Distributional Variants of the Aaronson-Ambainis Conjecture

Abstract

A longstanding conjecture in quantum complexity theory asserts that, under the uniform input distribution, quantum query algorithms can be polynomially simulated by classical query algorithms. More precisely, the acceptance probability of any quantum query algorithm can be approximated, on average over uniformly random inputs, by a classical query algorithm, with only polynomial query overhead. The conjecture is central to understanding whether exponential quantum advantages for decision problems necessarily rely on additional structure. We study analogues of this conjecture under other natural input distributions and prove that they are all equivalent to the original uniform-distribution conjecture. We first consider the product distribution ${\mu }_p$, where the input bits are independent Bernoulli variables with fixed bias $p$. We show that for every fixed $p\in (0,1)$, quantum query algorithms under the ${\mu }_p$ distribution admit polynomial-overhead classical simulations if and only if the same holds under the uniform distribution. Second, we consider the distribution ${\nu }_p$ that is uniform over the slice of strings with Hamming weight $\lfloor pn\rfloor$ and prove a similar equivalence for the ${\nu }_p$ distribution and the uniform distribution. The Aaronson-Ambainis conjecture is a stronger statement that implies the above-mentioned conjecture and is formulated in terms of bounded low-degree polynomials on the Boolean hypercube. It asserts that under the uniform distribution, any such polynomial with nonnegligible variance must have an influential variable. We formulate analogues of this conjecture, where the underlying distribution is a biased product distribution or a uniform distribution over a slice, and prove that all these variants are equivalent to the original Aaronson-Ambainis conjecture.