Robust Quantum Algorithms with $\eps$-Biased Oracles

TL;DR

This paper develops quantum algorithms for computing OR functions using $ ext{eps}$-biased oracles, improving query complexity bounds and addressing unknown bias scenarios, thus advancing quantum query complexity understanding.

Contribution

It presents the first quantum algorithms with optimal query bounds for $ ext{eps}$-biased oracles and handles cases with unknown bias, confirming a prior conjecture.

Findings

Quantum algorithms achieve $O(rac{ oot{2} }{ ext{eps}})$ query complexity.

The algorithms match the known lower bounds, confirming optimality.

Handling unknown $ ext{eps}$ improves quantum advantage over classical methods.

Abstract

This paper considers the quantum query complexity of {\it $\eps$-biased oracles} that return the correct value with probability only $1/2 + \eps$. In particular, we show a quantum algorithm to compute $N$-bit OR functions with $O(\sqrt{N}/{\eps})$ queries to $\eps$-biased oracles. This improves the known upper bound of $O(\sqrt{N}/{\eps}^2)$ and matches the known lower bound; we answer the conjecture raised by the paper by Iwama et al. affirmatively. We also show a quantum algorithm to cope with the situation in which we have no knowledge about the value of $\eps$. This contrasts with the corresponding classical situation, where it is almost hopeless to achieve more than a constant success probability without knowing the value of $\eps$.