Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

• URL: http://arxiv.org/abs/2512.20787v1
• Date: Tue, 23 Dec 2025 21:34:41 GMT
• Title: Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources
• Authors: Alejandro Borda, Julian Rincon, César Galindo,
• Abstract summary: We show that for high-dimensional quantum systems, the resources required to break the barrier are strictly governed by the Hilbert space dimension $d$.<n>For prime dimensions, the Clifford group is a maximal finite subgroup, and is robustly achieved by any non-Clifford gate.<n>For composite dimensions with coprime factors, we demonstrate that standard entangling operations alone generate the necessary non-Clifford resources.
• Score: 41.99844472131922
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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