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arXiv · 2512.20787

Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

Alejandro Borda·Julian Rincon·César Galindo
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Abstract

We show that single-qudit universality in Clifford-based gate sets follows a trichotomy determined by the prime factorization of the local dimension $d$. For prime $d$, any gate outside the Clifford group is universal. For prime-power dimensions $d=p^m$ with $m\ge 2$, not every non-Clifford gate is universal, but it can be achieved by suitable members of a family of diagonal phase gates, generalizing the qubit $T$ gate, as well as by permutations as simple as swapping $|0\rangle$ and $|1\rangle$ while leaving all other basis states unchanged. When $d$ decomposes into pairwise coprime prime powers, generalized CNOT-type gates between the corresponding factors already suffice for universality. In this composite case, universality can be obtained without introducing an explicit diagonal magic gate. Our results split non-Clifford resources for high-dimensional systems into two broad mechanisms: CNOT-type (permutations) or $T$-type (diagonal phases) gates.

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Alejandro Borda, Julian Rincon, César Galindo. 2025-12-23. Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources. https://arxiv.org/abs/2512.20787

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