Quantum Lower Bound for Recursive Fourier Sampling

TL;DR

This paper demonstrates that the Bernstein-Vazirani quantum algorithm for Recursive Fourier Sampling is nearly optimal, highlighting fundamental limits on quantum computation due to the necessity of uncomputing garbage, and introduces the nonparity coefficient as a new Boolean function parameter.

Contribution

It establishes a near-optimal lower bound for Recursive Fourier Sampling and introduces the nonparity coefficient to analyze quantum computational limits.

Findings

Bernstein-Vazirani algorithm is close to optimal for Recursive Fourier Sampling

Uncomputing garbage imposes fundamental quantum computational limits

Introduces the nonparity coefficient as a new Boolean function parameter

Abstract

One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called Recursive Fourier Sampling. This paper shows that the Bernstein-Vazirani algorithm is not far from optimal. The moral is that the need to "uncompute" garbage can impose a fundamental limit on efficient quantum computation. The proof introduces a new parameter of Boolean functions called the "nonparity coefficient," which might be of independent interest.