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In this work, we examine the topological phases of the spring-mass lattices when the spatial inversion symmetry of the system is broken and prove the existence of edge modes when two lattices with different topological phases are glued together. In particular, for the one-dimensional lattice consisting of an infinite array of masses connected by springs, we show that the Zak phase of the lattice is quantized, only taking the value 0 or . We also prove the existence of an edge mode when two semi-infinite lattices with distinct Zak phases are connected. For the two-dimensional honeycomb lattice, we characterize the valley Chern numbers of the lattice when the masses on the lattice vertices are uneven. The existence of edge modes is proved for a joint honeycomb lattice formed by gluing two semi-infinite lattices with opposite valley Chern numbers together.
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A Gelfand duality for continuous lattices
We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to [0,1], to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of [0,1] fixing 1. We also prove an analogous duality for completely distributive lattices. Moreover, we prove that these are essentially the only well-behaved "sound classes of joins Φ, dual to a class of meets'' for which "Φ-continuous lattice" and "Φ-algebraic lattice" are different notions, thus for which a 2-valued duality does not suffice.
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- Award ID(s):
- 2224709
- PAR ID:
- 10511128
- Publisher / Repository:
- Theory and applications of categories
- Date Published:
- Journal Name:
- Theory and applications of categories
- ISSN:
- 1201-561X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach to battle noise. In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice—a 4-dimensional dense-packing lattice—to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.more » « less
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Abstract Let $$f:[0,1]^{d}\to{\mathbb{R}}$$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $$\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$$ have a non-negative local discrepancy (NNLD) everywhere in $$[0,1]^{d}$$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $$f$$ over $$[0,1]^{d}$$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $$b\geqslant 2$$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $$b\geqslant 2$$ and any dimension $$d\geqslant 1$$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $$d\geqslant 3$$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $$d\geqslant 2$$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.more » « less
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