1. Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials Alina Dubrovska, Taisia Mischenko-Slatenkova University of Latvia

2. 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

3. Exact Quantum Query Algorithms Exact quantum algorithm with queries. Base function: Exact quantum algorithm with queries. Base function:

4. Low Degree Polynomials Problem: construct a polynomial p(x) for which deg(p) is much lower than number of variables. For each odd k>1 there exists 3k-variable Boolean function f with D(f)=3k and deg(f)=2(k-1). Generalization: Transform the set {0,1,2,3} to {0,1}  deg(f 9 )=4, D(f 9 )=9 where p represents Boolean function f. deg(p)=2 Approach: construction of a polynomial of degree 2 and non-Boolean range of values.