Influences of Fourier Completely Bounded Polynomials and Classical Simulation of Quantum Algorithms

We give a new presentation of the main result of Arunachalam, Bri\"et and Palazuelos (SICOMP'19) and show that quantum query algorithms are characterized by a new class of polynomials which we call Fourier completely bounded polynomials. We conjecture that all such polynomials have an influential variable. This conjecture is weaker than the famous Aaronson-Ambainis (AA) conjecture (Theory of Computing'14), but has the same implications for classical simulation of quantum query algorithms. We prove a new case of the AA conjecture by showing that it holds for homogeneous Fourier completely bounded polynomials. This implies that if the output of $d$-query quantum algorithm is a homogeneous polynomial $p$ of degree $2d$, then it has a variable with influence at least $Var[p]^2$. In addition, we give an alternative proof of the results of Bansal, Sinha and de Wolf (CCC'22 and QIP'23) showing that block-multilinear completely bounded polynomials have influential variables. Our proof is simpler, obtains better constants and does not use randomness

Influences of fourier completely bounded polynomials and classical simulation of quantum algorithms

We give a new presentation of the main result of Arunachalam, Briët and Palazuelos (SICOMP'19) and show that quantum query algorithms are characterized by a new class of polynomials which we call Fourier completely bounded polynomials. We conjecture that all such polynomials have an influential variable. This conjecture is weaker than the famous Aaronson-Ambainis (AA) conjecture (Theory of Computing'14), but has the same implications for classical simulation of quantum query algorithms. We prove a new case of the AA conjecture by showing that it holds for homogeneous Fourier completely bounded polynomials. This implies that if the output of d-query quantum algorithm is a homogeneous polynomial p of degree 2d, then it has a variable with influence at least Var[p]2. In addition, we give an alternative proof of the results of Bansal, Sinha and de Wolf (CCC'22 and QIP'23) showing that block-multilinear completely bounded polynomials have influential variables. Our proof is simpler, obtains better constants and does not use randomness

On the Fine-Grained Query Complexity of Symmetric Functions

This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to $1/2$. Our contributions include the following: i) An analysis of the optimal success probability of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. We prove that for any quantum algorithm computing these two functions using $T$ queries, there exist randomized algorithms using $\mathsf{poly}(T)$ queries that achieve the same success probability as the quantum algorithm, even if the success probability is arbitrarily close to 1/2. ii) We establish that for any total symmetric Boolean function $f$, if a quantum algorithm uses $T$ queries to compute $f$ with success probability $1/2+\beta$, then there exists a randomized algorithm using $O(T^2)$ queries to compute $f$ with success probability $1/2+\Omega(\delta\beta^2)$ on a $1-\delta$ fraction of inputs, where $\beta,\delta$ can be arbitrarily small positive values. As a corollary, we prove a randomized version of Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. iii) We present polynomial equivalences for several fundamental complexity measures of partial symmetric Boolean functions. Specifically, we first prove that for certain partial symmetric Boolean functions, quantum query complexity is at most quadratic in approximate degree for any error arbitrarily close to 1/2. Next, we show exact quantum query complexity is at most quadratic in degree. Additionally, we give the tight bounds of several complexity measures, indicating their polynomial equivalence.Comment: accepted in ISAAC 202