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This study quantifies the velocity deficit in continuous-time quantum walks on a hard-wall half-line when the initial state is restricted to a finite number of sites. By reducing the optimization problem to the principal eigenvalue of an $M \times M$ Hermitian matrix, the authors derive an inverse-square scaling law for the drift approaching the ballistic front. These analytical and numerical results provide precise bounds on how quickly quantum information can propagate under spatial confinement constraints.
Maximal-velocity deficit under a finite-support constraint in hard-wall half-line continuous-time quantum walk
arXiv:2609.01970v1 Announce Type: new Abstract: Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first $M$ sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit $M\times M$ Hermitian matrix, which yields the optimal state by direct diagonalization. For large $M$, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in $M$ and a constant prefactor $\pi^2/4$. Numerical results verify both the finite-$M$ spectral formulation and the predicted scaling.
Abstract:Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first $M$ sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit $M\times M$ Hermitian matrix, which yields the optimal state by direct diagonalization. For large $M$, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in $M$ and a constant prefactor $\pi^2/4$. Numerical results verify both the finite-$M$ spectral formulation and the predicted scaling.
16 pages, 4 figures. To be published in Physical Review A as a Regular Paper
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Spectral Theory (math.SP)
arXiv:2609.01970 [quant-ph]
(or arXiv:2609.01970v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.01970 arXiv-issued DOI via DataCite (pending registration)
https://doi.org/10.1103/dyyr-z1k8 DOI(s) linking to related resources
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