Quantum Lower Bound for Approximate Counting Via Laurent Polynomials

TL;DR

This paper establishes quantum lower bounds for approximate counting of set size using Laurent polynomials, showing that quantum sampling does not necessarily lead to efficient approximate counting in the black-box model.

Contribution

It introduces a novel Laurent polynomial method to prove quantum lower bounds for approximate counting, extending the polynomial method to negative exponents.

Findings

Quantum algorithms require many queries or copies of the superposition state for approximate counting.

Quantum sampling does not imply efficient approximate counting in the black-box setting.

The lower bounds depend on the size of the set and the total universe size.

Abstract

We consider the following problem: estimate the size of a nonempty set $S\subseteq\left[ N\right] $, given both quantum queries to a membership oracle for $S$, and a device that generates equal superpositions $\left\vert S\right\rangle $ over $S$ elements. We show that, if $\left\vert S\right\vert $ is neither too large nor too small, then approximate counting with these resources is still quantumly hard. More precisely, any quantum algorithm needs either $\Omega\left( \sqrt{N/\left\vert S\right\vert}\right) $ queries or else $\Omega\left( \min\left\{ \left\vert S\right\vert ^{1/4},\sqrt{N/\left\vert S\right\vert }\right\} \right)$ copies of $\left\vert S\right\rangle $. This means that, in the black-box setting, quantum sampling does not imply approximate counting. The proof uses a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents.