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arXiv:quant-ph/0411204 · IPSJ SIG Notes

Robust Quantum Algorithms for Oracle Identification

Andris Ambainis, Kazuo Iwama, Akinori Kawachi, Rudy Raymond, Shigeru Yamashita

Published Nov 30, 2004• 11 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0411204

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The oracle identification problem (OIP) was introduced by Ambainis et al. \cite{AIKMRY04}. It is given as a set SSS of MMM oracles and a blackbox oracle fff. Our task is to figure out which oracle in SSS is equal to the blackbox fff by making queries to fff. 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: (iii) For any oracle set SSS such that ∣S∣≤2Nd|S| \le 2^{N^d}∣S∣≤2Nd (d<1d < 1d<1), we design an algorithm whose query complexity is O(Nlog⁡M/log⁡N)O(\sqrt{N\log{M}/\log{N}})O(NlogM/logN​), matching the lower bound proved in \cite{AIKMRY04}. (iiiiii) Our algorithm also works for the range between 2Nd2^{N^d}2Nd and 2N/log⁡N2^{N/\log{N}}2N/logN (where the bound becomes O(N)), but the gap between the upper and lower bounds worsens gradually. (iiiiiiiii) 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.
The oracle identification problem (OIP) was introduced by Ambainis et.al. [4], which is given as a set SSS of MMM oracles and a hidden oracle fff .Our task is to figure out which oracle in SSS is equal to the hidden fff by doing queries to fff .OIP includes several problems such as Grover Search as special cases.In this paper, we design robust algorithms, i.e., those which are tolerant against noisy oracles, for OIP.Our results include: (i) For any oracle set SSS such that ∣S∣|S|∣S∣ is polynomial in NNN , O(N)O(\sqrt{N})O(N​) queries are enough to identify the hidden oracle, which is obviously optimal since this OIP includes Grover Search as a special case, (ii) For the case that ∣S∣≤2Nd(d<1)|S|\leq 2^{N^{d}}(d<1)∣S∣≤2Nd(d<1) , we design an algorithm whose query complexity is O(Nlog⁡M/log⁡N)O(\sqrt{N\log M/\log N})O(NlogM/logN​) and matches the lower bound proved in [4].(Hi) We can furthermore design a robust algorithm whose complexity changes smoothly between the complexity of (ii) and the complexity of recovering all information about the hidden oracle whose complexity is O(N)O(N)O(N) as showed by Buhrman et.al. in [11].Thus our new algorithms are not only robust but also their query complexities are even better than the previous noiseless case [4],
In the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set C of size M and our task is to identify x. We present a quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm considerably simplifies and improves the previous best algorithm due to Ambainis et al. Our algorithm also has applications in quantum learning theory, where it improves the complexity of exact learning with membership queries, resolving a conjecture of Hunziker et al. The algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered γ2γ_2γ2​-norm semidefinite program, which characterizes quantum query complexity. Our composition theorem is quite general and allows us to compose quantum algorithms with input-dependent query complexities without incurring a logarithmic overhead for error reduction. As an application of the composition theorem, we remove all log factors from the best known quantum algorithm for Boolean matrix multiplication.
This paper considers the quantum query complexity of {\it $\eps$-biased oracles} that return the correct value with probability only $1/2 + \eps$. In particular, we show a quantum algorithm to compute NNN-bit OR functions with $O(\sqrt{N}/{\eps})$ queries to $\eps$-biased oracles. This improves the known upper bound of $O(\sqrt{N}/{\eps}^2)$ and matches the known lower bound; we answer the conjecture raised by the paper by Iwama et al. affirmatively. We also show a quantum algorithm to cope with the situation in which we have no knowledge about the value of $\eps$. This contrasts with the corresponding classical situation, where it is almost hopeless to achieve more than a constant success probability without knowing the value of $\eps$.
We consider quantum search algorithms that have access to a noisy oracle that, for every oracle call, with probability p>0p>0p>0 completely depolarizes the query registers, while otherwise working properly. Previous results had not ruled out quantum O(n)\mathrm{O}(\sqrt{n})O(n​)-query algorithms in this setting, even for constant ppp. We show that, for all p≤0.99p\le 0.99p≤0.99, the quantum noisy-query complexity of the unstructured search is Θ~(max⁡{np,n})\tildeΘ(\max\{np,\sqrt{n}\})Θ~(max{np,n​}). The lower bound Ω(max⁡{np,n})Ω(\max\{np,\sqrt n\})Ω(max{np,n​}) holds also for the dephasing noise and even when, for every oracle call, the algorithm is provided with a flag indicating whether the error has occurred.
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.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}nx \in \{-1,1\}^nx∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlog⁡n)O(\sqrt{n} \log n)O(n​logn) and a near-matching lower bound of Ω(n)Ω(\sqrt{n})Ω(n​). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n)O(\sqrt{n})O(n​) upper bound. We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n]\cal{Q} \subseteq 2^{[n]}Q⊆2[n], where an algorithm may test whether xS=bx_S = bxS​=b for an arbitrary S∈QS \in \cal{Q}S∈Q and b∈{−1,1}Sb \in \{-1,1\}^Sb∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q\cal{Q}Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning xxx is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→Rf : \{-1,1\}^n \to \mathbb{R}f:{−1,1}n→R the ratio of the maximum value of fff to the maximum (over T∈QT \in \cal{Q}T∈Q) standard deviation of fff on a subcube whose free variables are exactly TTT.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.