scholarly journals Quantum conditional query complexity

2017 ◽  
Vol 17 (7&8) ◽  
pp. 541-567
Author(s):  
Imdad S.B. Sardharwalla ◽  
Sergii Strelchuk ◽  
Richard Jozsa
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Query Complexity ◽  
Equivalence Testing ◽  
Conditional Probabilities ◽  
Underlying Distribution ◽  
New Type ◽  
Testing Equivalence ◽  
Oracle Access

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals Following Forrelation – quantum algorithms in exploring Boolean functions' spectra

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Suman Dutta ◽  
Subhamoy Maitra ◽  
Chandra Sekhar Mukherjee
Keyword(s):  
Boolean Functions ◽  
Cross Correlation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Sampling Technique ◽  
Linear Functions ◽  
Query Complexity ◽  
Walsh Spectrum ◽  
Correlation Spectrum ◽  
The Cross

<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>

DERANDOMIZED LEARNING OF BOOLEAN FUNCTIONS OVER FINITE ABELIAN GROUPS

2001 ◽  
Vol 12 (04) ◽  
pp. 491-516
Author(s):  
M. SITHARAM ◽  
TIMOTHY STRANEY
Keyword(s):  
Boolean Functions ◽  
Partial Information ◽  
Abelian Groups ◽  
Query Complexity ◽  
Character Theory ◽  
Finite Abelian Groups ◽  
Training Set ◽  
Membership Queries ◽  
Approximation Classes ◽  
Query Model

We employ the Always Approximately Correct or AAC model defined in [35], to prove learnability results for classes of Boolean functions over arbitrary finite Abelian groups. This model is an extension of Angluin's Query model of exact learning. The Boolean functions we consider belong to approximation classes, i.e. functions that are approximable (in various norms) by few Fourier basis functions, or irreducible characters of the domain Abelian group. We contrast our learnability results to previous results for similar classes in the PAC model of learning with and without membership queries. In addition, we discuss new, natural issues and questions that arise when the AAC model is used. One such question is whether a uniform training set is available for learning any function in a given approximation class. No analogous question seems to have been studied in the context of Angluin's Query model. Another question is whether the training set can be found quickly if the approximation class of the function is completely unknown to the learner, or only partial information about the approximation class is given to the learner (in addition to the answers to membership queries). In order to prove the learnability results in this paper we require new techniques for efficiently sampling Boolean functions using the character theory of finite Abelian groups, as well as the development of algebraic algorithms. The techniques result in other natural applications closely related to learning, for example, query complexity of deterministic algorithms for testing linearity, efficient pseudorandom generators, and estimating VC dimensions for classes of Boolean functions over finite Abelian groups.

The limit behaviour of the maximum of random variables with applications to outlier-resistance

1984 ◽  
Vol 21 (03) ◽  
pp. 646-650 ◽  
Author(s):  
Rudolf Mathar
Keyword(s):  
Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Limit Laws ◽  
Underlying Distribution ◽  
Limit Behaviour

We consider degenerate limit laws for the sequence {Xn, n } n (N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.

scholarly journals Quantum Generative Adversarial Networks for learning and loading random distributions

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Christa Zoufal ◽  
Aurélien Lucchi ◽  
Stefan Woerner
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum States ◽  
Generative Adversarial Networks ◽  
Adversarial Networks

AbstractQuantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require $${\mathcal{O}}\left({2}^{n}\right)$$O2n gates to load an exact representation of a generic data structure into an $$n$$n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage. Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions - implicitly given by data samples - into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state. The loading requires $${\mathcal{O}}\left(poly\left(n\right)\right)$$Opolyn gates and can thus enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation. We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

Quantum algorithms for learning Walsh spectra of multi-output Boolean functions

2019 ◽  
Vol 18 (6) ◽  
Author(s):  
Jingyi Cui ◽  
Jiansheng Guo ◽  
Linhong Xu ◽  
Mingming Li
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Walsh Spectra

Quantum algorithms for testing and learning Boolean functions

2013 ◽  
Vol 23 (02) ◽  
pp. 386-398 ◽  
Author(s):  
DOMINIK FLOESS ◽  
ERIKA ANDERSSON ◽  
MARK HILLERY
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Learning Boolean Functions
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