Iwasawa-Type Spectral Resultant Growth Laws for Grover Walks on Graph Towers

Abstract: Let $X_0\leftarrow X_1\leftarrow\cdots$ be a $\mathbb Z_p<sup>d$-tower of finite graphs, and let $U_n$ be the Grover transition matrix on $X_n$. We study Iwasawa-type $p$-adic growth laws for the polynomial spectral quantities [ \det P(U_n), ] where $P(A)$ is a monic polynomial. The basic object is the spectral resultant [ \mathcal R_{X,P}(T)=\operatorname{Res}A(\mathcal F_X(A,T),P(A)), ] where $\mathcal F_X(A,T)$ is the universal Grover--Ihara spectral polynomial of the tower. In the integral setting, this resultant generates the zeroth Fitting ideal of a natural finite module over the Iwasawa algebra; when the resultant is nonzero, this module is torsion. The polynomial $P$ packages prescribed spectral values into a single spectral packet. If $P$ is coprime to the Bass factor $A<sup>2-1$ and $\mathcal R</em>{X,P}$ does not vanish at torsion characters, then $\det P(U_n)$ is nonzero for all $n$ and we prove a Cuoco--Monsky type leading asymptotic formula for $v_p(\det P(U_n))$. The leading terms are given explicitly by the $μ$- and $λ$-invariants of $\mathcal R_{X,P}$, with a separate correction coming from the Bass factor. For $P(A)=A-a$, with $a\ne\pm1$ and $a$ not an eigenvalue at any level, this recovers the leading invariants in the fixed non-eigenvalue formula for Grover characteristic polynomials. We also prove an equivariant factorization of spectral resultants for finite connected $p$-group covers. As a consequence, we obtain an unramified equivariant Kida formula under explicit integrality and nonzero-resultant assumptions. Finally, when $\gcd(P,A<sup>2-1)=1$, we show that torsion zeros of $\mathcal R_{X,P}$ correspond exactly to occurrences of roots of $P$ as Grover eigenvalues at finite levels. The examples include the $K_3$-tower, non-abelian Heisenberg $5$-group covers, and an explicit torsion-zero spectral packet.

• The paper introduces an Iwasawa-theoretic framework for analyzing spectral resultants of Grover quantum walks on graph towers.
• It derives explicit p-adic growth laws for determinants of polynomial-filtered spectral packets using μ- and λ-invariants.
• The paper establishes an equivariant Kida formula and identifies exceptional zeros that precisely capture eigenvalue occurrences in finite graph covers.

Iwasawa-Type Spectral Resultant Growth Laws for Grover Walks on Graph Towers

Introduction and Context

This paper develops an Iwasawa-theoretic framework for spectral invariants arising from Grover quantum walks on towers of finite graphs with Galois group of the form $\mathbb{Z}_p^d$. The Grover walk, fundamental in quantum computing and spectral graph theory, is encoded by a unitary transition operator whose spectrum exhibits deep, structured arithmetic when examined along such graph towers. Previous work connected Iwasawa theory and graph theory—especially via the study of spanning trees as analogues of class numbers—but this study introduces and analyzes global spectral data via the resultant of universal characteristic polynomials. Specifically, the focus is on the growth of spectral packets—constructed from determinants of prescribed polynomials of the Grover operator—along the tower, and their relation to $p$-adic invariants.

Universal Spectral Resultants and Polynomial Packets

Let $X_0 \leftarrow X_1 \leftarrow \cdots$ be a tower of finite graphs with abelian $p$-adic Galois group $\Gamma \simeq \mathbb{Z}_p^d$, and $U_n$ the Grover transition matrix on $X_n$. The primary object is the spectral resultant

$$\mathcal{R}_{X, P}(T) = \mathrm{Res}_A(\mathcal{F}_X(A, T), P(A)),$$

where $\mathcal{F}_X(A, T)$ is the universal Grover–Ihara spectral polynomial and $P$ is a chosen monic polynomial. $p$0 acts as a spectral filter, focusing on a prescribed "packet" of eigenvalues.

In the integral case, $p$1 generates the zeroth Fitting ideal of a natural spectral module over the Iwasawa algebra $p$2. When the resultant is nonzero, the module is torsion and its structure is controlled by the so-called $p$3- and $p$4-invariants.

A key structural result is a universal factorization: $p$5 where $p$6 is the number of unoriented edges, $p$7 the Euler characteristic, $p$8, and $p$9 the group of $X_0 \leftarrow X_1 \leftarrow \cdots$0-th roots of unity.

Iwasawa-Type Asymptotic Growth Laws

Let $X_0 \leftarrow X_1 \leftarrow \cdots$1 be a fixed monic polynomial coprime to $X_0 \leftarrow X_1 \leftarrow \cdots$2, and assume $X_0 \leftarrow X_1 \leftarrow \cdots$3 for all torsion characters. The principal result is an explicit $X_0 \leftarrow X_1 \leftarrow \cdots$4-adic asymptotic for the $X_0 \leftarrow X_1 \leftarrow \cdots$5-adic valuation of $X_0 \leftarrow X_1 \leftarrow \cdots$6 as $X_0 \leftarrow X_1 \leftarrow \cdots$7: $X_0 \leftarrow X_1 \leftarrow \cdots$8 where

$X_0 \leftarrow X_1 \leftarrow \cdots$9

This is a nontrivial generalization of known results for linear $p$0 to arbitrary polynomial packets, reflecting the entire spectral packet's arithmetic structure.

The paper establishes that the aforementioned growth law continues to hold for function fields and link covers, extending techniques from classical Iwasawa theory to quantum and combinatorial contexts. The results are enabled by an explicit control of the spectral Fitting module and the application of $p$1-adic methods à la Cuoco–Monsky.

Equivariant Factorization and Kida-Type Formulas

A central algebraic innovation is the formal equivariant factorization of spectral invariants under finite $p$2-group graph covers (including nonabelian cases). Given an unramified cover $p$3 with Galois group $p$4, the spectral resultant admits a factorization: $p$5 where the $p$6-twisted resultants refine the structure according to the representations of $p$7.

When $p$8 is a $p$9-group and integrality/nonvanishing conditions are satisfied, this leads to an unramified equivariant Kida formula for spectral invariants: $\Gamma \simeq \mathbb{Z}_p^d$0 and corresponding scaling for the Bass-corrected $\Gamma \simeq \mathbb{Z}_p^d$1-invariant. There is no $\Gamma \simeq \mathbb{Z}_p^d$2 correction term in this setting, in contrast to previous Laplacian-based Kida formulas.

Exceptional Zeros and Spectral Obstruction

A crucial structural dichotomy is established: if $\Gamma \simeq \mathbb{Z}_p^d$3 vanishes for some torsion character, then the growth law transitions from a $\Gamma \simeq \mathbb{Z}_p^d$4-adic asymptotic regime to a regime reflecting explicit eigenvalue occurrences—an analog of exceptional zeros in Iwasawa theory. In this case, the vanishing locus precisely captures when roots of $\Gamma \simeq \mathbb{Z}_p^d$5 occur in the Grover spectrum at finite levels of the tower and the precise multiplicity via a $\Gamma \simeq \mathbb{Z}_p^d$6-weighted spectral order.

If the resultant does not vanish at any torsion point, $\Gamma \simeq \mathbb{Z}_p^d$7 for all $\Gamma \simeq \mathbb{Z}_p^d$8; otherwise, the occurrences of exceptional zeros are determined by spectral packets harboring eigenvalues at finite stages.

Exemplifications and Applications

The theory is illustrated with several explicit and nontrivial examples:

• $\Gamma \simeq \mathbb{Z}_p^d$9-tower: The spectrum is described explicitly by cyclotomic data, with determinant identities $U_n$0 and tight connections to eigenvalue periodicity.
• Nonabelian Heisenberg 5-group covers: The full equivariant theory is demonstrated, showing the validity of the Kida scaling invariants and explicit computation of the $U_n$1-invariant for nonabelian $U_n$2-group covers.
• Spectral packets with torsion zeros: Specific polynomial packets are identified that detect the precise level at which certain eigenvalues enter the spectrum.

Theoretical and Practical Implications

This work links spectral graph invariants from quantum walks to deep algebraic structures, synthesizing $U_n$3-adic Iwasawa invariants, Fitting ideals, spectral resultants, and categorical representation theory of finite group covers. The results provide fine-grained arithmetic control over entire spectral packets of quantum walks along infinite graph towers, establishing new links between quantum/combinatorial invariants and algebraic number theory.

The explicit computation of spectral growth laws for polynomial packets opens avenues for further exploration in quantum algorithms, arithmetic geometry of discrete dynamical systems, and ultimately graph-theoretic counterparts of classical analytic number theory. The approach suggests analogous structures for other quantum or Markovian walks and can potentially be adapted to the analysis of higher-dimensional topological covers, random walks, and noncommutative geometry.

Conclusion

This paper provides a comprehensive algebraic and analytic description of Iwasawa-type growth for polynomial spectral invariants of Grover walks on graph towers. Key deliverables include the introduction and Fitting-theoretic analysis of the universal spectral resultant, the establishment of explicit $U_n$4-adic asymptotics via $U_n$5- and $U_n$6-invariants, and the proof of an equivariant Kida formula in spectral settings. The work also clarifies the dichotomy between growth and spectral occurrence via the behavior of resultants at torsion characters, and illustrates the theory through concrete examples, both abelian and nonabelian, highlighting the full scope of the methodology. The developed framework represents a significant step forward in extending Iwasawa-theoretic techniques and spectral packet analysis to quantum walk dynamics in graph towers (2607.02011).