Quantum walks driven by quantum coins with two multiple eigenvalues

TL;DR

We address the spectral analysis of coined quantum walks on graphs where local coins have spectrum in $\{κ,κ'\}$ and $\dim\ker(κ-C_u)=p$. The main idea is to decompose the evolution into an inherited subspace governed by a self-adjoint discriminant operator $T$ acting on $\mathcal{V}_p=\mathbb{C}^V\otimes\mathbb{C}^p$, and to show how the eigenpairs of $T$ lift to the full walk via a one-to-two mapping (with special handling for $±1$). A boundary-operator construction $K$ and a corresponding matrix-valued weight $W$ enable an explicit expression of $T$ and the lifting mechanism; this yields a transparent spectral map between $T$ and the time-evolution operator $U=SC$ on the invariant subspace $\mathcal{L}=K\mathcal{V}_p+SK\mathcal{V}_p$. As a key application, the authors derive the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with moving shift in Fourier space, illustrating how the framework handles non-Ihara-class quantum walks. Overall, the work extends Ihara-type spectral mapping to a broader class of quantum walks with internal degrees of freedom and moving shifts, offering exact spectral data and insights into localization and transport phenomena.

Abstract

We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $\{C_u\}_{u\in V}$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $κ,κ'$ and $p=\dim(\ker(κ-C_u))$ for any $u\in V$ with $1\leq p\leq δ(G)$, where $δ(G)$ is the minimum degrees of $G$. We show that this quantum walk can be decomposed into a cellular automaton on $\ell^2(V;\mathbb{C}^p)$ whose time evolution is described by a self adjoint operator $T$ and its remainder. We obtain how the eigenvalues and its eigenspace of $T$ are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with the moving shift in the Fourier space.

Figures (1)

• Figure 1: The way of the mapping from $\mathrm{Spec}(T)$ to $\mathrm{Spec}(U_{\mathcal{L}})$: The left figure shows how each eigenvalue of $T$; $\mu$, is mapped on the unit circle as $e^{\phi}e^{-(\xi-\eta)/2}$, where $e^{i\phi}$ is the eigenvalue of $U|_\mathcal{L}$, $e^{i\xi}=\kappa$ and $e^{i\eta}=\kappa'$. Each eigenvalue of $T$ except $\pm 1$ are mapped to two points of the unit circle, and each eigenvalue of $\pm 1$ is mapped to only one points. The right figure is obtained by the $(\xi+\eta)/2$ rotation of the left figure, and this shows the region including the support of the eigenvalues of $U|_{\mathcal{L}}$. The white two points on the unit circle, which represents the blanks as the support of the $\mathrm{Spec}(U|_\mathcal{L})$, are filled by the eigenvalues of $\mathcal{U}|_{\mathcal{L}^\perp}$ with the multiplicities $|E|-p|V|+m_\pm$.

Theorems & Definitions (26)

• Definition 1
• Proposition 2.1
• proof
• Lemma 3.1
• Definition 2
• Lemma 3.2
• proof
• Proposition 4.1
• proof
• Proposition 4.2