On restricted unitary Cayley graphs and symplectic transformations modulo n

Abstract: We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n, whose vertex set is the ring Z_n, and where residues a, b modulo n are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of such graphs, we show an upper bound on the number of elementary operations (symplectic scalar multiplications, symplectic row swaps, and row additions or subtractions) required to decompose a symplectic matrix over Z_n. We also characterize the conditions on n for G_n to be a perfect graph.

• The paper provides a complete characterization of the tensor product structure and perfectness of quadratic unitary Cayley graphs.
• It establishes explicit diameter bounds and path enumerations in Gₙ using modular arithmetic and quadratic residue properties.
• The work applies these findings to reduce the symplectic decomposition complexity in Sp₂ₘ(ℤₙ) to O(m²) operations, independent of n.

Summary of "On restricted unitary Cayley graphs and symplectic transformations modulo n"

Introduction and Main Contributions

This paper provides a detailed investigation of quadratic unitary Cayley graphs $G_n$ defined over the additive group of integers modulo $n$, with adjacency determined by differences in the subset $T_n = \pm Q_n$, where $Q_n$ is the group of quadratic units modulo $n$. Central results include a precise characterization of the tensor product structure of these graphs, bounds on their diameters, a complete classification of their perfectness, and applications to the decomposition complexity of symplectic matrices over $\mathbb{Z}_n$. The work synthesizes combinatorial, algebraic, and number-theoretic techniques to clarify structural properties and operational complexity within symplectic groups.

Quadratic Unitary Cayley Graphs: Structure and Tensor Decomposition

The graph $G_n = \operatorname{Cay}(\mathbb{Z}_n, T_n)$ generalizes the Paley graphs when $n$ is an odd prime congruent to $1 \pmod{4}$. Unlike classical unitary Cayley graphs, $G_n$ does not, in general, decompose as a tensor (Kronecker) product over the prime-power factors of $n$ due to the interaction between quadratic units and $-1$ in different cyclic group factors.

A key theorem specifies that $G_M \otimes G_N \cong G_{MN}$ if and only if $-1$ is a quadratic residue in $M$ or $N$. This sharp criterion distinguishes when the tensor factorization aligns with the Chinese Remainder decomposition at the graph level: when violated, minimal tensor decomposability of $T_n$ fails, and $G_n$ displays more intricate inter-factor adjacency structures. The paper provides explicit decompositions for powers of $2$ and odd primes, leveraging Chinese Remainder techniques and modular arithmetic. These structural results allow for the reduction of complexity in further analysis to the prime power case, modulo considerations of $-1$ and quadratic residues.

Path Structures, Diameters, and Reachability

Analyzing walk structures and induced cycles, the paper leverages strong number-theoretic enumeration results for representations of elements as sums (and differences) of quadratic units. Building on classical residue and non-residue distribution theory (e.g., as in Aladov and Gauss), explicit formulae are derived for the number of paths of length two between vertices in $G_n$, both in the cases where $n$ is odd or even.

Upper bounds on the diameter $\operatorname{diam}(G_n)$ are established, and exact values are provided in all cases. For odd $n$, the diameter is at most $4$, precisely determined by factorization properties, and depends on the presence or absence of "problematic" prime factors such as $3$ or $5$. For even $n$, the diameter can reach $12$ in the worst case (e.g., $n$ divisible by $24$), with precise values shown for all residue classes and factorizations of $n$. These bounds are optimal and directly reflect the algebraic constraints on sums and differences of quadratic units in $\mathbb{Z}_n$.

The paper also defines the uniform diameter of the associated digraphs $\Gamma_n$ and shows that for $n$ coprime to $6$, the uniform diameter is constant (either 2, 3, or 4), providing a mechanism for bounding reachability in tensor-decomposed cases and for constructing specific walks with controlled signatures.

Perfectness and Induced Odd Cycles

A full classification of perfectness is established: $G_n$ is perfect if and only if $n$ is even or a power of a prime $p \equiv 3 \pmod{4}$. The argument proceeds via analysis of forbidden induced odd cycles (holes) and leverages the self-complementary property of Paley graphs and strong perfect graph theorems. For odd composite $n$ with at least one $p \equiv 1 \pmod{4}$ or two distinct $p \equiv 3 \pmod{4}$ factors, the construction of explicit odd holes is given using the tensor product structure and number-theoretic properties associated with quadratic residues, establishing the non-perfectness of all other cases.

Application: Decomposition of Symplectic Operators Modulo $n$

The algebraic analysis of $G_n$ underpins a strong complexity result regarding the decomposition of symplectic operators in $\operatorname{Sp}_{2m}(\mathbb{Z}_n)$. Building on the framework of Hostens et al. and leveraging tight diameter bounds, the paper shows that the Gaussian elimination-like reduction for symplectic matrices, customarily requiring $O(m^2 \log n)$ symplectic row operations (due to repeated applications of the Euclidean algorithm), can be improved to $O(m^2)$ operations—independent of $n$. This is accomplished by expressing required combinations of elementary symplectic transforms using paths in $G_n$ of minimal bounded length, thus simulating necessary operations in constant time per step. The proof relies on a careful construction using Bézout coefficients, and the decomposition construction is fully constructive, with all bounds explicit.

Claim: The number of necessary symplectic row operations for arbitrary $S \in \operatorname{Sp}_{2m}(\mathbb{Z}_n)$ is bounded above by $O(m^2)$, independent of $n$.

This result has direct consequences for the efficient synthesis of Clifford operations in quantum computation with qudits of composite dimension, among other applications in modular group symplectic computations.

Computational Hardness and Open Problems

A significant caveat arises from the intrinsic computational difficulty of quadratic residuosity testing in $\mathbb{Z}_n$; thus, adjacency in $G_n$ parallels the hardness of factoring $n$ (classically), and efficient path finding is deemed unlikely unless quantum resources are assumed (by analogy to Shor’s algorithm). Consequently, algorithmic tractability is limited with respect to certain combinatorial problems in these graphs—enumeration and explicit path finding are constrained to small parameters or fixed modulus cases, and efficient algorithms for shortest paths are considered cryptographically hard.

Several further research questions are highlighted:

• Explicit classification and enumeration of small odd holes (e.g., $5$-holes) in imperfect quadratic unitary Cayley graphs.
• Investigation of energy and spectral properties transferring from the well-studied Paley graphs ($n$ prime, $n \equiv 1 \pmod{4}$) to general $G_n$, especially their extremal behavior.

Conclusion

The paper provides a rigorous structural theory for quadratic unitary Cayley graphs and extends these to practical implications in the algebraic complexity of symplectic group decompositions modulo composite moduli. The results establish sharp bounds for fundamental properties (diameter, perfectness, decomposability), and constructively leverage arithmetic graph structures to obtain efficient algorithms within quantum and modular settings. The interplay of graph theoretic and number-theoretic phenomena is exhaustively mapped, yielding implications for both pure combinatorics and applied computational algebra.