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 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 The quantum query complexity of AC0

2012 ◽  
Vol 12 (7&8) ◽  
pp. 670-676
Author(s):  
Paul Beame ◽  
Widad Machmouchi
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Input Function ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Quantum Query

We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from $[2n-2]$ to $[n]$ is surjective. These results yield a nearly linear $\Omega(n/\log n)$ lower bound on the quantum query complexity of $\cl{AC}^0$. The best previous lower bound known for any $\cl{AC^0}$ function was the $\Omega ((n/\log n)^{2/3})$ bound given by Aaronson and Shi's $\Omega(n^{2/3})$ lower bound for the element distinctness problem.

scholarly journals Explicit relation between all lower bound techniques for quantum query complexity

2015 ◽  
Vol 13 (04) ◽  
pp. 1350059
Author(s):  
Loïck Magnin ◽  
Jérémie Roland
Keyword(s):  
Lower Bound ◽  
Boolean Functions ◽  
Query Complexity ◽  
Polynomial Method ◽  
Quantum Query Complexity ◽  
State Generation ◽  
Adversary Method ◽  
Explicit Relation ◽  
Special Case ◽  
Quantum Query

The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.

scholarly journals On Exact Quantum Query Complexity

Algorithmica ◽  
2013 ◽  
Vol 71 (4) ◽  
pp. 775-796 ◽  
Author(s):  
Ashley Montanaro ◽  
Richard Jozsa ◽  
Graeme Mitchison
Keyword(s):  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Quantum Query

Relationship between Correlation Immune Order and Algebraic Immunity Order of Boolean Functions

2013 ◽  
Vol 740 ◽  
pp. 273-278
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Ya Jing Liu
Keyword(s):  
Boolean Function ◽  
Internal Structure ◽  
Boolean Functions ◽  
Necessary Conditions ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
Analytic Combinatorics ◽  
Sufficient And Necessary Conditions ◽  
Research Tools

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, we go deep into the internal structure of the Boolean function values. Additionally, by the methods of cascade calculations and analytic combinatorics, cryptographic properties such as correlation immune and algebraic immunity of H Boolean functions with Hamming weight of with diffusibility are studied. Then we prove the existing of m order correlation immune H Boolean functions ,and get the result of the sufficient and necessary conditions of algebraic immunity order is 1 of Boolean function with correlation immune order is m.

Quantum Query Complexity of Boolean Functions with Small On-Sets

2008 ◽  
pp. 907-918 ◽  
Author(s):  
Andris Ambainis ◽  
Kazuo Iwama ◽  
Masaki Nakanishi ◽  
Harumichi Nishimura ◽  
Rudy Raymond ◽  
...  
Keyword(s):  
Boolean Functions ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Quantum Query

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.

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