k-forrelation optimally separates Quantum and classical query complexity

Bansal, Nikhil; Sinha, Makrand

doi:10.1145/3406325.3451040

Cited by 18 publications

(50 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, here we obtain a constant factor improvement over the DJ algorithm using the algorithm A (3,3) (f, g, f ). The problem of checking whether a given Boolean function f is m-resilient or not, can also be improved upon using A (3,3) (f, g, f ) where g is designed such that g(x) = −1, ∀x : wt (x) ≤ m and 1, otherwise. Here wt(x) denotes the count of 1's in the bit pattern x.…”

Section: A (K Kmentioning

confidence: 76%

“…Here we set g to be a function such that g(x) = −1 if and only if x ∈ S and 1, otherwise. Then we observe that the sampling probability obtained from using the algorithm A (3,2) (f, g, f ) is equivalent to that of the Deutsch-Jozsa algorithm. However, here we obtain a constant factor improvement over the DJ algorithm using the algorithm A (3,3) (f, g, f ).…”

Section: A (K Kmentioning

confidence: 95%

“…Then we observe that the sampling probability obtained from using the algorithm A (3,2) (f, g, f ) is equivalent to that of the Deutsch-Jozsa algorithm. However, here we obtain a constant factor improvement over the DJ algorithm using the algorithm A (3,3) (f, g, f ). The problem of checking whether a given Boolean function f is m-resilient or not, can also be improved upon using A (3,3) (f, g, f ) where g is designed such that g(x) = −1, ∀x : wt (x) ≤ m and 1, otherwise.…”

Section: A (K Kmentioning

confidence: 95%

“…As we shall obtain in Section 4, the 3-fold Forrelation for three functions f 1 , f 2 and g can be written as Φ f1,g,f2 = 1 2 3n/2 x∈{0,1} n g(x)W f1 (x)W f2 (x) where the functions are defined from {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}, however this is a simple linear transformation of the output bits of the form 0 → 1 and 1 → −1). Further, note that the 3-fold Forrelation algorithms A (3,2) and A (3,3) need oracle access to all the three functions f 1 , f 2 and f 3 .…”

Section: A (K Kmentioning

confidence: 99%

“…In contrast, 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 quantum model is not as simple and recently resolved in [3].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Following Forrelation – quantum algorithms in exploring Boolean functions' spectra

Dutta¹,

Maitra

Mukherjee³

2024

AMC

View full text Add to dashboard Cite

<p style='text-indent:20px;'>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 introduce the existing 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 oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>

show abstract