arXiv subjects

Alp Atici

Publications and source records attributed to Alp Atici.

4 recordsLinked to original sources

arXiv2007

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $ε$ that uses $O(k/ε)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $Ω(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $ε$ that uses $O(ε^{-1} k\log k)$ quantum examples and $O(2^k \log(1/ε))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.

quant-ph↗

arXiv2007

Learning Unions of $ω(1)$-Dimensional Rectangles

We consider the problem of learning unions of rectangles over the domain $[b]^n$, in the uniform distribution membership query learning setting, where both b and n are "large". We obtain poly$(n, \log b)$-time algorithms for the following classes: - poly$(n \log b)$-way Majority of $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles. - Union of poly$(\log(n \log b))$ many $O(\frac{\log^2 (n \log b)} {(\log \log(n \log b) \log \log \log (n \log b))^2})$-dimensional rectangles. - poly$(n \log b)$-way Majority of poly$(n \log b)$-Or of disjoint $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles. Our main algorithmic tool is an extension of Jackson's boosting- and Fourier-based Harmonic Sieve algorithm [Jackson 1997] to the domain $[b]^n$, building on work of [Akavia, Goldwasser, Safra 2003]. Other ingredients used to obtain the results stated above are techniques from exact learning [Beimel, Kushilevitz 1998] and ideas from recent work on learning augmented $AC^{0}$ circuits [Jackson, Klivans, Servedio 2002] and on representing Boolean functions as thresholds of parities [Klivans, Servedio 2001].

cs.LG↗

arXiv2005

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hatγ}^{C}}})$, where $\hatγ^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\log |C| \log \log |C|}{\sqrt{{\hatγ}^{C}}})$ quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for $(ε,δ)$-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain $\{0,1\}^n$ from $Ω(\frac{d}{n})$ to $Ω(\frac{1}ε\log \frac{1}δ+d+\frac{\sqrt{d}}ε)$. This new lower bound comes closer to matching known upper bounds for classical PAC learning.

quant-ph↗

arXiv2004

Comparative Computational Strength of Quantum Oracles

It is an established fact that for many of the interesting problems quantum algorithms based on queries of the standard oracle bring no significant improvement in comparison to known classical algorithms. It is conceivable that there are other oracles of algorithmic importance acting in a less intuitive fashion to which such limitations do not apply. Thus motivated this article suggests a broader understanding towards what a general quantum oracle is. We propose a general definition of a quantum oracle and give a classification of quantum oracles based on the behavior of the eigenvalues and eigenvectors of their queries. Our aim is to determine the computational characteristics of a quantum oracle in terms of these eigenvalues and eigenvectors. Within this framework we attempt to describe the class of quantum oracles that are efficiently simulated by the standard oracle and compare the computational strength of different kinds of quantum oracles by an adversary argument using trigonometric polynomials.

quant-ph↗