Techniques algébriques en calcul quantique

Transcription

1 Techniques algébriques en calcul quantique E. Jeandel Laboratoire de l Informatique du Parallélisme LIP, ENS Lyon, CNRS, INRIA, UCB Lyon 8 Avril 25 E. Jeandel, LIP, ENS Lyon Techniques algébriques en calcul quantique /54

2 Algebraic Techniques in Quantum Computing E. Jeandel Laboratoire de l Informatique du Parallélisme LIP, ENS Lyon, CNRS, INRIA, UCB Lyon April 8th, 25 E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 2/54

3 Outline Combinatorial setting: Quantum gates Definitions Completeness and Universality 2 Algebraic setting Quantum gates are unitary matrices Computing the group Density 3 Conclusion Automata Conclusion E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

4 Introduction State Classical q Quantum αi q i The system may be in all states simultaneously Operators aps Unitary (hence reversible) maps E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 4/54

6 What is a quantum gate?.... E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

7 What is a quantum gate? β + iα.... α + β E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

8 What is a quantum gate? β + α β δ.... γ + α γ + δ E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

9 What can we do with quantum gates?.. N.. (a) The multiplication N (b) [σ] (c) The operation I A quantum circuit is everything we can obtain by applying these constructions. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 7/54

10 What we cannot do x x x Quantum mechanics implies no-cloning. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 8/54

12 Completeness A (finite) set of gates is complete if every quantum gate can be obtained by a quantum circuit built on these gates. How to show that some set of gates is complete? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing /54

13 Completeness A (finite) set of gates is complete if every quantum gate can be obtained by a quantum circuit built on these gates. How to show that some set of gates is complete? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing /54

14 Game: Design this gate R G G G B G B R G E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing /54

15 Toolkit R G R R R R R R E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 2/54

16 Toolkit : Universality Fact If there are two wires set to, we can make the gate G. This is called universality with ancillas. R G E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

17 Toolkit : Non-completeness Fact If among the additional wires, strictly less than 2 are set to, the gate G cannot be made. Any circuit, even the most intricate, cannot produce any using only the gate. R R R E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 4/54

18 Toolkit : Summary Theorem (8.7) There exists a set of gates B i such that B i is 2-universal but neither -universal nor k-complete. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 5/54

19 Toolkit 2 R G R G R x y z R x y z otherwise E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 6/54

20 Toolkit 2: Non-completeness Fact Without any additional wire, we cannot realise the gate G. If the three given wires are set to, and there is no mean to have three or three. R R E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 7/54

21 Toolkit 2: 2 additional wires We are given two additional /-wires. We have now five /-wires. 3 of them must be equal! R G Problem: The wiring depends on the 3 equal wires. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 8/54

22 Toolkit 2: 2 additional wires We are given two additional /-wires. We have now five /-wires. 3 of them must be equal! R G Problem: The wiring depends on the 3 equal wires. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 8/54

23 Toolkit 2: Solution Consider the following circuit: E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 9/54

24 Toolkit 2: Solution If 4 bits are equal: R R G B B B R G G G G E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 2/54

30 Toolkit 2: Solution If 3 bits are equal: R R R R R R G G G G G E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 2/54

31 Toolkit 2: Solution If all 5 bits are equal: R G B R G B R G B R G E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 22/54

32 Toolkit 2: Summary Fact The previous circuit simulates the gate G whatever the bits on the wires are. This is called 2-completeness (since we use 2 additional wires). Up to some technical details, we obtain: Theorem (8.8) There exists a set of gates B i such that B i is 3-complete but not complete. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 23/54

33 Toolkit 2: Summary Fact The previous circuit simulates the gate G whatever the bits on the wires are. This is called 2-completeness (since we use 2 additional wires). Up to some technical details, we obtain: Theorem (8.8) There exists a set of gates B i such that B i is 3-complete but not complete. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 23/54

35 What is a quantum gate? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

39 What is a quantum gate? A quantum gate over n qubits is a 2 n 2 n unitary matrix E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 25/54

40 Approximating Quantum Circuits Problem Given unitary matrices X... X n and a unitary matrix, is in the group generated by the X i? In the real life, we do not try to obtain quantum gates, but rather to approximate them. Problem Given unitary matrices X... X n and a unitary matrix, is in the euclidean closure of the group generated by the X i? (ore generally, investigate finitely generated compact groups) E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 26/54

41 Approximating Quantum Circuits Problem Given unitary matrices X... X n and a unitary matrix, is in the group generated by the X i? In the real life, we do not try to obtain quantum gates, but rather to approximate them. Problem Given unitary matrices X... X n and a unitary matrix, is in the euclidean closure of the group generated by the X i? (ore generally, investigate finitely generated compact groups) E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 26/54

42 Why compact groups? Property A compact group G of n (R) is algebraic. That is there exists polynomials p... p k such that X G i, p i (X ) = For instance, if G = O 2 (R), then G = { X = ( ) a b c d } : XX T = I = We can compute things! Now we focus on algebraic groups. ( ) a b : c d a 2 + b 2 = c 2 + d 2 = ac + bd = E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 27/54

44 Question Problem Given matrices X... X n, compute the algebraic group generated by the matrices X i. Computing the group means finding polynomials p i such that X G i, p i (X ) = Algebraic sets (defined by polynomials) are the closed sets of a topology called the Zariski topology. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 29/54

45 Irreducible groups Theorem If G and G 2 are irreducible algebraic groups given by polynomials, one may obtain polynomials for G, G 2 by the following algorithm: H := G G 2 2 While H H H do H := H H (A is the Zariski-closure of A, the smallest algebraic set containing A. A B may be obtained by using Groebner basis techniques) E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

46 Irreducible groups Theorem If G and G 2 are irreducible algebraic groups given by polynomials, one may obtain polynomials for G, G 2 by the following algorithm: H := G G 2 2 While H H H do H := H H Sketch of proof: At each step H is an irreducible algebraic variety. If H H H, H H is of a greater dimension, which proves that the algorithm terminates. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

47 General groups Fact Let G be an algebraic group generated by X... X k. Then G = S H with i, X i S H 2 H is an irreducible algebraic group 3 S H S H = S H 4 H is normal in G : S H S = H 5 S is finite Furthermore, if the conditions are satisfied by some S and H, then G = S H is the algebraic group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

48 General groups Fact Let G be an algebraic group generated by X... X k. Then G = S H with i, X i S H 2 H is an irreducible algebraic group 3 S S S H 4 H is normal in G : S H S = H 5 S is finite Furthermore, if the conditions are satisfied by some S and H, then G = S H is the algebraic group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 3/54

49 Sketch of an algorithm Define by induction S = {X i }, H = {I} 2 H n+ := H n H n 3 S n+ := S n. For X, Y in S n, if X Y S n H n then S n+ := S n+ {X Y } 4 For X in S n do H n+ := X H n+ X H n+ Then the limit S = S n, H = H n satisfies all conditions of the previous fact... except perhaps the last one. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 32/54

50 Sketch of an algorithm Define by induction S = {X i }, H = {I} 2 H n+ := H n H n 3 S n+ := S n. For X, Y in S n, if X Y S n H n then S n+ := S n+ {X Y } 4 For X in S n do H n+ := X H n+ X H n+ Then the limit S = S n, H = H n satisfies all conditions of the previous fact... except perhaps the last one. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 32/54

51 General groups revisited Fact Let G be an algebraic group generated by X... X k. Then G = S H with i, X i S H 2 H is an irreducible algebraic group 3 S S S H 4 H is normal in G : S H S = H 5 S is finite Furthermore, if the conditions are satisfied by some S and H, then S is finite and G = S H is the algebraic group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 33/54

52 General groups revisited Fact Let G be an algebraic group generated by X... X k. Then G = S H with i, X i S H 2 H is an irreducible algebraic group 3 S S S H 4 H is normal in G : S H S = H 5 X S there exists n > such that X n H. Furthermore, if the conditions are satisfied by some S and H, then S is finite and G = S H is the algebraic group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 33/54

53 Sketch of an algorithm, revisited Define by induction S = {X i }, H = {I} 2 H n+ := H n H n 3 S n+ := S n. For X, Y in S n, if X Y S n H n then S n+ := S n+ {X Y } 4 For X in S n do H n+ := X H n+ X H n+ 5 For X in S n, compute the group G X = S X H X generated by X and add H X to H n+ : H n+ := H X H n+ Then the limit S = S n, H = H n satisfies all conditions of the previous fact. In particular, S is finite. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 34/54

54 The new algorithm works Theorem The previous algorithm terminates and gives sets S, H such that G = S H is the algebraic group generated by the X i. We need only to know how to compute the group generated by one matrix. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 35/54

55 Group generated by one matrix : example β 2 X = β βγ 3 γ The group generated by X is β 2k X = β k β k γ 3k, k Z γ k The algebraic group generated by X is a b c, ab 2 =, b d 3 c = d E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 36/54

56 Group generated by one matrix A unitary matrix, up to a change of basis is of the form α... α n (ultiplicative) relationships between the α i is the key point: (m,..., m n ) Γ i αm i i = The algebraic group generated by X is then λ... : i λm i i = (m,..., m n ) Γ λ n To find Γ, we must find bounds for the m i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 37/54

57 Group generated by one matrix Theorem (Ge) There exists a polynomial-time algorithm which given the α i computes the multiplicative relations between the α i. Corollary There exists an algorithm which computes the compact group generated by a unitary matrix X. Theorem There exists an algorithm which computes the algebraic group generated by a matrix X. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 38/54

60 Summary Theorem (3.3) There exists an algorithm which given matrices X i computes the algebraic group generated by the X i. Due to the method (keep going until it stabilises), there is absolutely no bound of complexity for the algorithm. Theorem There exists an algorithm which given unitary matrices X i computes the compact group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 39/54

61 Summary Theorem (3.3) There exists an algorithm which given matrices X i computes the algebraic group generated by the X i. Due to the method (keep going until it stabilises), there is absolutely no bound of complexity for the algorithm. Theorem There exists an algorithm which given unitary matrices X i computes the compact group generated by the X i. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 39/54

63 Question Problem Given matrices X... X k, decide if the group generated by the matrices X i is dense in the algebraic group G. The good notion of density for an algebraic group is the Zariski-density. Problem Given unitary matrices X... X k of dimension n, decide if the group generated by the matrices X i is dense in U n E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 4/54

64 Question Problem Given matrices X... X k, decide if the group generated by the matrices X i is dense in the algebraic group G. The good notion of density for an algebraic group is the Zariski-density. Problem Given unitary matrices X... X k of dimension n, decide if the group generated by the matrices X i is dense in U n E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 4/54

65 Simple groups A simple group has no non-trivial normal irreducible subgroups. This gives an algorithm for a simple group: Theorem H is dense in a simple group G iff H is infinite and H is normal in G. There exists an algorithm from Babai, Beals and Rockmore to test if a finitely generated group is finite. We only have to find a way to show that H is normal in G. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 42/54

66 Normal groups H is normal in G X G, XHX = H Denote by K G the set { XX, X G}. K G is a set (in fact a group) of endomorphisms of n. H is normal in G φ K G, φ(h) = H Fact φ K H, φ(h) = H. Corollary If K H = K G then H is normal in G. Testing K H = K G is not that easy.. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 43/54

70 Normal groups Denote by Span(S) the vector space generated by S. Theorem (2.5) If Span(K H ) = Span(K G ), then H is normal in G. Proof. We use Lie algebras techniques. The condition implies that the Lie algebra of H is an ideal of the Lie algebra of G. Fact Testing whether Span(K H ) = Span(K G ) is easy. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 44/54

73 Computing Span(K H ) Let E be the vector space generated by the morphisms X i X i While E is not stable by multiplication (composition), let E := EE = {φ ψ : φ E, ψ E} Theorem For every simple group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 45/54

74 Computing Span(K H ) Let E be the vector space generated by the morphisms X i X i While E is not stable by multiplication (composition), let E := EE = {φ ψ : φ E, ψ E} Theorem For every simple group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 45/54

75 Generalisation Theorem (2.26) For every reductive group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is Zariski-dense in G. Theorem (2.27) For every compact group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 46/54

76 Generalisation Theorem (2.26) For every reductive group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is Zariski-dense in G. Theorem (2.27) For every compact group G, there exists a polynomial time algorithm which decides if a finitely generated subgroup H is dense in G. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 46/54

77 Back to circuits Theorem (8.5) There exists a polynomial time algorithm which decides if a set of gates is complete. Theorem (8.4) There exists an algorithm which decides if a set of gates is universal. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 47/54

78 Back to circuits Theorem (8.5) There exists a polynomial time algorithm which decides if a set of gates is complete. Theorem (8.4) There exists an algorithm which decides if a set of gates is universal. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 47/54

80 Automata (Sketch) We are given a gate for each letter a, b, c a.... b.... c.. The value (or probability) of a word ω is function of the result of the circuit corresponding to ω. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 49/54

81 Automata (Sketch).. a.. c.. c. α + β. acc is accepted with probability α 2... b.. b.. a. bbab is accepted with probability δ 2.. b. δ + ɛ. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 5/54

82 Theorems Some theorems about quantum automata : Theorem (5.4) We can decide given an automaton A and a threshold λ if there exists a word accepted with a probability strictly greater than λ. We use the algorithm which computes the group generated by some matrices. Theorem (7.) Non-deterministic quantum automata with an isolated threshold recognise only regular languages. The proof introduces a new model of automata, called topological automata. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 5/54

83 Theorems Some theorems about quantum automata : Theorem (5.4) We can decide given an automaton A and a threshold λ if there exists a word accepted with a probability strictly greater than λ. We use the algorithm which computes the group generated by some matrices. Theorem (7.) Non-deterministic quantum automata with an isolated threshold recognise only regular languages. The proof introduces a new model of automata, called topological automata. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 5/54

85 Conclusion Study of quantum objects using algebraic groups techniques. New algorithms about algebraic groups. any other potentially interesting things. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 53/54

88 Perspectives and open problems Problem What if the number of auxiliary wires depends on the gate to realise ( -universality)? Is it equivalent to m-universality for some m? Problem Find an efficient algorithm to decide whether some matrix X is in the algebraic group generated by the matrices X i. ore generally, use the structure of the algebraic groups more efficiently. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 54/54

89 Perspectives and open problems Problem What if the number of auxiliary wires depends on the gate to realise ( -universality)? Is it equivalent to m-universality for some m? Problem Find an efficient algorithm to decide whether some matrix X is in the algebraic group generated by the matrices X i. ore generally, use the structure of the algebraic groups more efficiently. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 54/54