Minimal instances for toric code ground states

A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and triangular lattice, such that the quasi-locality of the toric code hamiltonian becomes a distinctive feature. To this end, a purely geometric procedure to transform a given toric code setup into an LC-equivalent graph state is derived. In combination with an algorithmic computation of LC-equivalent graph states, we find the smallest non-trivial setup on the square lattice to contain 5 plaquettes and 16 qubits; on the triangular lattice the number of plaquettes and qubits is reduced to 4 and 9, respectively.Comment: 14 pages, 11 figure

Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.Comment: 102 pages, 17 figure

Locally finite graphs with ends: A topological approach, I. Basic theory

AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested

A geometric construction of isospectral magnetic graphs

We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number r of given length s (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph GG with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs (FFa)a∈N . A frame graph FFa is constructed contracting a copies of G along a subset of vertices V0 . In a second step, for any partition A=(a1,…,as) of length s of a natural number r (i.e., r=a1+⋯+as ) we construct a new graph FFA contracting now the frames FFa1,…,FFas selected by A along a proper subset of vertices V1⊂V0 . All the graphs obtained by different s-partitions of r≥4 (for any choice of V0 and V1 ) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given r and s for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block G and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices V0 and V1 with multiplicities determined by the numbers r and s of the partition.JSFC was supported by the Leverhulme Trust via a Research Project Grant (RPG-2020-158). FLl was supported by the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554) and from the Spanish National Research Council, through the Ayuda extraordinaria a Centros de Excelencia Severo Ochoa (20205CEX001) and by the Madrid Government under the Agreement with UC3M in the line of Research Funds for Beatriz Galindo Fellowships (C&QIG-BG-CM-UC3M), and in the context of the V PRICIT.Publicad

Metascientific aspects of topoi of spaces

This thesis presents a study of the importance of topoi for Science. It is argued that whenever the concept of space enters the practice of Science then formal (mathematical) theories should be interpreted in a topos of spaces. It is claimed that these topoi encode knowledge of space arising directly out of the needs of Science, in that the constitutive questions of the Sciences can be traced back to their leading knowledge interests and these determine the role of mathematics as a methodical device. In the Natural Sciences the constitutive questions involve the study of non-intentional objects, in terms of a causal nexus to be explained geometrically, and this facilitates the introduction of geometric objects as the methodical device for posing questions to Nature. Although the study of intentional subjects in the Human Sciences requires ordinary language, not mathematics, to pose questions to each other, secondary methodological objectifications permit a conception of geometric objects analogous to that of the Natural Sciences. Lawvere*s axioms for the gros and petit topoi illustrate attempts to formalise the idea of topoi of spaces, as a rational reconstruction of categories in which geometric objects satisfying the formal theories of Science can be found. The catalysing function of this knowledge of topoi of spaces can lead to a diagnosis of mathematical difficulties caused by a failure to align mathematical conceptions with these topoi. This is illustrated through Varela's use of self-reference in Biology and Atkin's use of algebraic topology in Social Studies