Quantum Recursion and Second Quantisation Basic Ideas and Examples

Transcription

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

2 Happy Birthday Prakash!

3 Happy Birthday Prakash! I m very grateful to Prakash for teaching me second quantisation method during his visit to UTS in 2013

4 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

5 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

6 Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004].

7 Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004]. Termination of quantum while-loops in finite-dimensional state spaces [Ying, Feng, Acta Informatica 2010].

8 Classical Recursion of Quantum Programs Recursive procedure in quantum programming language QPL [Selinger, Mathematical Structures in Computer Science 2004]. Termination of quantum while-loops in finite-dimensional state spaces [Ying, Feng, Acta Informatica 2010]. Selinger s slogan: Quantum data, classical control - control flow of the quantum recursions is classical because branchings are determined by the outcomes of certain quantum measurements.

9 Quantum Control Flow Quantum programming language QML [Altenkirch and Grattage, LICS 2005]: Two case constructs in the quantum setting: case, measure a qubit in the data it analyses - The control flow is determined by the outcome of a measurement and thus is classical.

10 Quantum Control Flow Quantum programming language QML [Altenkirch and Grattage, LICS 2005]: Two case constructs in the quantum setting: case, measure a qubit in the data it analyses - The control flow is determined by the outcome of a measurement and thus is classical. case, analyse quantum data without measuring - if then else statement.

11 Quantum Control Flow Hadamard gate: had x = if x then { 1 2 (qfalse qtrue)} else { 1 2 (qfalse + qtrue)}

12 Quantum Control Flow NOT gate: CNOT gate: not x = if x cnot c x = if c then qfalse else qtrue then (qtrue, not x) else (qfalse, x)

13 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

14 Coined Quantum Case Statement Introduce an external quantum coin c: The state Hilbert space H c = span{ 0, 1 }

15 Coined Quantum Case Statement Introduce an external quantum coin c: The state Hilbert space H c = span{ 0, 1 } U 0 and U 1 two unitary transformations on a quantum system q - the state Hilbert space H q.

16 Coined Quantum Case Statement Introduce an external quantum coin c: The state Hilbert space H c = span{ 0, 1 } U 0 and U 1 two unitary transformations on a quantum system q - the state Hilbert space H q. A quantum case statement employing quantum coin c: qif [c] 0 U 0 [q] 1 U 1 [q] fiq

17 coined Quantum Case Statement The semantics is an unitary operator U in H c H q - the state Hilbert space of the composed system of coin c and principal system q: U 0, ψ = 0 U 0 ψ, U 1, ψ = 1 U 1 ψ

18 coined Quantum Case Statement The semantics is an unitary operator U in H c H q - the state Hilbert space of the composed system of coin c and principal system q: U 0, ψ = 0 U 0 ψ, U 1, ψ = 1 U 1 ψ Matrix representation: ( U0 0 U = 0 0 U U 1 = 0 U 1 ).

19 Quantum Choice V a unitary operator in the state Hilbert space H c of the coin.

20 Quantum Choice V a unitary operator in the state Hilbert space H c of the coin. The quantum choice of U 0 [q] and 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] 1 U 1 [q] fiq

21 Quantum Choice V a unitary operator in the state Hilbert space H c of the coin. The quantum choice of U 0 [q] and 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] 1 U 1 [q] fiq Compare with probabilistic choice [McIver and Morgan, Abstraction, Refinement and Proof for Probabilistic Systems, 2005]

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

23 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].

24 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]. Unitary transformations U 0 [q], U 1 [q] are replaced by general quantum programs that may contain quantum measurements? [Ying, Yu, Feng, arxiv: ]

25 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

26 A new notion of quantum recursion can be defined based on quantum case statement and quantum choice

27 A new notion of quantum recursion can be defined based on quantum case statement and quantum choice Example - One-dimensional 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 of one-dimensional random walk.

28 A new notion of quantum recursion can be defined based on quantum case statement and quantum choice Example - One-dimensional 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 of one-dimensional random walk. Its state Hilbert space H d H p :

29 A new notion of quantum recursion can be defined based on quantum case statement and quantum choice Example - One-dimensional 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 of one-dimensional random walk. Its state Hilbert space H d H p : Hd = span{ L, R }, L, R indicate the direction Left and Right.

30 A new notion of quantum recursion can be defined based on quantum case statement and quantum choice Example - One-dimensional 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 of one-dimensional random walk. Its state Hilbert space H d H p : Hd = span{ L, R }, L, R indicate the direction Left and Right. Hp = span{ n : n Z}, n indicates the position marked by integer n.

31 Example - One-dimensional quantum walk One step of Hadamard walk W = T(H I):

32 Example - One-dimensional quantum walk One step of Hadamard walk W = T(H I): Translation T: T L, n = L, n 1, T R, n = R, n + 1 is unitary operator in H d H p.

33 Example - One-dimensional quantum walk One step of Hadamard walk W = T(H I): Translation T: T L, n = L, n 1, T R, n = R, n + 1 is unitary operator in H d H p. H = 1 2 ( ) is Hadamard transform in the direction space H d

34 Example - One-dimensional quantum walk Define the left and right translation operators T L and T R in the position space H p : T L n = n 1, T R n = n + 1

35 Example - One-dimensional quantum walk Define the left and right translation operators T L and T R in the position space H p : T L n = n 1, T R n = n + 1 Then the translation operator T is a quantum case statement: T = qif [d] L T L [p] R T R [p] fiq

36 Example - One-dimensional quantum walk Define the left and right translation operators T L and T R in the position space H p : T L n = n 1, T R n = n + 1 Then the translation operator T is a quantum case statement: T = qif [d] L T L [p] R T R [p] fiq The single-step walk operator W is a quantum choice: T L [p] H[d] T R [p]

37 Example - One-dimensional quantum walk Define the left and right translation operators T L and T R in the position space H p : T L n = n 1, T R n = n + 1 Then the translation operator T is a quantum case statement: T = qif [d] L T L [p] R T R [p] fiq The single-step walk operator W is a quantum choice: T L [p] H[d] T R [p] Hadamard walk repeated applications of operator W.

38 (Unidirectional) Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d], and then a quantum case statement:

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

40 (Unidirectional) Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d], and then a quantum case statement: if the direction 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.

41 (Unidirectional) Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d], and then a quantum case statement: if the direction 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. Recursive Hadamard walk program X declared by the recursive equation: X T L [p] H[d] (T R [p]; X)

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

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

44 Bidirectional Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d] and then a quantum case statement: if the direction 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.

45 Bidirectional Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d] and then a quantum case statement: if the direction 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. The walk Program X (or program Y) declared by the recursive equation: X (T L [p]; X) H[d] (T R [p]; X)

46 Bidirectional Recursive Hadamard Walk The walk first runs the coin-tossing Hadamard operator H[d] and then a quantum case statement: if the direction 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. The walk Program X (or program Y) declared by the recursive equation: X (T L [p]; X) H[d] (T R [p]; X) A variant of the bidirectional recursive Hadamard walk is declared by the following system of recursive equations: { X T L [p] H[d] (T R [p]; Y), Y (T L [p]; X) H[d] T R [p]

47 A More Interesting Recursive Quantum Walk Let n 2. Another variant of unidirectional recursive quantum walk is defined as the program declared by the following recursive equation: X ((T L [p]; X) H[d] (T R [p]; X)); (T L [p] H[d] T R [p]) n

48 A More Interesting Recursive Quantum Walk Let n 2. Another variant of unidirectional recursive quantum walk is defined as the program declared by the following recursive equation: X ((T L [p]; X) H[d] (T R [p]; X)); (T L [p] H[d] T R [p]) n How to solve these quantum recursive equations?

49 Syntactic Approximation A recursive program X declared by equation X F(X)

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

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

52 Example - (Unidirectional) 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))),...

53 Example - (Unidirectional) 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.

54 Example - (Unidirectional) 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 the coin particles that are needed in running the recursive walk is unknown beforehand.

55 Example - (Unidirectional) 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 the coin particles that are needed in running the recursive walk is unknown beforehand. We need to deal with quantum systems where the number of particles of the same type may vary.

56 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

57 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]

58 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] Let H be the state Hilbert space of one particle.

59 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] Let H be the state Hilbert space of one particle. For each permutation π of 1,..., n, define the permutation operator P π in H n by P π ψ 1... ψ n = ψ π(1)... ψ π(n)

60 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] Let H be the state Hilbert space of one particle. For each permutation π of 1,..., n, define the permutation operator P π in H n by P π ψ 1... ψ n = ψ π(1)... ψ π(n) Define the symmetrisation and antisymmetrisation operators in H n : S + = 1 n! P π, S = 1 π n! ( 1) π P π π

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

62 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. The state space of n bosons and that of fermions are H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H}

63 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. The state space of n bosons and that of fermions are H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Introduce the vacuum state 0 and the one-dimensional space = H 0 = span{ 0 }. H 0 v

64 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. The state space of n bosons and that of fermions are H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Introduce the vacuum state 0 and the one-dimensional space = H 0 = span{ 0 }. H 0 v The space of the states of variable particle number is the Fock space: F v (H) = Hv n n=0

65 Fock Spaces v = + for bosons, v = for fermions. Write ψ 1,..., ψ n v = S v ψ 1... ψ n. The state space of n bosons and that of fermions are H n v = S v H n = span{ ψ 1,..., ψ n v : ψ 1,..., ψ n are in H} Introduce the vacuum state 0 and the one-dimensional space = H 0 = span{ 0 }. H 0 v The space of the states of variable particle number is the Fock space: F v (H) = Hv n n=0 The free Fock space: F(H) = H n n=0

66 Evolution in the Fock Spaces Let the (discrete-time) evolution of one particle be unitary operator U.

67 Evolution in the Fock Spaces Let the (discrete-time) evolution of one particle be unitary operator U. The evolution of n particles without mutual interactions is operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n

68 Evolution in the Fock Spaces Let the (discrete-time) evolution of one particle be unitary operator U. The evolution of n particles without mutual interactions is operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n U ψ 1,..., ψ n v = Uψ 1,...Uψ n v.

69 Evolution in the Fock Spaces Let the (discrete-time) evolution of one particle be unitary operator U. The evolution of n particles without mutual interactions is operator U in H n : U ψ 1... ψ n = Uψ 1... Uψ n U ψ 1,..., ψ n v = Uψ 1,...Uψ n v. Extend to the Fock spaces F v (H) and F(H): U ( Ψ(n) n=0 ) = U Ψ(n) n=0

70 Creation and Annihilation of Particles The transitions between states of different particle numbers.

71 Creation and Annihilation of Particles The 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 the individual state ψ to the system of n particles without modifying their respective states.

72 Creation and Annihilation of Particles The 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 the individual state ψ to the system of n particles without modifying their respective states. Annihilation operator a(ψ) the 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 the state.

73 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

74 Second quantisation provides us with the necessary tool for defining the semantics of quantum recursion!

75 Second quantisation provides us with the necessary tool for defining the semantics of quantum recursion! Example - (Unidirectional) Recursive Hadamard Walk Semantics of the recursive Hadamard walk: X = i 1 R dj R L di L T L TR i (H I) i=0 j=0 An operator in F v (H d ) H p F(H d ) H p. The sign v is + or, depending on using bosons or fermions to implement the direction coins d, d 1, d 2,...

76 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 )

77 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 ) Mapping X, Ψ p is called the principal system semantics of X with coin initialisation Ψ.

78 Bidirectional 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 the symmetric Fock space F + (H) over H: ( ψ coh = exp 1 ) 2 ψ ψ n=0 [a (ψ)] n 0 n!

79 Bidirectional 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 the symmetric Fock space F + (H) over 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 )

80 Quantum while-loop Program X declared by the recursive equation X W[c, q]; qif[c] 0 skip fiq 1 U[q]; X where W a unitary operator in H c H q the interaction between the coin c and the principal system q.

81 Quantum while-loop Program X declared by the recursive equation X W[c, q]; qif[c] 0 skip fiq 1 U[q]; X where W a unitary operator in H c H q the interaction between the coin c and the principal system q. Semantics of X: X = k 1 k=1 j=0 W[c j, q] k 2 1 cj 1 0 ck 1 0 U k 1 [q] j=0 from the space F v (H 2 ) H q into F(H 2 ) H q.

82 Outline 1. Introduction 2. Quantum Case Statement and Quantum Choice 3. Motivating Example: Recursive Quantum Walks 4. Second Quantisation 5. Semantics of Quantum Recursion 7. Conclusion

83 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Sorting? [Høyer, Neerbek, Shi, ICALP 2001]

84 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Sorting? [Høyer, Neerbek, Shi, ICALP 2001] Hoare logic for quantum while-loops defined using quantum coins?

85 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Sorting? [Høyer, Neerbek, Shi, ICALP 2001] Hoare logic for quantum while-loops defined using quantum coins? Fock space can serve as a model of linear logic with exponential types [Blute, Panangaden, Seely, MFPS 1994].

86 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Sorting? [Høyer, Neerbek, Shi, ICALP 2001] Hoare logic for quantum while-loops defined using quantum coins? Fock space can serve as a model of linear logic with exponential types [Blute, Panangaden, Seely, MFPS 1994]. Combine linear logic with Hoare logic for quantum programs [Ying, TOPLAS 2011]?

87 Problems: What kind of problems can be solved more conveniently by using quantum recursion? Sorting? [Høyer, Neerbek, Shi, ICALP 2001] Hoare logic for quantum while-loops defined using quantum coins? Fock space can serve as a model of linear logic with exponential types [Blute, Panangaden, Seely, MFPS 1994]. Combine linear logic with Hoare logic for quantum programs [Ying, TOPLAS 2011]? What kind of physical systems can be used to implement quantum recursion where new coins must be continuously created?

88 Thank You!