Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina...
-
Upload
tiffany-farmer -
Category
Documents
-
view
216 -
download
1
description
Transcript of Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina...
![Page 1: Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina Dubrovska, Taisia Mischenko-Slatenkova University of Latvia.](https://reader036.fdocuments.us/reader036/viewer/2022082908/5a4d1ace7f8b9ab059970977/html5/thumbnails/1.jpg)
Computing Boolean Functions: Exact Quantum Query Algorithms
and Low Degree Polynomials
Alina Dubrovska, Taisia Mischenko-SlatenkovaUniversity of Latvia
![Page 2: Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina Dubrovska, Taisia Mischenko-Slatenkova University of Latvia.](https://reader036.fdocuments.us/reader036/viewer/2022082908/5a4d1ace7f8b9ab059970977/html5/thumbnails/2.jpg)
Research problem
• Query algorithms are used to compute the value of Boolean functions• Complexity = number of questions• We consider quantum query model• Every Boolean function can be represented by an algebraic polynomial, which is unique• Complexity of quantum query algorithm is related to degree of representing polynomial:
• We are searching for new efficient quantum query algorithms and functions with low polynomial degree
deg( )( )2E
fQ f
![Page 3: Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina Dubrovska, Taisia Mischenko-Slatenkova University of Latvia.](https://reader036.fdocuments.us/reader036/viewer/2022082908/5a4d1ace7f8b9ab059970977/html5/thumbnails/3.jpg)
Exact Quantum Query Algorithms
Exact quantum algorithm with queries. 2( )
3D f
3 1 2 3 1 2 1 3( , , ) ( ) ( ) F x x x x x x xBase function:
Exact quantum algorithm with queries. ( )2
D f
4 1 2 3 4 1 2 3 4( , , , )G x x x x x x x x Base function:
![Page 4: Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina Dubrovska, Taisia Mischenko-Slatenkova University of Latvia.](https://reader036.fdocuments.us/reader036/viewer/2022082908/5a4d1ace7f8b9ab059970977/html5/thumbnails/4.jpg)
Low Degree Polynomials
Problem: construct a polynomial p(x) for which deg(p) is much lower than number of variables.
][
),(,
:,1 ),,(
Ni
Sjiji
jijiiN xxxxxp
For each odd k>1 there exists 3k-variable Boolean function f with D(f)=3k and deg(f)=2(k-1).
Generalization:
1 2 3 4 5 6 7 8 9[9] , :
,( , )
( , , , , , , , , ) i i ji i j
i ji j S
p x x x x x x x x x x x x
}3,2,1,0{x
Transform the set {0,1,2,3} to {0,1} deg(f9)=4, D(f9)=9 where p represents Boolean function f.
deg(p)=2
Approach: construction of a polynomial of degree 2 and non-Boolean range of values.