Quantum Evaluation of Multi-Valued Boolean Functions

Our problem is to evaluate a multi-valued Boolean function F through oracle calls. If F is one-to-one and the size of its domain and range is the same, then our problem can be formulated as follows: Given an oracle f(a, x) : {0, 1}n × {0, 1}n → {0, 1} and a fixed (but hidden) value a0, we wish to obtain the value of a0 by querying the oracle f(a0, x). Our goal is to minimize the number of such oracle calls (the query complexity) using a quantum mechanism. Two popular oracles are the EQ-oracle defined as f(a, x) = 1 iff x = a and the IP-oracle defined as f(a, x) = a · x mod 2. It is also well-known that the query complexity is Θ( √ N) (N = 2n) for the EQ-oracle while only O(1) for the IP-oracle. The main purpose of this paper is to fill this gap or to investigate what causes this large difference. To do so, we introduce a parameter K as the maximum number of 1’s in a single column of Tf where Tf is the N × N truth-table of the oracle f(a, x). Our main result shows that the (quantum) query complexity is heavily governed by this parameter K: (i) The query complexity is Ω( √ N/K). (ii) This lower bound is tight in the sense that we can construct an explicit oracle whose query complexity is O( √ N/K). (iii) The tight complexity, Θ( K + logK), is also obtained for the classical case. Thus, the quantum algorithm needs a quadratically less number of oracle calls when K is small and this merit becomes larger when K is large, e.g., logK v.s. constant when K = cN .