
Quantum QuerytoCommunication Simulation Needs a Logarithmic Overhead
Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean f...
read it

Open Problems Related to Quantum Query Complexity
I offer a case that quantum query complexity still has loads of enticing...
read it

The Hardest Halfspace
We study the approximation of halfspaces h:{0,1}^n→{0,1} in the infinity...
read it

kForrelation Optimally Separates Quantum and Classical Query Complexity
Aaronson and Ambainis (SICOMP `18) showed that any partial function on N...
read it

Towards Optimal Separations between Quantum and Randomized Query Complexities
The query model offers a concrete setting where quantum algorithms are p...
read it

Hybrid Decision Trees: Longer Quantum Time is Strictly More Powerful
In this paper, we introduce the hybrid query complexity, denoted as Q(f;...
read it

New Separations Results for External Information
We obtain new separation results for the twoparty external information ...
read it
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 ℓ≥1 sum to at most c^ℓ√(dℓ(1+log n)^ℓ1), 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 conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ, becoming trivial already at ℓ=√(d). As an application, we obtain, for every integer k≥1, a partial Boolean function on n bits that has boundederror quantum query complexity at most ⌈ k/2⌉ and randomized query complexity Ω̃(n^11/k). This separation of boundederror quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: O(1) versus Ω(n^2/3ϵ) for any ϵ>0 (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of O(log n) versus Ω(n^1ϵ) for boundederror quantum versus randomized communication complexity, for any ϵ>0. The best previous separation was polynomially weaker: O(log n) versus Ω(n^2/3ϵ) (implicit in Tal, FOCS 2020).
READ FULL TEXT
Comments
There are no comments yet.