[Paper Review] One-dimensional discrete-time quantum walks on random environments

Norio Konno|arXiv (Cornell University)|Apr 2, 2009

Quantum Computing Algorithms and Architecture17 references5 citations

TL;DR

This paper establishes quenched and annealed weak limit theorems for one-dimensional discrete-time quantum walks on random environments using path counting techniques. It derives the limiting distribution of the walker's position, showing that the quenched limit depends on the local environment parameter ω₀, while the annealed limit averages over ω₀, generalizing the symmetric Hadamard walk's known limit to random environments.

ABSTRACT

We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.

Motivation & Objective

• To establish the first rigorous weak limit theorems for discrete-time quantum walks on random environments in one dimension.
• To analyze the asymptotic behavior of the walker's position distribution under quenched (conditional on a fixed environment) and annealed (averaged over environments) measures.
• To generalize the known limit law of the symmetric Hadamard walk to the case of spatially inhomogeneous, random unitary matrices.
• To derive explicit expressions for path amplitudes using combinatorial sums and relate them to Jacobi polynomials for asymptotic analysis.

Proposed method

• The model uses a sequence of i.i.d. random unitary matrices $ U_x = \frac{1}{\sqrt{2}} \begin{bmatrix} e^{i\omega_x} & 1 \\ 1 & -e^{-i\omega_x} \end{bmatrix} $, where $ \omega_x \in \mathbb{R} $, defining the local dynamics.
• Path counting is employed to compute the amplitude of the walker reaching position $ x = -l + m $ after $ n = l + m $ steps.
• The amplitudes $ p_n^{(H)}(l,m) $ and $ q_n^{(H)}(l,m) $ for the Hadamard walk are expressed via sums involving binomial coefficients and alternating signs.
• These sums are rewritten in terms of Jacobi polynomials $ P^{\nu,\mu}_{n}(x) $, enabling asymptotic analysis via hypergeometric function identities.
• The characteristic function $ E_n^\omega(e^{i\xi X_n/n}) $ is analyzed in the limit $ n \to \infty $, leading to the weak limit theorem.
• The quenched limit is derived by combining the deterministic Hadamard walk limit with a perturbation term proportional to $ \sin(\omega_0) $, yielding a modified density function.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the position of a discrete-time quantum walker in a one-dimensional random environment?
• RQ2 How does the presence of a random environment affect the weak limit law compared to the standard Hadamard walk?
• RQ3 Can the quenched and annealed distributions be explicitly characterized using path counting and special functions?
• RQ4 What role does the local environment parameter $ \omega_0 $ play in shaping the limiting distribution?
• RQ5 How do the amplitudes of paths with given left/right step counts relate to orthogonal polynomials in the asymptotic regime?

Key findings

• The quenched weak limit of the quantum walk converges to a probability density proportional to $ \frac{1 - \sin(\omega_0)x}{\pi(1 - x^2)\sqrt{1 - 2x^2}} $ on the interval $ [-1/\sqrt{2}, 1/\sqrt{2}] $.
• The annealed weak limit is obtained by averaging over $ \omega_0 $, recovering the standard Hadamard walk limit when $ \omega_0 = 0 $.
• The path amplitudes $ p_n^{(H)}(l,m) $ and $ q_n^{(H)}(l,m) $ are expressed in terms of Jacobi polynomials $ P^{0,n-2l}_{l-1}(0) $ and $ P^{1,n-2l}_{l-1}(0) $, enabling precise asymptotic analysis.
• The characteristic function $ E_n^\omega(e^{i\xi X_n/n}) $ converges to an integral involving $ \sin(\omega_0) $, showing that the environment introduces a linear correction to the standard density.
• The limiting density is singular at $ x = \pm 1/\sqrt{2} $, with inverse-square-root singularities, similar to the Hadamard walk but modulated by the environment parameter.
• The method successfully generalizes the Hadamard walk’s limit law to random environments, providing a rigorous foundation for future studies of quantum walks in disordered media.