C ) H) m, S (A 1,..., A m ) = ((C m P 1 )(A 1,..., A m ),..., (C m P m )(A 1,..., A m ))

110 Notation System S of recursive equations induces semantic functional: S : O(F(H C ) H) m O(F(H C ) H) m, S (A 1,..., A m ) = ((C m P 1 )(A 1,..., A m ),..., (C m P m )(A 1,..., A m )) Semantical functional S : (O(F(H C ) H) m, ) (O(F(H C ) H) m, ) is continuous.

111 Fixed point semantics Knaster-Tarski Fixed Point Theorem: S has the least fixed point µ S.

112 Fixed point semantics Knaster-Tarski Fixed Point Theorem: S has the least fixed point µ S. Definition The fixed point semantics of recursive program P declared by S: where µ S = (A 1,..., A m). P fix = P (A 1,..., A m) O(F(H C ) H)

113 Theorem Equivalence of Denotational Semantics and Operational Semantics 1. For each k, { X (n) k } n=0 is an increasing chain and exists in O(F(H C ) H). X ( ) k = lim X (n) n k = X (n) k n=0

114 Theorem Equivalence of Denotational Semantics and Operational Semantics 1. For each k, { X (n) k } n=0 is an increasing chain and exists in O(F(H C ) H). X ( ) k = lim X (n) n k = X (n) k n=0 2. ( X ( ) 1,..., X ( ) m ) = µ S is the least fixed point of semantic functional S.

115 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

116 Solutions of recursive equations in Boson/Fermion Fock space Symmetrisation/anti-symmetrisation of the solutions of recursive equations in free Fock space!

117 Solutions of recursive equations in Boson/Fermion Fock space Symmetrisation/anti-symmetrisation of the solutions of recursive equations in free Fock space! Principal System Semantics Each state Ψ in Fock space F v (H d ) induces mapping: X, Ψ p : pure states partial density operators in H p X, Ψ p ( ψ ) = tr F(Hd )( Φ Φ ) where Φ = X ( Ψ ψ ) Principal system semantics of X with coin initialisation Ψ : mapping X, Ψ p.

118 Example Recursive Quantum Walk { X T L [p] H[d] (T R [p]; Y), Y (T L [p]; X) H[d] T R [p] Coherent state of bosons in Boson Fock space F + (H): ( ψ coh = exp 1 ) 2 ψ ψ n=0 [a (ψ)] n 0 n!

119 Example Recursive Quantum Walk { X T L [p] H[d] (T R [p]; Y), Y (T L [p]; X) H[d] T R [p] Coherent state of bosons in Boson Fock space F + (H): ( ψ coh = exp 1 ) 2 ψ ψ n=0 [a (ψ)] n 0 n! The walk starts from position 0 and the coins are initialised in the coherent states of bosons corresponding to L : ( ) X, L coh p ( 0 ) = 1 1 e k=0 2 2k k=0 22k+2 = 1 ( 2 e )

120 Outline 1. Introduction 2. Quantum Case Statement 3. Syntax of Quantum Recursive Programs 4. Recursive Quantum Walks 5. Second Quantisation 6. Solving Recursive Equations in Free Fock Space 7. Quantum Recursion in Boson and Fermion Fock Spaces 8. Conclusion

121 Problems: What kind of problems can be solved more conveniently by using quantum recursion?

122 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Hoare logic for quantum while-loops defined using quantum coins?

123 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Hoare logic for quantum while-loops defined using quantum coins? Hoare logic for quantum while-programs with classical controls [Ying, TOPLAS 2011]

124 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Hoare logic for quantum while-loops defined using quantum coins? Hoare logic for quantum while-programs with classical controls [Ying, TOPLAS 2011] What kind of physical systems can be used to implement quantum recursion?

125 Thank You!