Query complexity of quantum biased oracles with explicit bias rate

We investigate the query complexity of quantum biased oracles. $\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\epsilon$ that the biased oracles answer queries correctly with probability at least $1/2+\epsilon$ .Given such an oracle, we present an algorithm to simulate a single query to an oracle that answers queries $\mathrm{C}\mathrm{O}\alpha \mathrm{e}\mathrm{c}\mathrm{u}_{\mathrm{y}}$ with probability at least 2/3, using $0(1/\epsilon)$ queries to the given oracle.For searching problems, combining the algorithm with a known result.we can obtain an optimal algorithm.The simulating algorithm works effectively when we know the value of $\epsilon$ .We also consider the situation where no knowledge about $\epsilon$ is given.We investigate the query complexity of quantum biased oracles.$\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\epsilon$ that the biased oracles answer queries correctly with probability at least $1/2+\epsilon$ .Given such an oracle, we present an algorithm to simulate a single query to an oracle that answers queries $\mathrm{C}\mathrm{O}\alpha \mathrm{e}\mathrm{c}\mathrm{u}_{\mathrm{y}}$ with probability at least 2/3, using $0(1/\epsilon)$ queries to the given oracle.For searching problems, combining the algorithm with a known result.we can obtain an optimal algorithm.The simulating algorithm works effectively when we know the value of $\epsilon$ .We also consider the situation where no knowledge about $\epsilon$ is given.

数理解析研究所講究録May 1, 20060 citations

General Bounds for Quantum Biased Oracles

Kazuo Iwama, Rudy Raymond, Shigeru Yamashita

An oracle with bias ε is an oracle that answers queries correctly with a probability of at least 1/2+ε. In this paper, we study the upper and lower bounds of quantum query complexity of oracles with bias ε. For general upper bounds, we show that for any quantum algorithm solving some problem with high probability using T queries of perfect quantum oracles, i.e., oracles with ε =1/2, there exists a quantum algorithm solving the same problem, also with high probability, using O(T/ε) queries of the corresponding biased quantum oracles. As corollaries we can show robust quantum algorithms and gaps between biased quantum and classical oracles, e.g., by showing a problem where the quantum query complexity is O(N/ε) but the classical query complexity is lower bounded by Ω(N logN/ε2). For general lower bounds, we generalize Ambainis' quantum adversary argument to biased quantum oracles and obtain the first lower bounds with explicit bias factor. As one of its applications we can provide another proof of the optimality of quantum algorithms for the so-called quantum Goldreich-Levin problem which was proved before by Adcock, et al. using different and more complicated methods.An oracle with bias ε is an oracle that answers queries correctly with a probability of at least 1/2+ε. In this paper, we study the upper and lower bounds of quantum query complexity of oracles with bias ε. For general upper bounds, we show that for any quantum algorithm solving some problem with high probability using T queries of perfect quantum oracles, i.e., oracles with ε =1/2, there exists a quantum algorithm solving the same problem, also with high probability, using O(T/ε) queries of the corresponding biased quantum oracles. As corollaries we can show robust quantum algorithms and gaps between biased quantum and classical oracles, e.g., by showing a problem where the quantum query complexity is O(N/ε) but the classical query complexity is lower bounded by Ω(N logN/ε2). For general lower bounds, we generalize Ambainis' quantum adversary argument to biased quantum oracles and obtain the first lower bounds with explicit bias factor. As one of its applications we can provide another proof of the optimality of quantum algorithms for the so-called quantum Goldreich-Levin problem which was proved before by Adcock, et al. using different and more complicated methods.

IPSJ Digital CourierJan 1, 20056 citations

We investigate the query complexity of quantum biased oracles.

\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\epsilon

that the biased oracles answer queries correctly with probability at least

1/2+\epsilon

.Given such an oracle, we present an algorithm to simulate a single query to an oracle that answers queries

\mathrm{C}\mathrm{O}\alpha \mathrm{e}\mathrm{c}\mathrm{u}_{\mathrm{y}}

with probability at least 2/3, using

0(1/\epsilon)

queries to the given oracle.For searching problems, combining the algorithm with a known result.we can obtain an optimal algorithm.The simulating algorithm works effectively when we know the value of

\epsilon

.We also consider the situation where no knowledge about

\epsilon

is given.