One-Dimensional Coinless Quantum Walks

TL;DR

This paper explores one-dimensional coinless quantum walks, demonstrating their equivalence to coined walks, analyzing their asymptotic behavior, and assessing their efficiency in quantum search algorithms.

Contribution

It introduces a method to transform one-dimensional coinless quantum walks into equivalent coined versions, highlighting their fundamental similarities and analyzing their properties.

Findings

Coinless walks are as efficient as coined walks in quantum search.

They can be transformed into coined walks with specific evolution operators.

The paper analyzes mixing times and limiting distributions on cycles.

Abstract

A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely-studied coined walk. Coined walks require the direct product of the site basis with the coin space, coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.