Qirui Ma | Search arXiv

arXiv2026

Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters

For positive integers $m,n$, let \[ A_{m,n}=(\{1,\ldots,m\}\times\Z_n)\cup\{0\} \] have flat addition and multiplication truncated at degree $m$. We prove that $A_{m,n}$ is finitely based exactly when $m\le 2$ or $(m,n)=(3,1)$. Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.

math.GR↗

arXiv2023

Design principles for >90% efficiency and >99% indistinguishability broadband quantum dot cavities

Quantum dots have the potential to be the brightest deterministic single photon source with plausible high end applications in quantum computing and cluster state generation. In this work, we re-examine the design of simple micropillars by meticulously examining the structural effects of the decay into leaky channels beyond the atom-like cavity estimation. We show that precise control of the side losses with the diameter and avoidance of propagating Bloch modes in the DBR structure can result in easy to manufacture broadband (Q$\approx750-2500$) micropillars and demonstrate extremely high internal efficiency ($90.5\%-96.4\%$). We also demonstrate that such cavities naturally decouple from the phonon sideband, with the phonon sideband reducing by a factor of $5-33$ allowing us to predict that the photons should show $99.2\%-99.8\%$ indistinguishability.

quant-ph↗