The Quantum Walk Characteristic Polynomial Distinguishes All Strongly Regular Graphs of Prime Orde

Abstract: Let $G$ be a strongly regular graph of prime order $p$ with connection degree $k \geq 6$. We prove that the \emph{quantum walk characteristic polynomial} $χ_q(G,λ) \coloneqq \det(λI - U_G)$, where $U_G$ is the coined quantum walk operator on $G$, completely determines $G$ up to isomorphism within the class of strongly regular graphs of the same order. The proof proceeds in three steps. First, we show that $U_G$ block-diagonalizes under the discrete Fourier transform over $\Z_p$, yielding $p$ blocks $U_G^{(j)}$ of size $k \times k$. Second, we prove an explicit formula [ χ_q!\bigl(U_G^{(j)}, λ\bigr) = (λ-1)^{(k-2)/2}(λ+1)^{(k-2)/2} !\left(λ^2 - \tfrac{2\widehat{A}_G(j)}{k}\,λ+ 1\right), ] from which the Fourier coefficient $\widehat{A}_G(j)$ is recovered as the unique real part of an eigenvalue of $U_G^{(j)}$ distinct from $\pm 1$. Third, the inverse discrete Fourier transform recovers the connection set $S$ of $G$, and Turner's theorem (1967) identifies $G$ up to isomorphism. As a consequence, graph isomorphism is decidable in polynomial time within this class using the quantum walk spectrum, without resorting to the general quasi-polynomial algorithm of Babai (2016).

• The paper shows that the quantum walk characteristic polynomial is a complete invariant for prime-order SRGs, uniquely distinguishing non-isomorphic graphs.
• It employs a Fourier block decomposition of the quantum walk operator to extract connection set data via a structured quadratic factor.
• The method yields a polynomial-time isomorphism test that outperforms classical spectral approaches, including Babai’s algorithm.

The Quantum Walk Characteristic Polynomial as a Complete Invariant for Strongly Regular Graphs of Prime Order

Introduction and Context

This work establishes that the quantum walk characteristic polynomial $\chi_q(G, \lambda) = \det(\lambda I - U_G)$, where $U_G$ is the coined quantum walk operator, forms a complete isomorphism invariant for all strongly regular graphs (SRGs) of prime order $p$ with degree $k \geq 6$. This resolves a central problem in algebraic graph theory concerning the effectiveness of spectral invariants for the graph isomorphism problem, a setting where the classical adjacency spectrum is known to fail on SRGs.

Vertex-transitive graphs of prime order are isomorphic to circulant graphs, and all non-trivial SRGs of prime order are captured in this family. Previous approaches for the isomorphism problem in SRGs—particularly in the regime of prime order—have not leveraged the refined perspective granted by quantum walks, for which the spectrum generally improves discriminatory power. The main contribution herein is a rigorous, complete characterization: for SRGs of the given class, $\chi_q$ distinguishes non-isomorphic graphs and, crucially, reconstructs the connection set $S$ up to isomorphism. Thus, the isomorphism problem that can otherwise require quasi-polynomial time via Babai’s algorithm now admits a polynomial-time solution within this regime using the quantum walk spectrum.

Technical Approach

Fourier Block Decomposition

Let $G = Cay(Z_p, S)$ be a strongly regular circulant graph. The quantum walk operator $U_G$ is defined on $\mathbb{C}^p \otimes \mathbb{C}^k$ by combining a Grover coin and a shift operator. The key structural insight is that $U_G$ admits a unitary block diagonalization under the discrete Fourier transform over $U_G$0, yielding $U_G$1 blocks $U_G$2 of size $U_G$3 each. This block structure is exploited to parametrize the quantum walk spectrum in terms of the graph’s arithmetic data.

Each block operator $U_G$4 takes the form $U_G$5, where $U_G$6 encodes the connection set and coin structure, and $U_G$7 is the Grover coin (reflection). One demonstrates that, for $U_G$8, $U_G$9 is diagonally decomposable such that all but two eigenvalues are $p$0, with the remaining two lying on the unit circle and their real part determined by the normalized Fourier coefficient $p$1.

Explicit Expression for the Block Characteristic Polynomial

A central lemma provides the explicit form:

$p$2

Here, $p$3 is the $p$4-th Fourier coefficient of the graph’s connection set indicator (with $p$5 a primitive $p$6-th root of unity). For prime $p$7, these coefficients are real due to the symmetry $p$8.

The proof involves linear algebra arguments about the action of $p$9 on two distinguished orthogonal vectors in $k \geq 6$0, showing that the quadratic term encodes the full information about $k \geq 6$1. The decomposition is feasible only for sufficiently large $k \geq 6$2, i.e., $k \geq 6$3.

Recovery of Graph Isomorphism Class

Given $k \geq 6$4, one extracts the coefficients of the degree-2 factors for each $k \geq 6$5 to obtain the full list of $k \geq 6$6. The entire connection set $k \geq 6$7 is reconstructed by direct inversion via the discrete Fourier transform. By a classical result (Turner's theorem), for prime $k \geq 6$8, the connection set uniquely identifies $k \geq 6$9 up to isomorphism (specifically, up to multiplication by units in $\chi_q$0).

Thus, the quantum walk spectrum not only distinguishes non-isomorphic prime-order SRGs, but also enables explicit reconstruction—an isomorphism test that is polynomial in $\chi_q$1.

Numerical Results

The paper numerically verifies these claims for Paley graphs with various prime orders. The explicit computation of the quantum walk characteristic polynomials, decomposition into irreducible factors, and inverse reconstruction of the connection sets aligns with the theoretical predictions, confirming the recoverability and discriminatory power on concrete instances.

Implications and Future Directions

This work demonstrates that quantum walk-based invariants, though typically associated with quantum information, yield powerful tools for classical combinatorics and algebraic graph theory. The construction achieves a separation over the classical adjacency spectrum for prime SRGs, which are long-standing hard instances for spectral methods.

From a computational standpoint, this provides a polynomial-time algorithm for isomorphism testing within this class, sidestepping the need for Babai's more general (and more complex) structure-theoretic machinery. On the quantum algorithmic side, the block diagonal structure suggests that such invariants may be accessible to quantum devices, opening the possibility of quantum speedup for graph isomorphism—or more general isomorphism-invariant problems—beyond current classical approaches.

The techniques are fundamentally reliant on the arithmetic of prime cyclic groups. Extension to composite orders or non-circulant SRGs is non-trivial; the proof depends crucially on unique factorization properties and the character group structure of $\chi_q$2. Whether quantum walk invariants can resolve isomorphism for larger families—especially for imprimitive or non-vertex-transitive SRGs—remains an open avenue. The question of whether quantum computation can achieve practical or even theoretical advantage for broader combinatorial isomorphism problems using these methods also persists as a substantive direction.

Conclusion

The quantum walk characteristic polynomial is shown to be a complete isomorphism invariant for all strongly regular graphs of prime order and degree at least six. This result establishes the supremacy of quantum walk-based spectral invariants over classical adjacency spectra for this notorious hard family in isomorphism testing, and supplies a constructive, polynomial-time isomorphism test grounded in Fourier analytic and linear algebraic techniques. Future research must resolve the extension to composite-order or more general SRG families and examine the practical utility of these invariants in classical and quantum algorithmics.