A comparison of Zeroes and Ones of a Boolean Polynomial

In this paper we consider the computational complexity of the following problem. Let f be a Boolean polynomial. What value of f , 0 or 1, is taken more frequently? The problem is solved in polynomial time for polynomials of degrees 1, 2. The next case of degree 3 appears to be PP-complete under polynomial reductions in the class of promise problems. The proof is based on techniques of quantum computation. The class PP was defined by J.Gill [4] as probabilistic polynomial time with unbounded error. The following problem represents all computational power of PP: A Boolean function f is given by a Boolean circuit computing this function. What value of f , 0 or 1, is taken more frequently? In fact, this value comparison problem is PP-complete under polynomial reductions, if promise problems are considered instead of decision problems. A promise problem is a decision problem in which some inputs are excluded. So, a promise problem F is described by a pair of disjoint sets (Fyes, Fno) of strings corresponding to “yes” and “no” instances. In this paper we restrict the value comparison problem to the case of Boolean polynomials of fixed degree. It is known that the problem of counting of zeroes for polynomials of degree 3 is #P-complete [2]. An easy corollary of this result is PP-completeness of the value comparison problem for polynomials of degree 4 (see Theorem 1 below). At other hand, the comparison problem for polynomials of degree 1 is trivial. In the case of degree 2 the problem is solved in polynomial time (by reduction to the canonical form, see [11, 10]). We address to the remaining case — polynomials of degree 3. It will be shown that it is PP-complete. Surprisingly enough, the proof will use techniques of quantum computation. We will apply the theorem of efficient approximation for unitary operators [5] and the results of [8]. It was shown in [8] that a problem of determination of sign of specific quadratically signed weight enumerators is BQP-complete (again, we mean the completeness in the class of promise problems). It is possible to use the results of [8] directly for the proof of our main theorem. Instead, we prefer to follow the arguments of [8] and present a slightly more restrictive form of the enumerators. 1 Preliminaries 1.1 An another definition of PP We will also use the definition of the class PP given by Fenner, Fortnow and Kurtz [3]. They introduced the class GapP functions consisting of the closure under substraction of the set of #P functions. In other words, for any GapP function f : B → Z there are predicates Q1(·, ·), Q2(·, ·) ∈ P and a polynomial q(·) such that for all x f(x) = Card{y : Q1(x, y)&|y| = q(|x|)} − Card{y : Q2(x, y)&|y| = q(|x|)}. (1) The class PP can be defined in these terms as follows: ∗Supported by RFBR grant 99-01-00122