Fourier 1-norm and quantum speed-up Sebasti´ an Alberto Grillo Universidad Aut´ onoma de Asunci´ on sgrillo@uaa.com August 19, 2019 Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 1 / 20
Overview Query Models 1 The Fourier expansion over Boolean functions 2 A classical simulation for functions on the Boolean cube 3 Bounding theorems 4 An application 5 Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 2 / 20
Decision trees We want to compute a Boolean function. The depth of the tree represents the complexity. A randomized tree is a probabilistic distribution over deterministic trees. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 3 / 20
The quantum query model I The states of our computer are described by unit vectors in a Hilbert space H , whose basis is | i � | j � , where i ∈ { 0 , 1 , .., n } and j ∈ { 1 , .., m } . We have a set of unitary operators { U i } over H . We denote a query operator O x , such that O x | i � | j � = ( − 1) x i | i � | j � , where x ≡ x 0 x 1 · · · x n is the input, and x 0 ≡ 0. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 4 / 20
The quantum query model II The initial state of the algorithm is | 0 � | 0 � . The final state of the algorithm over input x is defined as � � Ψ f � = U t O x U t − 1 ... O x U 0 | 0 � | 0 � . x We denote a query operator O x , such that O x | i � | j � = ( − 1) x i | i � | j � , where x ≡ x 0 x 1 · · · x n is the input, and x 0 ≡ 0. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 5 / 20
The quantum query model III CSOP An indexed set of pairwise orthogonal projectors { P z : z ∈ T } is called a Complete Set of Orthogonal Projectors if it satisfies � P z = I H . (1) z ∈ T The probability of obtaining the output z ∈ T is � 2 . � � Ψ f � �� π z ( x ) = � P z x An algorithm computes a function f : D → T within error ε if π f ( x ) ( x ) ≥ 1 − ε for all input x ∈ D ⊂ { 0 , 1 } n . Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 6 / 20
The Fourier basis I We consider the Fourier basis for the vector space of all functions f : { 0 , 1 } n → R given by the functions χ b : { 0 , 1 } n → { 1 , − 1 } , such that χ b ( x ) = ( − 1) b · x for b ∈ { 0 , 1 } n and b · x = � i b i x i . This family contains a constant function that we denote as χ 0 = 1. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 7 / 20
The Fourier basis II Any function f : { 0 , 1 } n → R has a unique representation as a linear combination � f = (2) α b χ b . b ∈{ 0 , 1 } n Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 8 / 20
Metrics 1-norm we denote the Fourier 1-norm of f as � L ( f ) = | α b | . (3) b ∈{ 0 , 1 } n Degree Another measure is the degree of f , which is defined as deg ( f ) = max | b | { b : α b � = 0 } . (4) Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 9 / 20
The intuition Any output probability can be decomposed in functions χ b . We define a classical simulation for each χ b . 1 1 0 1 - 1 0 e r r o r n n o 1 1 i o t 1 i n i t o s a i o l t i u s p o m m 0 i p o s - c m e b o d u c s 0 0 - 1 e r r o r 1 1 0 0 - 1 0 Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 10 / 20
Notation R ε ( f ) denotes the minimum number of queries that are necessary for computing f within error ε by a randomized decision tree. π 1 ( x ) is the probability of a quantum algorithm returning output 1 for a given input x . Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 11 / 20
First Bound Theorem Consider D ⊂ { 0 , 1 } n and a function f : D → { 0 , 1 } that is computed within error ε > 0 and t queries, by a quantum query algorithm. If we define � − 16 ln ( ε ) (1 + l ) (1 + l − ε ) � F ε ( l ) = (5) , (1 − 2 ε ) 2 then R ε ( f ) ≤ F ε ( L ( π 1 )) . (6) t Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 12 / 20
Second Bound Theorem Consider D ⊂ { 0 , 1 } n and a function f : D → { 0 , 1 } that is ε -approximated by a polynomial p : R n → R . If deg ( p ) ≤ 2 t , then R ε ( f ) ≤ F ε ( L ( p )) . (7) 2 t Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 13 / 20
A characterization between degree and query complexity. A partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by ǫ < 1/2 iff f can be approximated by a degree-2 polynomial with error bounded by ǫ ′ < 1/2. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 14 / 20
Fourier analysis of degree-2 polynomials. The Fourier spectrum is composed by Walsh functions of the form χ ( x ) = − 1( ax i + bx j ). Each Walsh function is affected by at most two values of the input x . Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 15 / 20
1-query algorithms as weighted dynamic graphs Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 16 / 20
Graphs that maximize L-1 norm How to maximize L-1 norm on graph based quantum query algorithms Low difference between weights. Minimizing the upper-bound value of the sum weight for every vertex and configuration. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 17 / 20
An special case Consider a regular graph with the following property: The edges of the graph intersect almost half of the edges of any complete bipartite graph that has the same vertices. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 18 / 20
Conclusions Weighted graphs are an alternative representation for 1-query quantum algorithms. Such representation gives a direct upper bound for quantum speed up over a similar classical algorithm. Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 19 / 20
The End Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 20 / 20
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