Quantum walks driven by quantum coins with two multiple eigenvalues

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.

💡 Research Summary

The paper develops a unified spectral framework for coined quantum walks on finite graphs that goes beyond the traditional Ihara class. In the standard setting, each local coin operator C_u is required to have spectrum {±1} with a one‑dimensional eigenspace for the eigenvalue +1. Under these constraints the walk’s unitary evolution U = S C (where S is the flip‑flop shift) can be reduced to a self‑adjoint “discriminant” operator T acting on ℓ²(V) and the spectrum of U is obtained from the spectrum of T by a simple two‑to‑one mapping onto the unit circle.

The authors relax the coin condition dramatically. For a fixed integer p with 1 ≤ p ≤ δ(G) (δ(G) being the minimum vertex degree) they assume that every C_u has exactly two distinct unit‑modulus eigenvalues κ and κ′, and that the eigenspace ker(κ − C_u) has dimension p, independent of u. Choosing an orthonormal basis {α_j(u)}_{j=1}^p of this eigenspace, they define a p‑component vector w(a) for each directed edge a = (o(a),t(a)) by collecting the conjugates of the α_j evaluated at a. The matrix‑valued weight W(a) = w(a) w(ā)† (where ā is the reverse edge) encodes the transition amplitudes between the p internal degrees of freedom.

A boundary operator K : ℓ²(V;ℂ^p) → ℓ²(A) is introduced, whose columns consist of the vectors w(a). Its adjoint K† maps edge amplitudes back to vertex amplitudes. The key identities are

1. K†K = I_{V^p},
2. T = K† S K,
3. C = (κ − κ′) K K† + κ′ I_p.

Thus T is a self‑adjoint operator on the enlarged vertex space V^p = ℂ^p⊗ℓ²(V), with matrix elements

 T_{u,v} = Σ_{a: o(a)=v, t(a)=u} W(a).

When p = 1, T reduces to the scalar weighted adjacency matrix familiar from the Ihara–Bass theory. For p > 1, T becomes a matrix‑valued walk operator that still retains self‑adjointness.

The main spectral mapping theorem states that if λ ∈