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Paper 1
Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources
Alejandro Borda, Julian Rincon, César Galindo
- Year
- 2025
- Journal
- arXiv preprint
- DOI
- arXiv:2512.20787
- arXiv
- 2512.20787
The Clifford group is efficiently classically simulable, and universality is obtained by supplementing it with non-Clifford resources. We determine which single-qudit gates suffice to achieve universality. We show that the structure of such resources is governed by the prime factorization of the qudit dimension $d$. Using the adjoint action on the space of complex trace-zero matrices, we relate density to irreducibility together with an infiniteness criterion, yielding a trichotomy based on the factorization of $d$. When $d$ is prime, any non-Clifford gate generates a dense subgroup of the determinant-one unitaries. If $d$ is a prime power, the adjoint action is reducible, and universality requires gates that couple the resulting invariant subspaces. For composite $d$ with pairwise coprime factors, generalized intra-qudit controlled-NOT gates connecting the factors already suffice. These findings suggest that ``composite architectures'' -- hybrid registers combining incommensurate dimensions -- offer a route to bypass the standard overhead associated with magic-state injection.
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Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems
Stavros Mouslopoulos
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2605.00952
- arXiv
- 2605.00952
In the ordered phase of a $\mathbb{Z}_2$-symmetric collective spin system, two natural bases -- localised pointer states $\{|P\rangle,|R\rangle\}$ and energy eigenstates $\{|E_0\rangle,|E_1\rangle\}$ -- yield Lindblad dephasing rates that differ by a factor approaching $2$ as $N\to\infty$ and reaching $2.42$ near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces $\langle E_i|\hat{J}_z|E_i\rangle=0$ exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: $η_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42$, where $m_*$ is the order parameter and $G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)$ (advantage over the classical mean-field estimate), and $η_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86$, where $J_{01}=\langle E_0|\hat{J}_z|E_1\rangle$ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the $\mathbb{Z}_2$ parity of the Lindblad jump operator.
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