Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

Abstract: The efficient classical simulation of Clifford circuits constitutes a fundamental barrier to quantum advantage, typically overcome by injecting explicit non-Clifford "magic" resources. We demonstrate that for high-dimensional quantum systems (qudits), the resources required to break this barrier are strictly governed by the number-theoretic structure of the Hilbert space dimension $d$. By analyzing the adjoint action of the Clifford group, we establish a classification of single-qudit universality as a trichotomy. (I) For prime dimensions, the Clifford group is a maximal finite subgroup, and universality is robustly achieved by any non-Clifford gate. (II) For prime-power dimensions, the group structure fragments, requiring tailored diagonal non-Clifford gates to restore irreducibility. (III) Most notably, for composite dimensions with coprime factors, we demonstrate that standard entangling operations alone -- specifically, generalized intra-qudit CNOT gates -- generate the necessary non-Clifford resources to guarantee a dense subgroup of $\mathrm{SU}(d)$ without explicit diagonal magic injection. Our proofs rely on a new geometric criterion establishing that a subgroup with irreducible adjoint action is infinite if it contains a non-scalar element with projective distance strictly less than $1/2$ from the identity. These results establish that "coprime architectures" -- hybrid registers combining subsystems with coprime dimensions -- can sustain universal computation using only classical entangling operations, rendering the explicit injection of magic resources algebraically unnecessary.