Boolean Functions with a Low Polynomial Degree and Quantum Query Algorithms Raitis Ozols1 , R¯ usi¸ nˇs Freivalds1, , Jevge¸nijs Ivanovs1 , El¯ına Kalni¸ na1 , na3 Lelde L¯ace1 , Masahiro Miyakawa2 , Hisayuki Tatsumi2 , and Daina Taimi¸ 1
Institute of Mathematics and Computer Science, University of Latvia, Raina bulv. 29, Riga, Latvia
[email protected] 2 Tsukuba College of Technology, 4-3-15 Amakubo, Tsukuba, Ibaraki, 305-0005 Japan
[email protected] 3 Department of Mathematics, Cornell University, 511 Malott Hall, Ithaca, NY, 14853, U.S.A.
[email protected]
Abstract. The complexity of quantum query algorithms computing Boolean functions is strongly related to the degree of the algebraic polynomial representing this Boolean function. There are two related difficult open problems. First, Boolean functions are sought for which the complexity of exact quantum query algorithms is essentially less than the complexity of deterministic query algorithms for the same function. Second, Boolean functions are sought for which the degree of the representing polynomial is essentially less than the complexity of deterministic query algorithms. We present in this paper new techniques to solve the second problem.
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Introduction
In the query model, the input x1 , . . . , xN is contained in a black box and can be accessed by queries to the black box. In each query, we give i to the black box and the black box outputs xi . The goal is to solve the problem with the minimum number of queries. The classical version of this model is known as decision trees The quantum version of this model is described in [2]. Deutsch [3] constructed an unexpected quantum query algorithm computing the 2-argument Boolean function PARITY with 1 query only such that this algorithm produces the correct result with probability 1. Such quantum query algorithms are called exact quantum query algorithms. It is a well-known open problem to construct exact quantum query algorithms with complexity
Research supported by Grant No.01.0354 from the Latvian Council of Science and by the European Commission, Contract IST-1999-11234 (QAIP).
M. Bielikov´ a et al. (Eds.): SOFSEM 2005, LNCS 3381, pp. 408–412, 2005. c Springer-Verlag Berlin Heidelberg 2005
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(number of queries) much smaller than the complexity of deterministic query algorithms for the same Boolean function. Deutsch’s algorithm allows to save half of queries needed for deterministic query algorithms to compute PARITY. Unfortunately, up to now this is the best proved advantage of exact quantum vs. deterministic query algorithms. On the other hand, the complexity of an exact quantum query algorithm QE (f ) can never be less than 3 D(f ) for the same function f (Midrij¯ anis [5], improving over Beals et al. [1]). However there is a huge gap between D(f )/2 and 3 D(f ). Many researchers have tried to bridge the gap but with no success. Every Boolean function can be expressed as an algebraic polynomial. For instance, x1 ∨ x2 can be expressed as x1 + x2 − x1 x2 , x1 ⊕ x2 can be expressed as x1 + x2 − 2x1 x2 , the function x1 ∨ x2 ∨ x3 can be expressed as x1 + x2 + x3 − x1 x2 −x1 x3 −x2 x3 −x1 x2 x3 . This representing polynomial is unique. The degree of the representing polynomial is called the degree of the Boolean function and it is denoted as deg(f ). Theorem 1. [1] For arbitrary Boolean function f , deg(f ) ≤ D(f ). Theorem 2. [1] For arbitrary Boolean function f , 12 deg(f ) ≤ QE (f ). Hence to find a Boolean function for which the exact quantum query complexity is essentially smaller than the deterministic query complexity, we are to consider Boolean functions with small deg(f ) and larger D(f ). However even this problem presents a big difficulty. We present in this paper new techniques to solve this problem but the concrete results are still moderate. We have improved the existing best upper bound but the open problem is still open.
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Graphs and Functions
We write variables x1 , x2 , . . . x7 where xi ∈ {0, 1} (Figure 1) in a circle and consider them as graph vertices. Let us compare variables x1 and x2 . If x1 = x2 then we connect them with a continuous line . If x1 = x2 then we connect them with dashed line. After that we compare x2 and x3 and again connect them with the appropriate line. We continue until we get variables x7 and x1 which we again connect. This is the way how we get closed cyclic graph with ”coloured” edges (Figure 2). Edges which are drawn with dashed lines let us call differences. Now we show that number of differences will always be an even number. Sum of (x1 − x2 )2 + (x2 − x3 )2 + . . . + (x6 − x7 )2 + (x7 − x1 )2 is always an even number 0, 2, 4 or 6. Hence the number of differences can only be 0, 2, 4 or 6. Moreover, it is easy to understand that the number of differences in the graph can be expressed by a function ϕ = (x1 − x2 )2 + (x2 − x3 )2 + . . . + (x6 − x7 )2 + (x7 − x1 )2 .
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Hence for all x1 , x2 , . . . , x7 ∈ {0, 1}, ϕ ∈ {0, 2, 4, 6}. From here ϕ − 3 ∈ {−3, −1, 1, 3} and (ϕ − 3)2 ∈ {9, 1, 1, 9}. (ϕ − 3)2 − 1 ∈ {8, 0, 0, 8} and f0 = ((ϕ − 3)2 − 1)/8 ∈ {1, 0, 0, 1}. Thus, the derived function f0 = f0 (x1 , x2 , . . . x7 ) is a Boolean function, because all of its variables xi ∈ {0, 1}, f0 ∈ {0, 1}. We have deg(f0 ) = deg(((ϕ − 3)2 − 1)/8) = 4. Obviously, 0 if d=2 or d=4 , (d is the number of differences). f0 = 1 if d=0 or d=6 Considering the case x1 = x2 = . . . = x7 we see that D(f0 ) = 7. The technique described above can be generalized: 1. We take any Boolean function h(x, y) with two variables which has no fictive variables (for example, in case of function f0 we consider function h(x, y) = (x − y)2 = x + y − 2xy. 2. We consider a graph G with n vertices. For each vertice we assign a variable xi . After then for each graph edge (xi , xj ) we calculate h(xi , xj ) and the sum for all edges. Then we get a function ϕ(x1 , x2 , . . . , xn ): ϕ(x1 , x2 , . . . , xn ) =
1≤i
