Gap sets for the spectra of cubic graphs

Kollár, Alicia J.; Sarnak, Peter

doi:10.1090/cams/3

Cited by 13 publications

(19 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Combining these numerical results with previous graphtheoretic ones examining large gaps in the spectra of regular graphs [47], we find a few general principles, which are indicated schematically in Fig. 6.…”

Section: B Gap Calculationssupporting

confidence: 74%

“…In order to gain understanding of the behavior of these two gaps, we undertook a study of a series of examples of free-fermion models which contain the solution to a set of exactly solvable spin models with coefficients all zero or one. In order to make use of previous graph theoretic results [46,47], we restrict our numerical simulations primarily to root graphs which are 3-regular as well as translation invariant and whose corresponding spinmodels have the anticommutation relations of 4-regular line graphs. We generate the root graphs directly using two methods: first, applying one-dimensional periodic boundary conditions on graphene to produce nanotubes, and second using the method of Abelian covers to stitch copies of a finite graph together to form a lattice.…”

Section: Heuristics and Numerical Study Of Example Latticesmentioning

confidence: 99%

“…A numerical search of all possible lattices is not possible, so attention was focused on canonical lattices such as graphene and the square lattice and two sets of examples known from Ref. [47] to give rise to large λ (τ ) 1 : first, one and two-dimensional square-lattice Abelian covers of small 3-regular graphs, and second, carbon nanotubes of small diameter.…”

Section: A Lattice Constructionmentioning

confidence: 99%

“…The remaining two edges may either be {1, x}, {x, y}, {x, x}, or {x, x}. This is a severely restricted set of covers; however, it was shown in previous work [47] that it contains examples which exhibit the largest possible gaps for symmetric adjacency matrices of arbitrary unoriented 3-regular graphs. The numerical studies described below show that it also contains examples with extremely large values of λ = 2 [47,51,52].…”

Section: A Lattice Constructionmentioning

confidence: 99%

“…is dictated by the singleparticle spectrum, and so a fair amount is known about its behavior from both mathematical studies of graph spectra [47,49] and also from the study of band structures in the solid state, e.g. [53].…”

Section: B Gap Calculationsmentioning

confidence: 99%

See 4 more Smart Citations

Free-Fermion Subsystem Codes

Chapman¹,

Flammia²,

Kollár³

2022

Preprint

View full text Add to dashboard Cite

We consider quantum error-correcting subsystem codes whose gauge generators realize a translationinvariant, free-fermion-solvable spin model. In this setting, errors are suppressed by a Hamiltonian whose terms are the gauge generators of the code and whose exact spectrum and eigenstates can be found via a generalized Jordan-Wigner transformation. Such solutions are characterized by the frustration graph of the Hamiltonian: the graph whose vertices are Hamiltonian terms, which are neighboring if the terms anticommute. We provide methods for embedding a given frustration graph in the anticommutation relations of a spin model and present the first known example of an exactly solvable spin model with a two-dimensional free-fermion description and exact topological qubits. This model can be viewed as a free-fermionized version of the two-dimensional Bacon-Shor code. Using graph-theoretic tools to study the unit cell, we give an efficient algorithm for deciding if a given translation-invariant spin model is solvable, and explicitly construct the solution. Further, we examine the energetics of these exactly solvable models from the graph-theoretic perspective and show that the relevant gaps of the spin model correspond to known graph-theoretic quantities: the skew energy and the median eigenvalue of an oriented graph. Finally, we numerically search for models which have large spectral gaps above the ground state spin configuration and thus exhibit particularly robust thermal suppression of errors. These results suggest that optimal models will have low dimensionality and odd coordination numbers, and that the primary limit to energetic error suppression is the skew energy difference between different symmetry sectors rather than single-particle excitations of the free fermions.

show abstract

Section: B Gap Calculationssupporting

confidence: 74%

Section: Heuristics and Numerical Study Of Example Latticesmentioning

confidence: 99%

Section: A Lattice Constructionmentioning

confidence: 99%

Section: A Lattice Constructionmentioning

confidence: 99%

Section: B Gap Calculationsmentioning

confidence: 99%

See 3 more Smart Citations

Free-Fermion Subsystem Codes

Chapman¹,

Flammia²,

Kollár³

2022

Preprint

View full text Add to dashboard Cite

show abstract

Petals and books: The largest Laplacian spectral gap from 1

Jost

Mulas

Zhang

2023

Journal of Graph Theory

View full text Add to dashboard Cite

We prove that, for any connected graph on vertices, the spectral gap from the value 1 with respect to the normalized Laplacian is at most 1/2. Moreover, we show that equality is achieved if and only if the graph is either a petal graph (for odd) or a book graph (for even). This implies that is a maximal gap interval for the normalized Laplacian on connected graphs. This is closely related to the Alon–Boppana bound on regular graphs and a recent result by Kollár and Sarnak on cubic graphs. Our result also provides a sharp bound for the convergence rate of some eigenvalues of the Laplacian on neighborhood graphs.

show abstract