Sure-Success Quantum Algorithms on Weight Decision Problem

TL;DR

This paper investigates conditions for sure-success quantum algorithms in weight decision problems, showing they reduce to algebraic equations and scale favorably compared to classical algorithms, with potential efficiency gains in limited iterations.

Contribution

It introduces algebraic conditions for sure-success quantum algorithms and analyzes their iteration scaling compared to classical probabilistic algorithms.

Findings

Quantum algorithms' iteration number scales as the square root of classical algorithms for large problems.

Quantum algorithms can outperform classical ones in efficiency with fewer iterations.

Decidability reduces to solving a single-variable algebraic system.

Abstract

Conditions on sure-success decidability of weights of Boolean functions are presented for a given number of generalized Grover iterations. It is shown that the decidability problem reduces to a system of algebraic equations of a single variable. For problems that require a large number of iterations, it is observed that the iteration number of sure-success quantum algorithms scale as the square root of the iteration number of the corresponding classical probabilistic algorithms. It is also demonstrated that for a few iterations, quantum algorithms can be more efficient than this.