Equitable decompositions of graphs with symmetries

We present an algorithm to design networks that feature pretty good state transfer (PGST), which is of interest for high-fidelity transfer of information in quantum computing. Realizations of PGST networks have so far mostly relied either on very special network geometries or imposed conditions such as transcendental on-site potentials. However, it was recently shown [Eisenberg et al., arXiv:1804.01645] that PGST generally arises when a network's eigenvectors and the factors P± of its characteristic polynomial P fulfill certain conditions, where P± correspond to eigenvectors which have ±1 parity on the input and target sites. We combine this result with the so-called isospectral reduction of a network to obtain P± from a dimensionally reduced form of the Hamiltonian. Equipped with the knowledge of the factors P±, we show how a variety of setups can be equipped with PGST by proper tuning of P±. Having demonstrated a method of designing networks featuring pretty good state transfer of single site excitations, we further show how the obtained networks can be manipulated such that they allow for robust storage of qubits. We hereby rely on the concept of compact localized states, which are eigenstates of a Hamiltonian localized on a small subdomain, and whose amplitudes completely vanish outside of this domain. Such states are natural candidates for the storage of quantum information, and we show how certain Hamiltonians featuring pretty good state transfer of single site excitation can be equipped with compact localized states such that their transfer is made possible.

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“…By means of the 'equitable partition theorem' [26,46,47] its N + 4 eigenstates can then be shown to be can then be evaluated as…”

Section: Brief Conclusionmentioning

confidence: 99%

Designing pretty good state transfer via isospectral reductions

et al. 2020

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“…Here quotient graphs will be formed with respect to an equitable partition. The definition given below uses the approach of Barrett, Francis and Webb [2] to extend the definition of an equitable partition from the familiar setting of simple graphs to the more general setting of weighted directed graphs. We also follow the thorough treatment of the simple graph case in Chapter 5 of Godsil [7].…”

Section: Figures 3 and 4 Show Two Examples Of Orbigraphsmentioning

confidence: 99%

Orbigraphs: a graph-theoretic analog to Riemannian orbifolds

Gavin¹,

Ochoa

et al. 2019

Involve

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A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on R n modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have examined the link between the Laplace spectrum of an orbifold and the singularities of the orbifold. One open question in this field is whether or not a singular orbifold and a manifold can be Laplace isospectral. Motivated by the connection between spectral geometry and spectral graph theory, we define a graph theoretic analogue of an orbifold called an orbigraph. We obtain results about the relationship between an orbigraph and the spectrum of its adjacency matrix. We prove that the number of singular vertices arXiv:1901.03743v1 [math.CO]

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“…In [1] both equitable partitions and equitable decompositions are defined for matrices beyond the adjacency matrix of a graph, including the various Laplacian matrices, distance matrices, etc. (see Proposition 3.4).…”

Section: Introductionmentioning

confidence: 99%

“…The particular types of automorphisms for which equitable decompositions are defined are the so-called uniform, basic, and separable automorphisms. A uniform automorphism φ is an automorphism in which all orbits of the automorphism have the same cardinality (see [1]). A basic automorphism φ is an automorphism for which all orbits of size greater than one have the same cardinality (see [1]).…”

Section: Introductionmentioning

confidence: 99%

“…A uniform automorphism φ is an automorphism in which all orbits of the automorphism have the same cardinality (see [1]). A basic automorphism φ is an automorphism for which all orbits of size greater than one have the same cardinality (see [1]). A separable automorphism is an automorphism whose order is the product of distinct primes (see [2]).…”

Section: Introductionmentioning

confidence: 99%

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General equitable decompositions for graphs with symmetries

Francis

Smith

Webb

2019

Linear Algebra and its Applications

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Using the theory of equitable decompositions it is possible to decompose a matrix M appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M.

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Equitable decompositions of graphs with symmetries

Cited by 30 publications

References 8 publications

Designing pretty good state transfer via isospectral reductions

Designing pretty good state transfer via isospectral reductions

Orbigraphs: a graph-theoretic analog to Riemannian orbifolds

General equitable decompositions for graphs with symmetries

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