A strong direct product theorem for quantum query complexity

TL;DR

This paper proves a strong direct product theorem for quantum query complexity, showing that attempting to compute multiple copies of a function with fewer queries results in exponentially small success probability, and establishes an XOR lemma for boolean functions.

Contribution

It demonstrates that the multiplicative adversary method always exceeds the additive adversary method, leading to a strong direct product theorem in quantum query complexity.

Findings

Quantum query complexity satisfies a strong direct product theorem.

An XOR lemma for boolean functions is established.

The multiplicative adversary method is shown to be at least as large as the additive adversary method.

Abstract

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing $k$ copies of a function with less than $k$ times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in $k$. For a boolean function $f$ we also show an XOR lemma---computing the parity of $k$ copies of $f$ with less than $k$ times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, is always at least as large as the additive adversary method, which is known to characterize quantum query complexity.