[Paper Review] Large Qudit Limit of One-dimensional Quantum Walks

Mitsunori Sato, Naoki Kobayashi+2 more|ArXiv.org|Feb 14, 2008

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper investigates the large-qudit limit of one-dimensional discrete-time quantum walks with spin-j systems, showing that as the spin quantum number $j \to \infty$, the pseudovelocities' limit distribution transitions from a multi-peak structure to a universal, monotone convex shape centered at zero. The analysis reveals a quantum-to-classical crossover, where the system approaches diffusive behavior in the large-$j$ limit, suggesting a route from quantum walks to classical diffusion via the large-qudit limit.

ABSTRACT

We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et al.}, each of which the walker's wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j o \infty$ limit.

Motivation & Objective

• To understand the asymptotic behavior of pseudovelocities in one-dimensional quantum walks as the spin quantum number $j$ becomes large.
• To analyze the limit distribution of pseudovelocities in the $j \to \infty$ limit for $2j+1$-component quantum walks.
• To investigate whether the large-$j$ limit of quantum walks exhibits classical diffusive behavior, suggesting a quantum-to-classical transition.
• To derive and analyze the asymptotic form of the weight functions $\mathcal{M}^{(j,m)}(x)$ in the superposition of Konno's density functions.

Proposed method

• The study employs a family of discrete-time quantum walks on $\mathbb{Z}$, parameterized by half-integer spin $j = 1/2, 1, 3/2, \dots$, with $2j+1$ internal states.
• The quantum coin is implemented via Wigner's rotation matrices with Euler angles $\alpha, \beta, \gamma$, ensuring unitarity and spin-$j$ structure.
• The limit distribution of pseudovelocities is expressed as a superposition of scaled Konno's density functions $\mu(v; \cos(\beta/2))$, weighted by polynomials $\mathcal{M}^{(j,m)}(v/2m)$, plus a possible delta function at zero for odd $2j+1$.
• Asymptotic analysis of $\mathcal{M}^{(j,m)}(x)$ is performed for large $j$, using generating functions and combinatorial identities derived from Wigner $6j$-symbols.
• The behavior of the weight functions is studied in the $j \to \infty$ limit to determine the shape of the resulting pseudovelocities' distribution.
• A scaling ansatz $X_t^{(j)} \sim jt F(t/j^\theta)$ is proposed to explore a crossover from ballistic to diffusive scaling in the large-$j$ regime.

Experimental results

Research questions

• RQ1 How does the limit distribution of pseudovelocities evolve as the spin quantum number $j$ increases in $2j+1$-component quantum walks?
• RQ2 Do the $2j+1$ pikes in the limit distribution—present for finite even $2j+1$—disappear in the $j \to \infty$ limit?
• RQ3 Does the large-$j$ limit of the quantum walk exhibit a universal, monotone convex structure in the pseudovelocities' distribution?
• RQ4 Can the large-$j$ limit be interpreted as a classical diffusive process, and if so, what scaling regime realizes this?

Key findings

• For finite $j$, the pseudovelocities' limit distribution exhibits $2j+1$ pikes when $2j+1$ is even, due to the superposition of Konno's density functions with non-trivial weight functions.
• As $j \to \infty$, the pikes vanish and the limit distribution develops a universal, monotone convex structure centered at zero, indicating a loss of discrete peak structure.
• The weight functions $\mathcal{M}^{(j,m)}(x)$ become highly delocalized and smooth in the large-$j$ limit, leading to a broadened, convex profile in the pseudovelocities' distribution.
• The large-$j$ limit suggests a quantum-to-classical crossover, where the system's behavior approaches diffusive scaling under an appropriate time-space scaling.
• The asymptotic form of the limit distribution is consistent with classical diffusion when the scaling exponent $\theta$ satisfies $\theta = 2$, leading to $X_t^{(j)} \sim j^{3/2} \sqrt{t}$ in the large-$t$ regime.
• The emergence of a convex, smooth distribution in the large-$j$ limit supports the idea that large spin systems can simulate classical diffusion, even though individual quantum walks are ballistic.