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 $f(a,x): \{0,1\}^n\times\{0,1\}^n \to \{0,1\}$ and a fixed (but hidden) value $a_0$, we wish to obtain the value of $a_0$ by querying the oracle $f(a_0,x)$. 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 $f(a,x)= a\cdot x \mod 2$. It is also well-known that the query complexity is $\Theta(\sqrt{N})$ ($N=2^n$) 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 $T_f$ where $T_f$ is the $N\times N$ 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 $\Omega(\sqrt{N/K})$. ($ii$) This lower bound is tight in the sense that we can construct an explicit oracle whose query complexity is $O(\sqrt{N/K})$. ($iii$) The tight complexity, $\Theta(\frac{N}{K}+\log{K})$, 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., $\log{K}$ v.s. constant when $K = cN$.