A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~\Theta-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie

Error-Correcting Data Structures

We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been corrupted by a constant fraction of errors. This new model is the common generalization of (static) data structures and locally decodable error-correcting codes. The main issue is the tradeoff between the space used by the data structure and the time (number of probes) needed to answer a query about the encoded object. We prove a number of upper and lower bounds on various natural error-correcting data structure problems. In particular, we show that the optimal length of error-correcting data structures for the Membership problem (where we want to store subsets of size s from a universe of size n) is closely related to the optimal length of locally decodable codes for s-bit strings.Comment: 15 pages LaTeX; an abridged version will appear in the Proceedings of the STACS 2009 conferenc

Group privacy management strategies and challenges in Facebook : a focus group study among Flemish youth organizations

A large body of research has studied young people’s privacy practices and needs in Facebook. Less is known about group privacy. In this study 12 focus groups were organized with a total of 78 adolescents and young adults of local Flemish youth organizations to discuss their privacy practices. Findings describe how different strategies are used to coordinate the group information flow. The study also shows how online group privacy management can be challenging because ‘implicit’ privacy rules need to be made ‘explicit’, personal boundaries may conflict with those of the group one belongs to and privacy turbulence is difficult to define

Search for QCD-instantons at HERA

Signals of QCD instanton induced processes are searched for in deep-inelastic ep scattering at HERA in a kinematic region defined by the Bjorken scaling variables x>0.001, 0.1156 degrees. Upper limits are derived from the expected instanton-induced final state properties based on the QCDINS Monte Carlo model.Comment: 3 pages, 4 figures, World Scientific Doc. class (included); For the H1 Collaboration; to be publ. in Proc. ICHEP 2000, Osak

Genuine Correlations in Hadronic Z0^0 Decays

Correlations among hadrons with the same electric charge produced in Z$^0$ decays are studied using the high statistics data collected from 1991 through 1995 with the OPAL detector at LEP. Normalized factorial cumulants up to fourth order are used to measure genuine pa rticle correlations as a function of the size of phase space domains in rapidity, azimuthal angle and transverse momentum. tein correlations. Some of the recently proposed algorithms to simulate Bose-Einstein effects, implemented in the Monte Carlo model \PYTHIA, reproduce reasonably well the me asured second- and higher-order correlations between particles with the same charge as well as those in all-charge particle multiplets.Comment: 6 pages, 4 figures (in ps), talk given at XXXI International Symposium on Multiparticle Dynamics, Sept 1-7, 2001, Datong China. See http://202.114.35.18

A Survey of Quantum Learning Theory

This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation. This version will appear as Complexity Theory Column in SIGACT News in June 2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a referenc

Average-Case Quantum Query Complexity

We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.Comment: 14 pages, LaTeX. Some parts rewritten. This version to appear in the Journal of Physics

Locally Decodable Quantum Codes

We study a quantum analogue of locally decodable error-correcting codes. A q-query locally decodable quantum code encodes n classical bits in an m-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most q qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a classical q-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that q-query quantum codes are not significantly better than q-query classical codes, at least for constant or small q.Comment: 15 pages, LaTe

Rational approximations and quantum algorithms with postselection

We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a factor of 2) the minimal query complexity of the latter. We give optimal (up to constant factors) quantum algorithms with postselection for the Majority function, slightly improving upon an earlier algorithm of Aaronson. Finally we show how Newman's classic theorem about low-degree rational approximation of the absolute-value function follows from these algorithms.Comment: v2: 12 pages LaTeX, to appear in Quantum Information and Computation. Compared to version 1, the writing has been improved but the results are unchange