Speed from Repetition

TL;DR

This paper introduces the Repeated Randomness problem, demonstrating a quantum advantage with a single query versus multiple classical queries, highlighting potential for quantum speedup in oracle problems.

Contribution

It defines a new oracle problem that showcases exponential quantum speedup over classical algorithms, connecting to existing problems like parity and NAND trees.

Findings

Quantum algorithm solves the problem in one query.

Classical algorithms require logarithmic number of queries.

The problem illustrates quantum speedup in oracle-based computation.

Abstract

We present an oracle problem, which we call the Repeated Randomness problem, that a quantum algorithm can solve in one query, while any classical algorithm requires $\Omega(\log n)$ queries, where the oracle function has $2^n$ inputs. This problem has connections to the parity problem introduced by Bernstein and Vazirani \cite{Bernstein1993} and to a variation of the NAND tree described by Zhan et al \cite{Zhan}.