Quantum Evaluation of Multi-Valued Boolean Functions
Kazuo Iwama, Akinori Kawachi, Hiroyuki Masuda, Raymond H. Putra, Shigeru Yamashita
Abstract
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
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⋅xmod2
. It is also well-known that the query complexity is
Θ(
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×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(
N
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
.