In Quantum Computing Speedup Illusory?: The False Coin of "Counting Function Evaluations"

TL;DR

This paper presents a quantum algorithm over Z_2 that claims to solve the Grover Search Problem with only one function evaluation, challenging the validity of counting function evaluations as a measure of quantum speedup.

Contribution

It introduces a new encoding of Boolean functions in reversible gates and demonstrates a quantum algorithm that appears to outperform classical and standard quantum algorithms in function evaluations.

Findings

The QC/2 algorithm solves Grover's problem with a single function evaluation.

Classical algorithms require about 2^{m-1} evaluations, and standard Grover's algorithm requires ~√2^m evaluations.

The perceived speedup is due to reinterpreting classical calculations, not genuine quantum advantage.

Abstract

By using a new way to encode Boolean functions in a reversible gate, an algorithm is developed in quantum computing over Z_2, symbolized QC/2, (as opposed to QC over C) that needs only one function evaluation to solve the Grover Database Search Problem of finding a designated record among 2^m records for any m. In the usual Grover algorithm in quantum computing over C, one needs essentially Sqrt(2^m) function evaluations as opposed to the average of (2^m)/2 functions evaluations needed in the classical algorithm. The one function evaluation of the QC/2 algorithm (for any m) represents such a super speedup, even over the Grover algorithm in QC/C, that one feels something has gone awry. Indeed, our analysis of the transparent calculations of Boolean functions over Z_2 shows that the classical algorithm is just repackaged in a rather obvious way in the single function evaluation of the QC/2 algorithm--whereas the calculations are hidden and non-transparent in the Grover QC/C algorithm using C. The conclusion in both cases (which is rather obvious in the QC/2 case) is that "counting function evaluations" is a false coin to measure speedup in the comparison between quantum and classical computing.