scholarly journals Quantum Random Access Codes for Boolean Functions

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 402
Author(s):  
João F. Doriguello ◽  
Ashley Montanaro
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Success Probability ◽  
Random Access ◽  
Fixed Size ◽  
Multiplicative Constant ◽  
Noise Stability ◽  
Access Code ◽  
Quantum Protocols

An n↦pm random access code (RAC) is an encoding of n bits into m bits such that any initial bit can be recovered with probability at least p, while in a quantum RAC (QRAC), the n bits are encoded into m qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function f on any subset of fixed size of the initial bits, which we call f-random access codes. We study and give protocols for f-random access codes with classical (f-RAC) and quantum (f-QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement (f-EARAC) and Popescu-Rohrlich boxes (f-PRRAC). The success probability of our protocols is characterized by the noise stability of the Boolean function f. Moreover, we give an upper bound on the success probability of any f-QRAC with shared randomness that matches its success probability up to a multiplicative constant (and f-RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.

A generalization of Shannon function

2018 ◽  
Vol 28 (5) ◽  
pp. 309-318 ◽  
Author(s):  
Nikolay P. Redkin
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Shannon Function ◽  
Conceptual Content

Abstract When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the considered basis. In this paper we propose a generalization of such a Shannon function in the form of an upper bound that is taken over all functionally complete bases. This generalization gives an idea of the complexity of implementing Boolean functions in the “worst” bases for them. The conceptual content of the proposed generalization is demonstrated by the example of a conjunction.

scholarly journals Two new results about quantum exact learning

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 587
Author(s):  
Srinivasan Arunachalam ◽  
Sourav Chakraborty ◽  
Troy Lee ◽  
Manaswi Paraashar ◽  
Ronald de Wolf
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Quantum Computers ◽  
Concept Class ◽  
Membership Queries ◽  
Exact Learning ◽  
Uniform Quantum

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k1.5(log⁡k)2) uniform quantum examples for that function. This improves over the bound of Θ~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O~(k1.5) upper bound by proving an improvement of Chang's lemma for k-Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O(Q2log⁡Qlog⁡|C|)classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log⁡Q-factor.

Canalyzing minors of Boolean functions

2020 ◽  
Vol 13 (08) ◽  
pp. 2050160
Author(s):  
Ivo Damyanov
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Disjunctive Normal Form ◽  
Function Identification

Canalyzing functions are a special type of Boolean functions. For a canalyzing function, there is at least one argument, in which taking a certain value can determine the value of the function. Identification of variables can also shrink the resulting function into constant or function depending on one variable. In this paper, we discuss a particular disjunctive normal form for representation of Boolean function with its identification minors. Then an upper bound of the number of canalyzing minors is obtained. Finally, the number of canalyzing minors for Boolean functions with five essential variables is discussed.

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.

On Multiple Access of a Large Number of Machine-Type Devices in Cellular Networks

2018 ◽  
pp. 105-114
Author(s):  
O. S. Galinina ◽  
S. D. Andreev ◽  
A. M. Tyurlikov
Keyword(s):  
Success Probability ◽  
Random Access ◽  
Time Interval ◽  
Short Time Interval ◽  
Control Mechanisms ◽  
Network Access ◽  
Smart Meters ◽  
Machine To Machine ◽  
Machine To Machine Communication ◽  
Machine Communication

Introduction: Machine-to-machine communication assumes data transmission from various wireless devices and attracts attention of cellular operators. In this regard, it is crucial to recognize and control overload situations when a large number of such devices access the network over a short time interval.Purpose:Analysis of the radio network overload at the initial network entry stage in a machine-to-machine communication system.Results: A system is considered that features multiple smart meters, which may report alarms and autonomously collect energy consumption information. An analytical approach is proposed to study the operation of a large number of devices in such a system as well as model the settings of the random-access protocol in a cellular network and overload control mechanisms with respect to the access success probability, network access latency, and device power consumption. A comparison between the obtained analytical results and simulation data is also offered. 

On 1-stable perfectly balanced Boolean functions

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Stanislav V. Smyshlyaev
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Correlation Immunity ◽  
Other Hand

AbstractThe paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the 1-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is 1-stable.

Clique Here: On the Distributed Complexity in Fully-Connected Networks

2016 ◽  
Vol 26 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Benny Applebaum ◽  
Dariusz R. Kowalski ◽  
Boaz Patt-Shamir ◽  
Adi Rosén
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Message Passing ◽  
Constant Number ◽  
Running Time ◽  
Message Size ◽  
Explicit Function ◽  
Randomized Protocols ◽  
Fully Connected ◽  
Fully Connected Networks

We consider a message passing model with n nodes, each connected to all other nodes by a link that can deliver a message of B bits in a time unit (typically, B = O(log n)). We assume that each node has an input of size L bits (typically, L = O(n log n)) and the nodes cooperate in order to compute some function (i.e., perform a distributed task). We are interested in the number of rounds required to compute the function. We give two results regarding this model. First, we show that most boolean functions require ‸ L/B ‹ − 1 rounds to compute deterministically, and that even if we consider randomized protocols that are allowed to err, the expected running time remains [Formula: see text] for most boolean function. Second, trying to find explicit functions that require superconstant time, we consider the pointer chasing problem. In this problem, each node i is given an array Ai of length n whose entries are in [n], and the task is to find, for any [Formula: see text], the value of [Formula: see text]. We give a deterministic O(log n/ log log n) round protocol for this function using message size B = O(log n), a slight but non-trivial improvement over the O(log n) bound provided by standard “pointer doubling.” The question of an explicit function (or functionality) that requires super constant number of rounds in this setting remains, however, open.

scholarly journals On diagnostic tests of contact break for contact circuits

2020 ◽  
Vol 30 (2) ◽  
pp. 103-116 ◽  
Author(s):  
Kirill A. Popkov
Keyword(s):  
Boolean Function ◽  
Diagnostic Test ◽  
Boolean Functions ◽  
Diagnostic Tests ◽  
Almost All

AbstractWe prove that, for n ⩾ 2, any n-place Boolean function may be implemented by a two-pole contact circuit which is irredundant and allows a diagnostic test with length not exceeding n + k(n − 2) under at most k contact breaks. It is shown that with k = k(n) ⩽ 2n−4, for almost all n-place Boolean functions, the least possible length of such a test is at most 2k + 2.

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 On the confusion coefficient of Boolean functions

2021 ◽  
Vol 16 (1) ◽  
pp. 1-13
Author(s):  
Yu Zhou ◽  
Jianyong Hu ◽  
Xudong Miao ◽  
Yu Han ◽  
Fuzhong Zhang
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Power Analysis ◽  
Signal To Noise Ratio ◽  
Sum Of Squares ◽  
Hamming Weight ◽  
Signal To Noise ◽  
Trade Offs ◽  
Cryptographic Algorithms ◽  
Transparency Order

Abstract The notion of the confusion coefficient is a property that attempts to characterize confusion property of cryptographic algorithms against differential power analysis. In this article, we establish a relationship between the confusion coefficient and the autocorrelation function for any Boolean function and give a tight upper bound and a tight lower bound on the confusion coefficient for any (balanced) Boolean function. We also deduce some deep relationships between the sum-of-squares of the confusion coefficient and other cryptographic indicators (the sum-of-squares indicator, hamming weight, algebraic immunity and correlation immunity), respectively. Moreover, we obtain some trade-offs among the sum-of-squares of the confusion coefficient, the signal-to-noise ratio and the redefined transparency order for a Boolean function.

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