scholarly journals Testing Boolean Functions Properties

2021 ◽  
Vol 182 (4) ◽  
pp. 321-344
Author(s):  
Xie Zhengwei ◽  
Qiu Daowen ◽  
Cai Guangya ◽  
Jozef Gruska ◽  
Paulo Mateus
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Randomized Algorithm ◽  
Quantum Algorithm ◽  
Black Box ◽  
Query Complexity ◽  
Testing Problem ◽  
Amplification Technique ◽  
Query Algorithm ◽  
Amplitude Amplification

The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is ɛ-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function f, of n variables, is identical with a given function g or is ɛ-far from g, where ɛ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f, g) denote the correlation of f and g. We propose an exact quantum algorithm for making distinction between |C(f, g)| = ɛ and |C(f, g)| = 1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using O(1ε) queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of O(1/ɛ2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

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 Partial Boolean Functions With Exact Quantum Query Complexity One

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 189
Author(s):  
Guoliang Xu ◽  
Daowen Qiu
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Necessary Conditions ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Sufficient And Necessary Conditions ◽  
Quantum Query

We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.

scholarly journals EXPONENTIAL IMPROVEMENT IN PRECISION FOR SIMULATING SPARSE HAMILTONIANS

2017 ◽  
Vol 5 ◽  
Author(s):  
DOMINIC W. BERRY ◽  
ANDREW M. CHILDS ◽  
RICHARD CLEVE ◽  
ROBIN KOTHARI ◽  
ROLANDO D. SOMMA
Keyword(s):  
Discrete Model ◽  
Quantum Algorithm ◽  
Small Error ◽  
Query Complexity ◽  
Input State ◽  
Time Varying ◽  
Correction Procedure ◽  
Complex Fault ◽  
New Form ◽  
Amplitude Amplification

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\unicode[STIX]{x1D716}$ using $O(\unicode[STIX]{x1D70F}(\log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})))$ queries and $O(\unicode[STIX]{x1D70F}(\log ^{2}(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716}))n)$ additional 2-qubit gates, where $\unicode[STIX]{x1D70F}=d^{2}\Vert H\Vert _{\max }t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification relies on a new form of ‘oblivious amplitude amplification’ that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

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>

scholarly journals Using Variable-Entered Karnaugh Maps in Determining Dependent and Independent Sets of Boolean Functions

2012 ◽  
Vol 1 (2) ◽  
pp. 45-67
Author(s):  
Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Independent Sets ◽  
Solution Procedure ◽  
Black Box ◽  
Important Class ◽  
Time Saving ◽  
Prime Implicants ◽  
Simultaneous Processing ◽  
Karnaugh Map

An important class for Boolean reasoning problems involves interdependence among the members of a set T of Boolean functions. Two notable problems among this class are (a) to establish whether a given subset of T is dependent, and (b) to produce economical representations for the complementary families of all dependent subsets and independent subsets of T. This paper solves these two problems via a powerful manual pictorial tool, namely, the variableentered Karnaugh map (VEKM). The VEKM is utilized in executing a Label-and-Eliminate procedure for producing certain prime implicants or consequents used in tackling the two aforementioned problems. The VEKM procedure is a time-saving short cut indeed, since it efficiently handles the three basic tasks demanded by the solution procedure, which are: (a) To combine several Boolean relations into a single one, (b) to compute conjunctive eliminants of a Boolean function, and (c) to derive the complete sum (CS) of a Boolean function. The VEKM procedure significantly reduces the complexities of these tasks by introducing useful shortcuts and allowing simultaneous processing. The VEKM procedure is described in detail, and then demonstrated via two illustrative examples, which previously had only black-box computer solutions as they were thought to be not amenable to manual solution. The first example deals with switching or bivalent functions while the second handles 'big' Boolean functions. Both examples indicate that the VEKM procedure proposed herein enjoys the merits of insightfulness, simplicity and efficiency

scholarly journals THE DEUTSCH–JOZSA ALGORITHM REVISITED IN THE DOMAIN OF CRYPTOGRAPHICALLY SIGNIFICANT BOOLEAN FUNCTIONS

2005 ◽  
Vol 03 (02) ◽  
pp. 359-370 ◽  
Author(s):  
SUBHAMOY MAITRA ◽  
PARTHA MUKHOPADHYAY
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Block Cipher ◽  
Quantum Algorithm ◽  
Building Blocks ◽  
Linear Functions ◽  
Constant Time ◽  
Exponential Time ◽  
Classical Algorithm ◽  
Walsh Spectra

Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems, one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let f be an n-variable Boolean function and its Walsh spectra is denoted by Wf(ω) at the point ω ∈ {0, 1}n. The Boolean function is available in the form of an oracle. We like to find a ω such that Wf(ω) ≠ 0 as this will provide one of the linear functions which are correlated to f. We show that the quantum algorithm proposed by Deutsch and Jozsa7 solves this problem in constant time. However, the best known classical algorithm to solve the problem requires exponential time in n. We also analyze certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch–Jozsa algorithm performs on them.

DE-QUANTIZING THE SOLUTION OF DEUTSCH'S PROBLEM

2007 ◽  
Vol 05 (03) ◽  
pp. 409-415 ◽  
Author(s):  
CRISTIAN S. CALUDE
Keyword(s):  
Quantum Computing ◽  
Boolean Function ◽  
Quantum Algorithm ◽  
Black Box ◽  
Affirmative Answer ◽  
Comparison Of Results ◽  
Famous Paper ◽  
Complete Probability ◽  
Quantum Solution

Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called Deutsch's problem. Consider a Boolean function f: {0,1} → {0,1} and suppose that we have a (classical) black box to compute it. The problem asks whether f is constant [that is, f(0) = f(1)] or balanced [f(0) ≠ f(1)]. Classically, to solve the problem seems to require the computation of f(0) and f(1), and then the comparison of results. Is it possible to solve the problem with only one query on f? In a famous paper published in 1985, Deutsch posed the problem and obtained a "quantum" partial affirmative answer. In 1998, a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello and Mosca. Here we will show that the quantum solution can be de-quantized to a deterministic simpler solution which is as efficient as the quantum one. The use of "superposition," a key ingredient of quantum algorithm, is — in this specific case — classically available.

scholarly journals Quantum query complexity of symmetric oracle problems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 403
Author(s):  
Daniel Copeland ◽  
Jamie Pommersheim
Keyword(s):  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Random Permutation ◽  
Query Complexity ◽  
Learning Problems ◽  
Unitary Matrices ◽  
Query Algorithm ◽  
Quantum Query

We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn. More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G, we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.

Exact Quantum 1-Query Algorithms and Complexity

SPIN ◽  
2021 ◽  
pp. 2140001
Author(s):  
Daowen Qiu ◽  
Guoliang Xu
Keyword(s):  
Boolean Function ◽  
Decision Trees ◽  
Future Study ◽  
Boolean Functions ◽  
Query Complexity ◽  
Symmetric Boolean Functions ◽  
Exact Quantum ◽  
Algorithms And Complexity ◽  
Special Case ◽  
Query Algorithms

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

scholarly journals Quantum and Randomised Algorithms for Non-linearity Estimation

2021 ◽  
Vol 2 (2) ◽  
pp. 1-27
Author(s):  
Debajyoti Bera ◽  
Sapv Tharrmashastha
Keyword(s):  
Boolean Function ◽  
Linear Function ◽  
Lower Bounds ◽  
Quantum Algorithm ◽  
Walsh Spectrum ◽  
Absolute Value ◽  
Additive Error ◽  
Amplitude Estimation ◽  
Walsh Coefficient ◽  
Amplitude Amplification

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires Θ (2 n ) queries to an n -bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted λ, with query complexities that depend polynomially on λ. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in n , already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of n . Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.

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