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REPORTS > KEYWORD > FORRELATION:
Reports tagged with Forrelation:
TR19-179 | 7th December 2019
Avishay Tal

#### Towards Optimal Separations between Quantum and Randomized Query Complexities

Revisions: 1

The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of $O(1)$ vs. ... more >>>

TR20-101 | 7th July 2020
Uma Girish, Ran Raz, Wei Zhan

#### Lower Bounds for XOR of Forrelations

The Forrelation problem, first introduced by Aaronson [AA10] and Aaronson and Ambainis [AA15], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [AA15]; the first separation between ... more >>>

TR20-127 | 21st August 2020
Nikhil Bansal, Makrand Sinha

#### $k$-Forrelation Optimally Separates Quantum and Classical Query Complexity

Revisions: 2

Aaronson and Ambainis (SICOMP '18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured ... more >>>

TR20-128 | 3rd September 2020
Alexander A. Sherstov, Andrey Storozhenko, Pei Wu

#### An Optimal Separation of Randomized and Quantum Query Complexity

Revisions: 1

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{{d\choose\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a ... more >>>

TR21-164 | 19th November 2021
Scott Aaronson, DeVon Ingram, William Kretschmer

#### The Acrobatics of BQP

We show that, in the black-box setting, the behavior of quantum polynomial-time (${BQP}$) can be remarkably decoupled from that of classical complexity classes like ${NP}$. Specifically:

-There exists an oracle relative to which ${NP}^{{BQP}}\not \subset {BQP}^{{PH}}$, resolving a 2005 problem of Fortnow. Interpreted another way, we show that ${AC^0}$ circuits ... more >>>

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