An improved quantum query algorithm for computing AND Boolean function

2010 ◽  
Author(s):  
Alina Vasilieva ◽  
Taisia Mischenko-Slatenkova
Keyword(s):  
Boolean Function ◽  
Query Algorithm ◽  
Quantum Query

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 Quantum Speedup Based on Classical Decision Trees

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 241
Author(s):  
Salman Beigi ◽  
Leila Taghavi
Keyword(s):  
Decision Tree ◽  
Main Tool ◽  
Upper Bounds ◽  
Query Complexity ◽  
Bipartite Matching ◽  
Quantum Query Complexity ◽  
Maximum Bipartite Matching ◽  
Query Algorithm ◽  
Binary Functions ◽  
Quantum Query

Lin and Lin \cite{LL16} have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function f:{0,1}n→[m] whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for f which makes at most O(GT) quantum queries where T is the depth of the decision tree and G is the maximum number of mistakes of the guessing algorithm. In this paper we give a simple proof of and generalize this result for functions f:[ℓ]n→[m] with non-binary input as well as output alphabets. Our main tool for this generalization is non-binary span program which has recently been developed for non-binary functions, and the dual adversary bound. As applications of our main result we present several quantum query upper bounds, some of which are new. In particular, we show that topological sorting of vertices of a directed graph G can be done with O(n3/2) quantum queries in the adjacency matrix model. Also, we show that the quantum query complexity of the maximum bipartite matching is upper bounded by O(n3/4m+n) in the adjacency list model.

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 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.

Identifying Generalized Reed-Muller Codewords by Quantum Queries

2017 ◽  
Vol 28 (02) ◽  
pp. 185-194
Author(s):  
Stefan Arnold
Keyword(s):  
Finite Field ◽  
Probability Of Error ◽  
Exact Quantum ◽  
Multilinear Polynomials ◽  
Query Algorithm ◽  
Quantum Query

We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qn over the finite field of size q. When d is constant, the algorithm needs 𝒪(nd-1) quantum queries, which is optimal. Classically, Ω(nd) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.

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.

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