Fourier 1-norm and quantum speed-up Sebasti an Alberto Grillo - - PowerPoint PPT Presentation

Fourier 1-norm and quantum speed-up

Sebasti´ an Alberto Grillo

Universidad Aut´

• noma de Asunci´
• n

sgrillo@uaa.com

August 19, 2019

Sebasti´ an Alberto Grillo (UAA) Quantum Computing August 19, 2019 1 / 20

Overview

1

Query Models

2

The Fourier expansion over Boolean functions

3

A classical simulation for functions on the Boolean cube

4

Bounding theorems

5

An application

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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.

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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 {Ui} over H. We denote a query operator Ox, such that Ox |i |j = (−1)xi |i |j, where x ≡ x0x1 · · · xn is the input, and x0 ≡ 0.

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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

• Ψf

x

• = UtOxUt−1...OxU0 |0 |0.

We denote a query operator Ox, such that Ox |i |j = (−1)xi |i |j, where x ≡ x0x1 · · · xn is the input, and x0 ≡ 0.

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The quantum query model III

CSOP

An indexed set of pairwise orthogonal projectors {Pz : z ∈ T} is called a Complete Set of Orthogonal Projectors if it satisfies

• z∈T

Pz = IH. (1) The probability of obtaining the output z ∈ T is πz (x) =

• Pz
• Ψf

x

• 2.

An algorithm computes a function f : D → T within error ε if πf (x) (x) ≥ 1 − ε for all input x ∈ D ⊂ {0, 1}n.

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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 χb : {0, 1}n → {1, −1} , such that χb(x) = (−1)b·x for b ∈ {0, 1}n and b · x =

i bixi. This family

contains a constant function that we denote as χ0 = 1.

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The Fourier basis II

Any function f : {0, 1}n → R has a unique representation as a linear combination f =

• b∈{0,1}n

αbχb. (2)

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Metrics

1-norm

we denote the Fourier 1-norm of f as L (f ) =

• b∈{0,1}n

|αb| . (3)

Degree

Another measure is the degree of f , which is defined as deg (f ) = max

|b| {b : αb = 0} .

(4)

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The intuition

Any output probability can be decomposed in functions χb. We define a classical simulation for each χb.

d e c

• m

p

• s

i t i

• n

1 1

• 1

1

• 1

1

• 1

s u b

• s

i m u l a t i

• n

1 1 1

c

• m

p

• s

i t i

• n

1

e r r

• r

e r r

• r

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Notation

Rε (f ) denotes the minimum number of queries that are necessary for computing f within error ε by a randomized decision tree. π1(x) is the probability of a quantum algorithm returning output 1 for a given input x.

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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 Fε (l) = −16 ln (ε) (1 + l) (1 + l − ε) (1 − 2ε)2

• ,

(5) then Rε (f ) t ≤ Fε (L (π1)) . (6)

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Second Bound

Theorem

Consider D ⊂ {0, 1}n and a function f : D → {0, 1} that is ε-approximated by a polynomial p : Rn → R. If deg (p) ≤ 2t, then Rε (f ) 2t ≤ Fε (L (p)) . (7)

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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.

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Fourier analysis of degree-2 polynomials.

The Fourier spectrum is composed by Walsh functions of the form χ (x) = −1(axi+bxj). Each Walsh function is affected by at most two values of the input x.

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1-query algorithms as weighted dynamic graphs

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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.

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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.

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Conclusions

Weighted graphs are an alternative representation for 1-query quantum algorithms. Such representation gives a direct upper bound for quantum speed up

• ver a similar classical algorithm.

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The End

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