On the Ambainis-Bach-Nayak-Vishwanath-Watrous Conjecture

TL;DR

This paper identifies a flaw in a previous theorem related to quantum walks, corrects it for a specific case, and demonstrates that a notable conjecture in the field is false, advancing understanding of quantum walk properties.

Contribution

It uncovers a flaw in a prior theorem, provides a correction for finite Hadamard walks, and disproves a significant conjecture in quantum walk theory.

Findings

Theorem in [3] is flawed and corrected for Finite Hadamard walk.

The conjecture by Ambainis et al. is shown to be false.

Provides insights into the limitations of certain quantum walk models.

Abstract

We show the flaw in a theorem of Konno, Namiki, Soshi, and Sudbury in [3] and provide the necessary correction in the case of the Finite Hadamard walk and use it to show that a conjecture of Ambainis, Bach, Nayak, Vishwanath, and Watrous in [1] is false.