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arXiv:2103.07934 · ACM Transactions on Quantum Computing

Quantum and Randomised Algorithms for Non-linearity Estimation

Debajyoti Bera, Tharrmashastha Sapv

Published Jun 30, 2021• 2 citations
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Abstract

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.

DOI
10.1145/3456509

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Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is εεε-far from having that property. The efficiency of the algorithm is judged by the number of calls to the oracle so that one can decide, with high probability, between these two alternatives. The best known quantum algorithm for testing whether a function is linear or εεε-far (0<ε<12)(0 < ε< \frac{1}{2})(0<ε<21​) from linear functions requires O(ε−23)O(ε^{-\frac{2}{3}})O(ε−32​) many calls [Hillery and Andersson, Physical Review A 84, 062329 (2011)]. We show that this can be improved to O(ε−12)O(ε^{-\frac{1}{2}})O(ε−21​) by using the Deutsch-Jozsa and the Grover Algorithms.
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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 ε\varepsilonε-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 fff, of nnn variables, is identical with a given function ggg or is ε\varepsilonε-far from ggg, where ε\varepsilonε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε)O(\frac{1}{\sqrt{\varepsilon}})O(ε​1​). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε)\Theta(\frac{1}{\varepsilon})Θ(ε1​). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f,g)C(f,g)C(f,g) denote the correlation of fff and ggg. We propose an exact quantum algorithm for making distinction between ∣C(f,g)∣=ε|C(f,g)|=\varepsilon∣C(f,g)∣=ε and ∣C(f,g)∣=1|C(f,g)|=1∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n)\Theta(2^{n})Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε\varepsilonε-far balanced using O(1ε)O(\frac{1}{\varepsilon})O(ε1​) 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/ε2)O(1/\varepsilon^{2})O(1/ε2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case. Comment: 24 pages, comments are welcome. It is accepted for publishing in Fundamenta Informaticae
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 ε\varepsilonε-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 fff, of nnn variables, is identical with a given function ggg or is ε\varepsilonε-far from ggg, where ε\varepsilonε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε)O(\frac{1}{\sqrt{\varepsilon}})O(ε​1​). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε)\Theta(\frac{1}{\varepsilon})Θ(ε1​). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f,g)C(f,g)C(f,g) denote the correlation of fff and ggg. We propose an exact quantum algorithm for making distinction between ∣C(f,g)∣=ε|C(f,g)|=\varepsilon∣C(f,g)∣=ε and ∣C(f,g)∣=1|C(f,g)|=1∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n)\Theta(2^{n})Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε\varepsilonε-far balanced using O(1ε)O(\frac{1}{\varepsilon})O(ε1​) 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/ε2)O(1/\varepsilon^{2})O(1/ε2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case. Comment: 24 pages, comments are welcome. It is accepted for publishing in Fundamenta Informaticae