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}
}
  • N. Bansal, Makrand Sinha
  • Published 16 August 2020
  • Computer Science, Mathematics
  • Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
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… 
Analyzing XOR-Forrelation through stochastic calculus
TLDR
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
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
TLDR
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
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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).
...
1
2
...

References

SHOWING 1-10 OF 48 REFERENCES
Degree vs. approximate degree and Quantum implications of Huang’s sensitivity theorem
TLDR
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).
Forrelation: A Problem That Optimally Separates Quantum from Classical Computing
TLDR
This work achieves essentially the largest possible separation between quantum and classical query complexities using a property-testing problem called Forrelation, where one needs to decide whether a classical query or a quantum one is correct.
BQP and the Polynomial Hierarchy (STOC ’10)
  • 2010
An optimal separation of randomized and Quantum query complexity
We prove that for every decision tree, the absolute values of the Fourier coefficients of given order t≥1 sum to at most (cd/t)t/2(1+logn)(t−1)/2, where n is the number of variables, d is the tree
Oracle separation of BQP and PH
We present a distribution D over inputs in {−1,1}2N, such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O(logN), that distinguishes between
2019. Krivine diffusions attain the Goemans- Williamson approximation ratio. (June 2019)
  • 1906
  • 2020
  • 2020
  • 2020
  • 2020
...
1
2
3
4
5
...