[Paper Review] The stationary measure for diagonal quantum walk with one defect

Takako Endo, Hikari Kawai+1 more|arXiv (Cornell University)|Mar 29, 2016

Quantum Computing Algorithms and Architecture20 references3 citations

TL;DR

This paper derives the stationary measure for one-dimensional quantum walks with diagonal unitary matrices and one defect using the scattering Green's function (SGF) method. It shows that the stationary measure is generally non-uniform and depends on the defect's structure and initial state, but becomes uniform when the defect is diagonal, providing a classification framework for stationary measures in diagonal quantum walks.

ABSTRACT

This study is motivated by the previous work [14]. We treat 3 types of the one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrix, and diagonal unitary matrices with one defect. In this paper, we call the QW defined by diagonal unitary matrices, "the diagonal QW", and we consider the stationary distributions of generally 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of the purposes of our study is to characterize the QWs by the stationary measure, which may lead to answer the basic and natural question, "What the stationary measure is for one-dimensional QWs ?". In order to analyze the stationary distribution, we focus on the corresponding eigenvalue problems and the definition of the stationary measure.

Motivation & Objective

• To characterize one-dimensional quantum walks by their stationary measures, addressing the fundamental question: What is the stationary measure for 1D quantum walks?
• To analyze the stationary distributions of 2-state and 3-state diagonal quantum walks with one defect, extending prior work on uniform and non-uniform measures.
• To clarify the conditions under which the stationary measure becomes uniform, particularly in relation to the diagonality of the defect and the quantum coin structure.
• To generalize previous results on stationary measures in diagonal quantum walks, especially those from Konno et al. [14], by incorporating defect effects.
• To lay the groundwork for classifying diagonal quantum walks based on their stationary measure properties, including future extension to multiple defects and topological invariants.

Proposed method

• Applies the scattering Green's function (SGF) method to solve the eigenvalue problem associated with the time evolution operator of the quantum walk.
• Models the time evolution using diagonal unitary matrices for the bulk and a non-diagonal defect matrix at x=0, with phase parameters σ± and general complex entries.
• Derives the stationary measure μ(x) as a function of the wavefunction amplitudes Ψ^L(0), Ψ^R(0), Ψ^L(1), Ψ^R(-1), and their complex conjugates.
• Uses the eigenvalues λ = ±√(Δ^(±)e^{iσ±}) to determine the stationary measure components, showing independence from the specific phase values σ± when Δ^(±) are fixed.
• Applies the same method to the 3-state space-homogeneous diagonal QW with a single diagonal matrix U, obtaining a simplified measure expression.
• Compares results across 2-state and 3-state cases to identify structural dependencies of the stationary measure on defect type and number of internal states.

Experimental results

Research questions

• RQ1 Under what conditions does the stationary measure of a diagonal quantum walk with one defect become uniform?
• RQ2 How does the presence of a non-diagonal defect influence the form and spatial dependence of the stationary measure?
• RQ3 What is the role of the determinant and phase parameters of the diagonal unitary matrices in shaping the stationary measure?
• RQ4 How does the number of internal states (2 vs 3) affect the structure of the stationary measure in diagonal quantum walks?
• RQ5 Can the SGF method be systematically extended to classify all stationary measures in diagonal quantum walks, including those with multiple defects?

Key findings

• The stationary measure for the 2-state diagonal quantum walk with one defect is generally non-uniform and depends on the complex entries of the defect matrix and the initial state amplitudes.
• The measure does not exhibit exponential decay and is strongly influenced by the defect structure and initial coin state, even when the bulk coins are diagonal.
• For the 3-state space-homogeneous diagonal QW, the stationary measure is μ(x) = |α|² + |β|² for x ≠ 0 and μ(0) = |α|² + |γ|² + |β|², showing explicit position dependence.
• When the defect is diagonal, the stationary measure becomes uniform regardless of the bulk parameters, confirming a key result from Konno et al. [14].
• The eigenvalues λ = ±√(Δ^(±)e^{iσ±}) determine the stationary measure, and the measure is independent of the specific values of σ± if Δ^(±) are fixed.
• The method successfully generalizes previous results on stationary measures, showing that the diagonality of the defect is a sufficient condition for uniformity of the stationary measure.