arXiv subjects

Kazuo Iwama

Publications and source records attributed to Kazuo Iwama.

23 records · Page 2Linked to original sources

arXiv2006

Robust Quantum Algorithms for Oracle Identification

The oracle identification problem (OIP) was introduced by Ambainis et al. \cite{AIKMRY04}. It is given as a set $S$ of $M$ oracles and a blackbox oracle $f$. Our task is to figure out which oracle in $S$ is equal to the blackbox $f$ by making queries to $f$. OIP includes several problems such as the Grover Search as special cases. In this paper, we improve the algorithms in \cite{AIKMRY04} by providing a mostly optimal upper bound of query complexity for this problem: ($i$) For any oracle set $S$ such that $|S| \le 2^{N^d}$ ($d < 1$), we design an algorithm whose query complexity is $O(\sqrt{N\log{M}/\log{N}})$, matching the lower bound proved in \cite{AIKMRY04}. ($ii$) Our algorithm also works for the range between $2^{N^d}$ and $2^{N/\log{N}}$ (where the bound becomes O(N)), but the gap between the upper and lower bounds worsens gradually. ($iii$) Our algorithm is robust, namely, it exhibits the same performance (up to a constant factor) against the noisy oracles as also shown in the literatures \cite{AC02,BNRW03,HMW03} for special cases of OIP.

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arXiv2006

(4,1)-Quantum Random Access Coding Does Not Exist

An (n,1,p)-Quantum Random Access (QRA) coding, introduced by Ambainis, Nayak, Ta-shma and Vazirani in ACM Symp. on Theory of Computing 1999, is the following communication system: The sender which has n-bit information encodes his/her information into one qubit, which is sent to the receiver. The receiver can recover any one bit of the original n bits correctly with probability at least p, through a certain decoding process based on positive operator-valued measures. Actually, Ambainis et al. shows the existence of a (2,1,0.85)-QRA coding and also proves the impossibility of its classical counterpart. Chuang immediately extends it to a (3,1,0.79)-QRA coding and whether or not a (4,1,p)-QRA coding such that p > 1/2 exists has been open since then. This paper gives a negative answer to this open question.

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arXiv2004

Quantum Identification of Boolean Oracles

The oracle identification problem (OIP) is, given a set $S$ of $M$ Boolean oracles out of $2^{N}$ ones, to determine which oracle in $S$ is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to $S$. The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper and lower bounds. They are quite general and mostly optimal: (i) The query complexity of OIP is $O(\sqrt{N\log M \log N}\log\log M)$ for {\it any} $S$ such that $M = |S| > N$, which is better than the obvious bound $N$ if $M < 2^{N/\log^{3}N}$. (ii) It is $O(\sqrt{N})$ for {\it any} $S$ if $|S| = N$, which includes the upper bound for the Grover search as a special case. (iii) For a wide range of oracles ($|S| = N$) such as random oracles and balanced oracles, the query complexity is $Θ(\sqrt{N/K})$, where $K$ is a simple parameter determined by $S$.

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arXiv2003

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 $Θ(\sqrt{N})$ ($N=2^n$) 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 $Ω(\sqrt{N/K})$. ($ii$) 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, $Θ(\frac{N}{K}+\log{K})$, 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 $K = cN$.

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arXiv2002

Exploiting the Difference in Probability Calculation between Quantum and Probabilistic Computations

The main purpose of this paper is to show that we can exploit the difference ($l_1$-norm and $l_2$-norm) in the probability calculation between quantum and probabilistic computations to claim the difference in their space efficiencies. It is shown that there is a finite language $L$ which contains sentences of length up to $O(n^{c+1})$ such that: ($i$) There is a one-way quantum finite automaton (qfa) of $O(n^{c+4})$ states which recognizes $L$. ($ii$) However, if we try to simulate this qfa by a probabilistic finite automaton (pfa) \textit{using the same algorithm}, then it needs $Ω(n^{2c+4})$ states. It should be noted that we do not prove real lower bounds for pfa's but show that if pfa's and qfa's use exactly the same algorithm, then qfa's need much less states.

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