Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

TL;DR

This paper classifies the conditions under which Clifford-based gate sets achieve universality in quantum systems based on the prime factorization of the local dimension, revealing a trichotomy in resource requirements.

Contribution

It introduces a comprehensive classification of quantum universality in composite systems, extending the understanding of non-Clifford resources beyond qubits.

Findings

For prime dimensions, any non-Clifford gate is universal.

In prime-power dimensions, specific diagonal phase gates and permutations suffice for universality.

In composite dimensions, CNOT-type gates between factors are enough for universality without additional magic gates.

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.