# k-forrelation optimally separates Quantum and classical query complexity

@article{Bansal2020kforrelationOS, title={k-forrelation optimally separates Quantum and classical query complexity}, author={Nikhil Bansal and Makrand Sinha}, journal={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing}, year={2020} }

Aaronson and Ambainis (SICOMP ‘18) showed that any partial function on N bits that can be computed with an advantage δ over a random guess by making q quantum queries, can also be computed classically with an advantage δ/2 by a randomized decision tree making Oq(N1−1/2qδ−2) queries. Moreover, they conjectured the k-Forrelation problem — a partial function that can be computed with q = ⌈ k/2 ⌉ quantum queries — to be a suitable candidate for exhibiting such an extremal separation. We prove their…

## 16 Citations

Analyzing XOR-Forrelation through stochastic calculus

- MathematicsSTACS
- 2022

This note presents a simplified analysis of the quantum and classical complexity of the k-XOR Forrelation problem by a stochastic interpretation of the Forrelation distribution.

Boolean functions with small approximate spectral norm

- Mathematics
- 2022

The sum of the absolute values of the Fourier coefficients of a function f : F2 → R is called the spectral norm of f . Green and Sanders’ quantitative version of Cohen’s idempotent theorem states…

Fourier Growth of Regular Branching Programs

- Mathematics, Computer Science
- 2022

It is proved that every read-once regular branching program B of width w with s accepting states on n-bit inputs must have its L1,k bounded by min { Pr[B(Un) = 1](w − 1), s ·O ( (n log n)/k ) k−1 2 } .

Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms

- MathematicsArXiv
- 2022

The Aaronson-Ambainis conjecture (Theory of Computing ’14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply…

Beyond quadratic speedups in quantum attacks on symmetric schemes

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2021

In this paper, we report the first quantum key-recovery attack on a symmetric block cipher design, using classical queries only, with a more than quadratic time speedup compared to the best classical…

Classical algorithms for Forrelation

- Computer Science
- 2021

It is shown that the graph-based forrelation problem can be solved on a classical computer in time O(n2) for any bipartite graph, any planar graph, or, more generally, any graph which can be partitioned into two subgraphs of constant treewidth.

Degree vs. approximate degree and Quantum implications of Huang’s sensitivity theorem

- MathematicsSTOC
- 2021

It is shown that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f)=Ω(n), which is also optimal, and the approximate degree of any read-once formula on n variables is Θ(√n).

Following Forrelation - Quantum Algorithms in Exploring Boolean Functions' Spectra

- Computer ScienceArXiv
- 2021

This work revisits the quantum algorithms for obtaining Forrelation 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, and tweaks the quantum algorithm with superposition of linear functions to obtain a cross-Correlation sampling technique.

Fourier Growth of Structured 𝔽2-Polynomials and Applications

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2021

The main structural results on Fourier growth are shown to show that any symmetric degree- d F 2 -polynomial p has L 1 ,k ( p ) ≤ Pr [ p = 1] · O ( d ) k , and this is tight for any constant k, which quadratically strengthens an earlier bound that was implicit in [RSV13].

Fourier growth of parity decision trees

- Computer Science, MathematicsEmpir. Softw. Eng.
- 2021

It is proved that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level l is at most d ·O(l · log(n)) and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021).

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It is shown that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f)=Ω(n), which is also optimal, and the approximate degree of any read-once formula on n variables is Θ(√n).

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