[Paper Review] Limit distribution with a combination of density functions for a 2-state quantum walk

Takuya Machida|arXiv (Cornell University)|Oct 18, 2010

Quantum Computing Algorithms and Architecture19 references3 citations

TL;DR

This paper investigates a time-inhomogeneous 2-state quantum walk on the integer line, where unitary evolution matrices alternate periodically, enabling localization at the origin—a phenomenon absent in standard time-homogeneous walks. Using Fourier analysis and eigenvalue decomposition, the authors derive the limit distribution as a combination of density functions, explicitly showing ballistic spreading with non-uniform, multi-component probability density arising from periodic alternation of evolution operators.

ABSTRACT

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk in certain intervals. In the usual QWs starting from the origin, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present some limit distributions for the walk.

Motivation & Objective

• To analyze the asymptotic behavior of a 2-state quantum walk with time-inhomogeneous evolution governed by two unitary matrices.
• To investigate whether localization can occur at the origin in such time-inhomogeneous models, contrary to standard time-homogeneous walks.
• To derive the limit distribution of the walker's position as a combination of multiple density functions using Fourier analysis.
• To establish convergence theorems for the walk's distribution at specific time scales, particularly at times of the form $(m+n) au + m$.
• To characterize the limiting probability density as a function of initial state parameters and evolution parameters such as $ heta$.

Proposed method

• The walk is defined on $ℤ$ using two unitary matrices $U$ and $H$, with time-dependent evolution alternating between $U$ and $H$ at regular intervals of length $\tau+1$.
• The time evolution is analyzed via the Fourier transform of the amplitude, leading to a matrix recurrence in the Fourier domain involving $\hat{U}(k) = R(k)U$ and $\hat{H}(k) = R(k)H$, where $R(k)$ is a phase shift matrix.
• Eigenvalues and eigenvectors of $\hat{U}(k)$ are used to diagonalize the evolution operator, enabling exact expression of the amplitude at time $(m+n)\tau + m$.
• The $r$-th moment of the position distribution is computed asymptotically as $\tau \to \infty$, with leading-order terms derived from the eigenvalue dynamics and transition matrix elements.
• The limit distribution is obtained by taking the characteristic function limit and identifying the density as a sum of weighted components involving $f_K(x;c)$, $M_2(x)$, $B_2(x;n)$, and $B_3(x;n)$.
• The final limit density is expressed as a sum of three components: one from the dominant eigenmode, and two from subdominant contributions scaled by $n/(n+2)$ and $(n+2)/(n-2)$, respectively.

Experimental results

Research questions

• RQ1 Can localization occur at the origin in a time-inhomogeneous 2-state quantum walk, despite its absence in standard time-homogeneous models?
• RQ2 What is the asymptotic limit distribution of the walker's position when the evolution alternates periodically between two unitary matrices?
• RQ3 How does the limit distribution depend on the initial state $[\alpha, \beta]^T$ and the parameter $\theta$?
• RQ4 Can the limit distribution be expressed as a combination of multiple density functions, and if so, what are their functional forms?
• RQ5 What is the scaling behavior of the moments and how does it relate to the ballistic spreading of the walk?

Key findings

• The limit distribution of the quantum walk is expressed as a sum of three distinct density components, each corresponding to different eigenmode contributions from the time-evolution operator.
• The dominant component is proportional to $f_K(x;c) = \frac{|s|}{\pi(1-x^2)\sqrt{c^2 - x^2}}I_{(-|c|,|c|)}(x)$, which is a semi-circle-like density scaled by $|s|$.
• Two subdominant components arise from mixed eigenvector contributions, with densities weighted by $B_2(x;n)$ and $B_3(x;n)$, which depend on the initial state and the parameter $n$.
• The limit distribution exhibits non-uniform, multi-modal structure due to the periodic alternation of evolution matrices, leading to a combination of densities rather than a single simple form.
• The characteristic function converges to a limit that corresponds to a density function composed of three weighted terms, each involving scaled versions of $f_K(x;c)$ with different scaling factors $n/(n+2)$ and $(n+2)/(n-2)$.
• Localization at the origin is possible due to the time-inhomogeneous structure, as the periodic alternation of $U$ and $H$ breaks the symmetry that prevents localization in homogeneous walks.