Distributional Variants of the Aaronson-Ambainis Conjecture

2026-09-28

AI Summary

This study addresses whether the equivalence of classical simulation for quantum query algorithms holds under biased product and sliced uniform distributions. By leveraging low-degree polynomial analysis on the Boolean hypercube alongside probability distribution transformation techniques, the work overcomes prior theoretical limitations confined to uniform distributions. It rigorously proves that a variant of the Aaronson–Ambainis conjecture under these distributions is fully equivalent to the original uniform-distribution formulation, thereby establishing theoretical equivalence across multiple natural input distributions. This result reveals a unified theoretical structure underlying the problem of simulating quantum advantage across different distributions, providing a generalizable framework for investigating the boundary between quantum and classical computation.

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.

Problem

• Aaronson-Ambainis conjecture
• quantum query complexity
• input distributions
• classical simulation
• Boolean hypercube

Innovation

• Aaronson-Ambainis conjecture
• quantum query complexity
• distributional variants
• Boolean hypercube
• polynomial simulation