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arXiv:2510.01584 · Physical Review A

Directionality and quantum backfire in continuous-time quantum walks from delocalized states: Exact results

Jefferson J. Ximenes, Marcelo A. Pires, J. M. Villas-Bôas

Published Jan 5, 2026• 0 citations
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Abstract

We derive analytical results for continuous-time quantum walks from a new class of initial states with tunable delocalization. The dynamics are governed by a Hamiltonian with complex hopping amplitudes. We provide closed-form equations for key observables, revealing three notable findings: (1) the emergence of directed quantum transport from completely unbiased initial conditions; (2) a quantum backfire effect, where greater initial delocalization enhances short-time spreading but counterintuitively induces a comparatively smaller long-time spreading after a crossing time $t_{\mathrm{cross}}$; and (3) an exact characterization of survival probability, showing that the transition to an enhanced $t^{-3}$ decay is a fine-tuned effect. Our work establishes a comprehensive framework for controlling quantum transport through the interplay between intermediate initial delocalization and Hamiltonian phase.

DOI
10.1103/h26t-g6c7

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We derive analytical results for continuous-time quantum walks from a new class of initial states with tunable delocalization. The dynamics are governed by a Hamiltonian with complex hopping amplitudes. We provide closed-form equations for key observables, revealing three notable findings: (1) the emergence of directed quantum transport from completely unbiased initial conditions; (2) a quantum backfire effect, where greater initial delocalization enhances short-time spreading but counterintuitively induces a comparatively smaller long-time spreading after a crossing time tcrosst_{\mathrm{cross}}tcross​; and (3) an exact characterization of survival probability, showing that the transition to an enhanced t−3t^{-3}t−3 decay is a fine-tuned effect. Our work establishes a comprehensive framework for controlling quantum transport through the interplay between intermediate initial delocalization and Hamiltonian phase.
We investigate the quantum transport of delocalized states in continuous-time quantum walks (CTQWs) on a one-dimensional lattice containing a single defect. The defect is modeled by assigning complex-valued hopping amplitudes to the edges that connect the site corresponding to the mean position of the delocalized initial state to its nearest neighbors. We find that this single defective site is sufficient to enhance the ballistic spreading of an initially Gaussian wave packet. Extending these results, we implement a time-dependent alternation protocol between two distinct defect configurations, each individually yielding poor propagation of the state. The combination of these two unfavorable configurations improves the transport efficiency of the quantum walker, revealing a manifestation of Parrondo's paradox in CTQWs with delocalized initial states. This study provides insights into the role of complex-phase defects and time-dependent protocols in CTQWs, demonstrating that the interplay between quantum interference and graph engineering can effectively enhance quantum transport in discrete lattices.
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 (Khalique et al 2021 New J. Phys. 23 083005). 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 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.
We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-1/21/21/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\ensuremath{\alpha}$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes NNN, time ttt, and the model parameter $\ensuremath{\alpha}$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\ensuremath{\alpha}=\ensuremath{\pi}/2$. By proving two mirror symmetries around $\ensuremath{\alpha}=0$ and $\ensuremath{\pi}/2$ we show that these quantum walks have a period of $\ensuremath{\pi}$ in $\ensuremath{\alpha}$. We show that undirected edges lose their effect on the quantum walk at $\ensuremath{\alpha}=\ensuremath{\pi}/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.
We present a manifestation of a Parrondo's effect in a continuous-time quantum walk (CTQW). In our protocol we consider a CTQW in the presence of a time-dependent transition defect. Our results show that the alternation between defects, that individually are detrimental to the wavepacket spreading, can paradoxically enhance overall wavepacket propagation. Our findings pave the way for the exploration of unconventional mechanisms that can potentially harness the adverse effects of defects to enhance quantum transport.
In this paper we study the quantum transport on networks with a temporal evolution governed by the fractional Schrödinger equation. We generalize the dynamics based on continuous-time quantum walks, with transitions to nearest neighbors on the network, to the fractional case that allows long-range displacements. By using the fractional Laplacian matrix of a network, we establish a formalism that combines a long-range dynamics with the quantum superposition of states; this general approach applies to any type of connected undirected networks, including regular, random, and complex networks, and can be implemented from the spectral properties of the Laplacian matrix. We study the fractional dynamics and its capacity to explore the network by means of the transition probability, the average probability of return, and global quantities that characterize the efficiency of this quantum process. As a particular case, we explore analytically these quantities for circulant networks such as rings, interacting cycles, and complete graphs.