Computing Boolean Functions: Exact Quantum Query Algorithms

and Low Degree Polynomials

Alina Dubrovska, Taisia Mischenko-SlatenkovaUniversity of Latvia

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

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:

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.