Controlled transport in chiral quantum walks on graphs

TL;DR

This paper explores how chiral quantum walks on graphs exhibit controllable asymmetric and complete transport properties, with a focus on Y-junction graphs, using gauge transformations and phase analysis to understand and optimize quantum transport.

Contribution

It introduces a gauge transformation approach to analyze chiral CTQWs, and develops a phase-based control method for asymmetric and complete transport on Y-junction graphs.

Findings

Chiral CTQWs are equivalent to non-chiral ones with initial momentum shifts.

Phases control asymmetric and complete transport in Y-junction graphs.

Explicit conditions for 100% quantum transport efficiency are derived.

Abstract

We investigate novel transport properties of chiral continuous-time quantum walks (CTQWs) on graphs. By employing a gauge transformation, we demonstrate that CTQWs on chiral chains are equivalent to those on non-chiral chains, but with additional momenta from initial wave packets. This explains the novel transport phenomenon numerically studied in [New J. Phys. 23, 083005(2021)]. Building on this, we delve deeper into the analysis of chiral CTQWs on the Y-junction graph, introducing phases to account for the chirality. The phase plays a key role in controlling both asymmetric transport and directed complete transport among the chains in the Y-junction graph. We systematically analyze these features through a comprehensive examination of the chiral continuous-time quantum walk (CTQW) on a Y-junction graph. Our analysis shows that the CTQW on Y-junction graph can be modeled as a combination of three wave functions, each of which evolves independently on three effective open chains. By constructing a lattice scattering theory, we calculate the phase shift of a wave packet after it interacts with the potential-shifted boundary. Our results demonstrate that the interplay of these phase shifts leads to the observed enhancement and suppression of quantum transport. The explicit condition for directed complete transport or 100% efficiency is analytically derived. Our theory has applications in building quantum versions of binary tree search algorithms.