Techniques algébriques en calcul quantique


 Kathlyn Kennedy
 2 years ago
 Views:
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
5 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 5/54
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 nocloning. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 8/54
11 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 9/54
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 : Noncompleteness 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 2universal but neither universal nor kcomplete. 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: Noncompleteness 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
25 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
26 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
27 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
28 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
29 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 2completeness (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 3complete 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 2completeness (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 3complete but not complete. E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 23/54
34 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 24/54
35 What is a quantum gate? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 25/54
36 What is a quantum gate? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 25/54
37 What is a quantum gate? E. Jeandel, LIP, ENS Lyon Algebraic Techniques in Quantum Computing 25/54
38 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
43 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 28/54
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 Zariskiclosure 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 polynomialtime 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
58 Group generated by one matrix Theorem (Ge) There exists a polynomialtime 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
59 Group generated by one matrix Theorem (Ge) There exists a polynomialtime 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
62 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 4/54
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 Zariskidensity. 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 Zariskidensity. 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 nontrivial 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
67 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
68 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
69 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
71 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
72 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 Zariskidense 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 Zariskidense 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
79 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 48/54
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.) Nondeterministic 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.) Nondeterministic 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
84 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 52/54
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
86 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
87 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 muniversality 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 muniversality 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
arxiv:quantph/0603009v3 8 Sep 2006
Deciding universality of quantum gates arxiv:quantph/0603009v3 8 Sep 2006 Gábor Ivanyos February 1, 2008 Abstract We say that collection of nqudit gates is universal if there exists N 0 n such that for
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationsome algebra prelim solutions
some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More information0.1 Phase Estimation Technique
Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design
More informationArithmetic complexity in algebraic extensions
Arithmetic complexity in algebraic extensions Pavel Hrubeš Amir Yehudayoff Abstract Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension R of F than over F?
More informationCanonical Heights, Nef Divisors, and Arithmetic Degrees Joseph H. Silverman
Canonical Heights, Nef Divisors, and Arithmetic Degrees Joseph H. Silverman Brown University Joint work with Shu Kawaguchi Conference on Diophantine Problems and Arithmetic Dynamics Academia Sinica, Taipei,
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationNotes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
More information9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ?
9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? Clearly a necessary condition is that H is normal in G.
More informationThe Ergodic Theorem and randomness
The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationORDERS OF ELEMENTS IN A GROUP
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationIntroduction to finite fields
Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at
More informationDeterministic Finite Automata
1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationGalois Fields and Hardware Design
Galois Fields and Hardware Design Construction of Galois Fields, Basic Properties, Uniqueness, Containment, Closure, Polynomial Functions over Galois Fields Priyank Kalla Associate Professor Electrical
More informationfg = f g. 3.1.1. Ideals. An ideal of R is a nonempty ksubspace I R closed under multiplication by elements of R:
30 3. RINGS, IDEALS, AND GRÖBNER BASES 3.1. Polynomial rings and ideals The main object of study in this section is a polynomial ring in a finite number of variables R = k[x 1,..., x n ], where k is an
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationRevision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic
More informationALGEBRA HW 5 CLAY SHONKWILER
ALGEBRA HW 5 CLAY SHONKWILER 510.5 Let F = Q(i). Prove that x 3 and x 3 3 are irreducible over F. Proof. If x 3 is reducible over F then, since it is a polynomial of degree 3, it must reduce into a product
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationIntroduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationSection 3 Sequences and Limits, Continued.
Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More information2.1 Complexity Classes
15859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationOn the Unique Games Conjecture
On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationOmega Automata: Minimization and Learning 1
Omega Automata: Minimization and Learning 1 Oded Maler CNRS  VERIMAG Grenoble, France 2007 1 Joint work with A. Pnueli, late 80s Summary Machine learning in general and of formal languages in particular
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More information3. Prime and maximal ideals. 3.1. Definitions and Examples.
COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationAlgebraic and Transcendental Numbers
Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationRepresenting Reversible Cellular Automata with Reversible Block Cellular Automata
Discrete Mathematics and Theoretical Computer Science Proceedings AA (DMCCG), 2001, 145 154 Representing Reversible Cellular Automata with Reversible Block Cellular Automata Jérôme DurandLose Laboratoire
More informationAppendix A. Appendix. A.1 Algebra. Fields and Rings
Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationOn Winning Conditions of High Borel Complexity in Pushdown Games
Fundamenta Informaticae (2005) 1 22 1 IOS Press On Winning Conditions of High Borel Complexity in Pushdown Games Olivier Finkel Equipe de Logique Mathématique U.F.R. de Mathématiques, Université Paris
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationInformatique Fondamentale IMA S8
Informatique Fondamentale IMA S8 Cours 1  Intro + schedule + finite state machines Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytechlille.fr Université Lille 1  Polytech Lille
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationMA106 Linear Algebra lecture notes
MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationsconvexity, model sets and their relation
sconvexity, model sets and their relation Zuzana Masáková Jiří Patera Edita Pelantová CRM2639 November 1999 Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical
More information3515ICT Theory of Computation Turing Machines
Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 90 Overview Turing machines: a general model of computation
More informationTuring Machines: An Introduction
CIT 596 Theory of Computation 1 We have seen several abstract models of computing devices: Deterministic Finite Automata, Nondeterministic Finite Automata, Nondeterministic Finite Automata with ɛtransitions,
More informationCourse Manual Automata & Complexity 2015
Course Manual Automata & Complexity 2015 Course code: Course homepage: Coordinator: Teachers lectures: Teacher exercise classes: Credits: X_401049 http://www.cs.vu.nl/~tcs/ac prof. dr. W.J. Fokkink home:
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationClassical Information and Bits
p. 1/24 Classical Information and Bits The simplest variable that can carry information is the bit, which can take only two values, 0 or 1. Any ensemble, description, or set of discrete values can be quantified
More informationIntroduction to Scheduling Theory
Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling
More informationDegrees of freedom in (forced) symmetric frameworks. Louis Theran (Aalto University / AScI, CS)
Degrees of freedom in (forced) symmetric frameworks Louis Theran (Aalto University / AScI, CS) Frameworks Graph G = (V,E); edge lengths l(ij); ambient dimension d Length eqns. pi  pj 2 = l(ij) 2 The p
More informationDegree Bounds for Gröbner Bases in Algebras of Solvable Type
Degree Bounds for Gröbner Bases in Algebras of Solvable Type Matthias Aschenbrenner University of California, Los Angeles (joint with Anton Leykin) Algebras of Solvable Type... introduced by KandriRody
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationGalois Theory III. 3.1. Splitting fields.
Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationAlgebra 3: algorithms in algebra
Algebra 3: algorithms in algebra Hans Sterk 20032004 ii Contents 1 Polynomials, Gröbner bases and Buchberger s algorithm 1 1.1 Introduction............................ 1 1.2 Polynomial rings and systems
More informationFormal Languages and Automata Theory  Regular Expressions and Finite Automata 
Formal Languages and Automata Theory  Regular Expressions and Finite Automata  Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationNoncatastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes
Noncatastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes Markus Grassl Institut für Algorithmen und Kognitive Systeme Fakultät für Informatik, Universität Karlsruhe T Am Fasanengarten
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationExplicit inverses of some tridiagonal matrices
Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,
More informationCONICS ON THE PROJECTIVE PLANE
CONICS ON THE PROJECTIVE PLANE CHRIS CHAN Abstract. In this paper, we discuss a special property of conics on the projective plane and answer questions in enumerative algebraic geometry such as How many
More informationCellular objects and Shelah s singular compactness theorem
Cellular objects and Shelah s singular compactness theorem Logic Colloquium 2015 Helsinki Tibor Beke 1 Jiří Rosický 2 1 University of Massachusetts tibor beke@uml.edu 2 Masaryk University Brno rosicky@math.muni.cz
More informationTopological properties of Rauzy fractals
Topological properties of Rauzy fractals Anne Siegel (IRISACNRS, Rennes) Joerg Thuswaldner (Univ. Leoben, Autriche) Kanazawa April 2008 Many shapes for Rauzy fractals... Several properties on examples
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationDownloaded 09/24/14 to 128.104.46.196. Redistribution subject to SIAM license or copyright; see
SIAMJ.COMPUT. c 1997 Society for Industrial and Applied Mathematics Vol. 26, No. 5, pp. 1411 1473, October 1997 007 QUANTUM COMPLEXITY THEORY ETHAN BERNSTEIN AND UMESH VAZIRANI Abstract. In this paper
More information