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1. Curvature Sets Over Persistence Diagrams

   https://doi.org/10.1007/s00454-024-00634-0  

   Gómez, Mario; Mémoli, Facundo (April 2024, Discrete & Computational Geometry)

   Abstract We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers$$k\ge 0$$ $k\ge 0$and$$n\ge 1$$ $n\ge 1$we consider the dimensionkVietoris–Rips persistence diagrams ofallsubsets of a given metric space with cardinality at mostn. We call these invariantspersistence setsand denote them as$${\textbf{D}}_{n,k}^{\textrm{VR}}$$ $D_{n,k}^{\text{VR}}$. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parametersnandk, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which$${\textbf{D}}_{4,1}^{\textrm{VR}}$$ $D_{4,1}^{\text{VR}}$fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a spaceXwith cardinality$$2k+2$$ $2k+2$with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. 

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2. The Gromov–Wasserstein Distance Between Spheres

   https://doi.org/10.1007/s10208-024-09678-3  

   Arya, Shreya; Auddy, Arnab; Clark, Ranthony A; Lim, Sunhyuk; Mémoli, Facundo; Packer, Daniel (September 2024, Foundations of Computational Mathematics)

   Abstract The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family$$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ ${\left\{d_{\text{GW}p,q}\right\}}_{p,q=1}^{\infty }$of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameterspandqand the metric of the underlying spaces, we are able to determine the exact value of the distance$$d_{{{\text {GW}}}4,2}$$ $d_{\text{GW}4,2}$between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure. 

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3. A substitutional quantum defect in WS2 discovered by high-throughput computational screening and fabricated by site-selective STM manipulation

   https://doi.org/10.1038/s41467-024-47876-3  

   Thomas, John C.; Chen, Wei; Xiong, Yihuang; Barker, Bradford A.; Zhou, Junze; Chen, Weiru; Rossi, Antonio; Kelly, Nolan; Yu, Zhuohang; Zhou, Da; et al (April 2024, Nature Communications)

   Abstract Point defects in two-dimensional materials are of key interest for quantum information science. However, the parameter space of possible defects is immense, making the identification of high-performance quantum defects very challenging. Here, we perform high-throughput (HT) first-principles computational screening to search for promising quantum defects within WS₂, which present localized levels in the band gap that can lead to bright optical transitions in the visible or telecom regime. Our computed database spans more than 700 charged defects formed through substitution on the tungsten or sulfur site. We found that sulfur substitutions enable the most promising quantum defects. We computationally identify the neutral cobalt substitution to sulfur (Co$${}_{{{{{{{{\rm{S}}}}}}}}}^{0}$$ ${}_S^0$) and fabricate it with scanning tunneling microscopy (STM). The Co$${}_{{{{{{{{\rm{S}}}}}}}}}^{0}$$ ${}_S^0$electronic structure measured by STM agrees with first principles and showcases an attractive quantum defect. Our work shows how HT computational screening and nanoscale synthesis routes can be combined to design promising quantum defects. 

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4. Persistent cup product structures and related invariants

   https://doi.org/10.1007/s41468-023-00138-5  

   Mémoli, Facundo; Stefanou, Anastasios; Zhou, Ling (October 2023, Journal of Applied and Computational Topology)

   Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant. 

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5. Isotope engineering for spin defects in van der Waals materials

   https://doi.org/10.1038/s41467-023-44494-3  

   Gong, Ruotian; Du, Xinyi; Janzen, Eli; Liu, Vincent; Liu, Zhongyuan; He, Guanghui; Ye, Bingtian; Li, Tongcang; Yao, Norman Y; Edgar, James H; et al (December 2024, Nature Communications)

   Abstract Spin defects in van der Waals materials offer a promising platform for advancing quantum technologies. Here, we propose and demonstrate a powerful technique based on isotope engineering of host materials to significantly enhance the coherence properties of embedded spin defects. Focusing on the recently-discovered negatively charged boron vacancy center ($${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$ $V_B^{-}$) in hexagonal boron nitride (hBN), we grow isotopically purified h¹⁰B¹⁵N crystals. Compared to$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$ $V_B^{-}$in hBN with the natural distribution of isotopes, we observe substantially narrower and less crowded$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$ $V_B^{-}$spin transitions as well as extended coherence timeT₂and relaxation timeT₁. For quantum sensing,$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$ $V_B^{-}$centers in our h¹⁰B¹⁵N samples exhibit a factor of 4 (2) enhancement in DC (AC) magnetic field sensitivity. For additional quantum resources, the individual addressability of the$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$ $V_B^{-}$hyperfine levels enables the dynamical polarization and coherent control of the three nearest-neighbor¹⁵N nuclear spins. Our results demonstrate the power of isotope engineering for enhancing the properties of quantum spin defects in hBN, and can be readily extended to improving spin qubits in a broad family of van der Waals materials. 

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