Tuning for Quantum Speedup in Directed Lackadaisical Quantum Walks

TL;DR

This paper investigates directed lackadaisical quantum walks, revealing how tuning the self-loop parameter $l$ can induce quantum speedup and identifying different regimes and scaling behaviors on line and binary tree structures.

Contribution

It introduces the study of directed lackadaisical quantum walks and demonstrates how tuning the self-loop parameter $l$ can lead to quantum speedup and different scaling regimes.

Findings

Existence of classical and quantum dominated regimes depending on $l$

Identification of two distinct scaling regimes for quantum walks on line and binary tree

Quantum speedup can be achieved and manipulated by tuning initial states

Abstract

Quantum walks constitute an important tool for designing quantum algorithms and information processing tasks. In a lackadaisical walk, in addition to the possibility of moving out of a node, the walker can remain on the same node with some probability. This is achieved by introducing self-loops, parameterized by self-loop strength $l$, attached to the nodes such that large $l$ implies a higher likelihood for the walker to be trapped at the node. In this work, {\it directed}, lackadaisical quantum walks is studied. Depending on $l$, two regimes are shown to exist -- one in which classical walker dominates and the other dominated by the quantum walker. In the latter case, we also demonstrate the existence of two distinct scaling regimes with $l$ for quantum walker on a line and on a binary tree. Surprisingly, a significant quantum-induced speedup is realized for large $l$. By tuning the initial state, the extent of this speedup can be manipulated.