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Universal entanglement dynamics now possess precisely defined stationary locations, ‘corners’ existing at 2ⁿ⁻¹ discrete points on its governing landscape. Previously reliant upon case-by-case calculations, entangling power extrema are revealed by this work to be predictable through seven inherent invariants. These corners represent generalised reflections with properties independent of specific interactions or projectors.

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**Authors:** Berta Casas, Diego García-Martín, Yuxuan Zhang Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of $2n$ Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an $n$-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in $\mathcal{O}\left(n^2\right)$ time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to $n=80$ qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements $\times2.57$ and $\times2.28$ in the total number of gates, respectively.

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Previously, certifying sufficient signal propagation in quantum circuits required full computation; now, a classical check, assessing data’s ‘purity’ against a defined threshold, guarantees minimum output variance of order 1/n for off-diagonal circuits before they even run. This preprocessing step uses only $2n$ coordinates and a constant number of trials, offering potential gains as barren plateaus limit scalability in quantum machine learning. Independent uniform angles yield comparable variances to matchgate circuits too.

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"Abans de res que quedi clar: nosaltres estem a favor de l'escola inclusiva, però en aquestes condicions és impossible". Ho explica la Martina, tutora de 2n de primària d'una escola pública de la província de Barcelona. Han accedit a deixar que l'ARA passi un dia amb ells per comprovar com funciona l'escola inclusiva des de dintre. Sense filtres. Demanen, però, l'anonimat. Per això tots els noms d'aquest reportatge són ficticis.