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Quantum Algorithms in Exploring Boolean Functions' Spectra

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Quantum Algorithms in Exploring Boolean Functions' Spectra 1 Following Forrelation – Quantum Algorithms in Exploring Boolean Functions’ Spectra Suman Dutta, Indian Statistical Institute, Kolkata Email: [email protected] Subhamoy Maitra, Indian Statistical Institute, Kolkata Email: [email protected] Chandra Sekhar Mukherjee, Indian Statistical Institute, Kolkata Email: [email protected] Abstract 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 first study the 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 an oracle access, one can sample from the Walsh Spectrum of f. Using this, we obtain improved results in resiliency checking. Furthermore, we use 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 superposition of linear functions to obtain a cross-correlation sampling technique. To the best of our knowledge, this is the first cross-correlation sampling algorithm with constant query complexity. This also provides a strategy to check if two functions are uncorrelated of degree m. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of m. Index Terms Boolean Functions, Cross-correlation Spectrum, Forrelation, Quantum Algorithm, Walsh Spectrum. I. INTRODUCTION Quantum computing is one of the fundamental aspects in Quantum mechanics with many exciting possibilities. However, showing separation between classical and quantum paradigm still remains a very hard and interesting problem. In this regard, the black-box model is widely used to show the separation in respective domains. Some of the most well known quantum algorithms, such as Shor’s factoring [18], Grover’s search [9] and Simon’s hidden shift [17] are all defined in this black-box paradigm. Naturally, this black-box model (also known as the query model) is one of the most studied domains of quantum computer science. In this domain, problems can be studied in the exact quantum and bounded error quantum model as well as probabilistic and deterministic classical model. One of the simplest yet fundamental algorithms in this domain is the Deutsch-Jozsa (DJ) algorithm [7]. Informally speaking, given access to a Boolean function f, it creates the Walsh spectrum in the form of a superposition, where the amplitudes of the final states are the normalized Walsh spectrum value of the function at the respective points. If f is promised to be either constant or balanced, the DJ algorithm deterministically concludes which one it is with a single quantum query, whereas any deterministic classical algorithm requires exponentially many queries, which is asymptotically the maximum possible separation between these two models. However obtaining the separation between the probabilistic classical model and the bounded error arXiv:2104.12212v1 [quant-ph] 25 Apr 2021 quantum model is not as simple and the maximum possible separation is still not known. In this direction the “Forrelation” formulation is of great interest. This concept was first introduced in the work by Aaronson et al [1] and then was used in the seminal paper by Aaronson et al [2]. They showed that the Forrelation problem has constant versus exponential (in n) query complexity separation in the bounded error quantum and the probabilistic classical model. Recently this concept was modified by Tal et al [20] to obtain even higher separation between these two. It is quite easy to see that the Forrelation problem is efficiently solvable in the quantum paradigm and the main contribution and focus of those papers were towards showing that no probabilistic classical algorithm could solve this problem efficiently. In this backdrop, before proceeding further, let us start by introducing the k-fold Forrelation set-up. The k-fold Forrelation of k Boolean functions f1; : : : ; fk, each defined on n variables can be expressed as 1 X x x1·x2 x xk−1·xk x Φf1;:::;fk = (k+1)n=2 f1( 1)( 1) f2( 2) ::: ( 1) fk( k): 2 n − − x1;:::;xk2f0;1g n Specifically, Aaronson et al [2] showed that the problem of identifying whether Φ 1 or Φ 3 has Ω 2 2 j f1;:::;fk j ≤ 100 f1;:::;fk ≥ 5 n query complexity in the probabilistic classical model as opposed to constant query complexity in the bounded error quantum 2 model. To prove those results, they designed the following two quantum algorithms assuming the oracle access to f1 through fk. 1) (k;k)(f ; : : : f ): It outputs the state 0n with probability Φ2 upon measuring the n predetermined qubits. 1 k f1;:::;fk A k (k;d e) 1+Φf1;:::;fk 2) 2 (f ; : : : f ): It outputs the state 0 with probability upon measuring a predetermined qubit. A 1 k 2 In this paper, we analyze the Forrelation problem from a different perspective, in terms of studying and evaluating different spectra of Boolean functions in the black-box (query) model. Boolean functions are one the most fundamental combinatorial objects in the domain of computer science, with applications in coding theory, cryptography as well as in understanding power and limitations of different computational models. The spectra that we consider here have many applications and in particular in the domain of cryptology. When a Boolean function is used as a primitive in a cryptosystem, its quality is studied by different parameters related to the spectra based on certain transformations [3], [5], [6], [10], [15], [16]. In this paper, we sometimes refer to Boolean functions simply as functions. While k-fold Forrelation can be defined for any k functions on n variables, we concentrate on the case of k = 2 and k = 3 in this study. Initial part of the paper is oriented around observing how certain promise problems can be reduced to different desirable instantiations of the 2-fold Forrelation problem, where the Forrelation of two functions f and g is denoted by Φf;g. We start by studying the scenario when Φf;g is equal to 1, which is the maximum value it can assume. We informally observe that Φf;g can be 1 ( 1) if and only if f and g are two bent functions, where f and g are each other’s dual (anti dual). On the other hand if there is− a third function g0 such that g g0 is almost balanced, then the Forrelation between f and g0 is close to 0. We further observe that one obvious instantiation⊕ of this occurs when g g0 is a bent function, which happens when they are both members of Kerdock code. Using these results we obtain certain desirable⊕ instantiations of the 2-fold Forrelation problems. We also observe that in certain situations the simple DJ algorithm works as efficiently as the Forrelation set-up, while in the other (majority of the) cases there is no obvious way of doing the same using the DJ algorithm. In summary, we note that the techniques in solving the Forrelation problem extend the idea of the DJ algorithm much further. We like to point out that in [14], certain quantum algorithms related to bent functions have been studied, though that is not related to the problems we consider here. In the next part we concentrate on the 3-fold Forrelation formulation which we use as a unifying framework for evaluating different spectra of functions, and this is where the most interesting results lie. Our initial approach in this direction is driven by an observation about the 3-fold Forrelation formulation. As we shall obtain in Section IV, the 3-fold Forrelation for three 1 P functions and can be written as n x x x where the functions are defined from f1; f2 g Φf1;g;f2 = 23n=2 x2f0;1g g( )Wf1 ( )Wf2 ( ) 0; 1 n to 1; 1 (note that the definition deviates from the usual realization of a Boolean function as f : 0; 1 n 0; 1 , howeverf g thisf is− ag simple linear transformation of the output bits of the form 0 1 and 1 1). Further, notef thatg the! f3-foldg Forrelation algorithms (3;2) and (3;3) need oracle access to all the three functions! f ; f!and − f . A A 1 2 3 Against this backdrop we set f1 = f3 = f and cleverly design a function g that we set as f2 to estimate the Walsh Spectrum n (3;2) value (Wf (u)) of a function at any particular point u 0; 1 using the algorithm (f; g; f). Here we set g to be a function such that g(x) = 1 if and only if x = u and2 f1, otherwise.g Then we studyA the problem of checking whether a − function f is m-resilient or not, given oracle access to it. We set f1 = f2 = f and g to be a symmetric function such that f(x) = 1 if wt(x) > m and 1 otherwise, where wt(x) denotes the count of 1’s in the bit pattern x. We observe that in − (3;2) such a case Φf;g;f is 1 if and only if the function is m-resilient, and less than 1, otherwise. Using we obtain identical outcomes with similar circuit costs as compared to the presently known most efficient algorithm forA this problem [6]. We use (3;3) to improve upon this result and obtain a constant improvement on Deutsch-Jozsa sampling probability for checking resiliency.A The same improvement is carried over during the application of amplitude amplification.
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