Eigenvalue confinement and spectral gap for random simplicial complexes

Knowles, Antti; Rosenthal, Ron

doi:10.1002/rsa.20710

Cited by 13 publications

(15 citation statements)

References 62 publications

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“…• Spectrum and expansion: Similarly to graphs, one can define an adjacency and Laplacian operators for simplicial complexes, as well as different types of expansion properties. These were studied mainly in the the context of the random d-complex Y d (n, p) [48,[105][106][107].…”

Section: Other Directionsmentioning

confidence: 99%

Random Simplicial Complexes: Models and Phenomena

Bobrowski,

Krioukov

2021

Preprint

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We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

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Section: Other Directionsmentioning

confidence: 99%

Random Simplicial Complexes: Models and Phenomena

Bobrowski,

Krioukov

2021

Preprint

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“…However, we can identify many specific examples of simplicial complexes for which δ can be large. Indeed, several articles [43,44,45,46,47] have studied the spectra of the Laplacian of different simplicial complexes, including random simplicial complexes [48,49,50,51,52]. Some specific complexes: First, let us consider a few specific types of simplicial complexes.…”

Section: # Qubits # Gates Depth Timementioning

confidence: 99%

Quantum Topological Data Analysis with Linear Depth and Exponential Speedup

Ubaru,

Akhalwaya,

Squillante

et al. 2021

Preprint

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Quantum computing offers the potential of exponential speedups for certain classical computations. Over the last decade, many quantum machine learning (QML) algorithms have been proposed as candidates for such exponential improvements. However, two issues unravel the hope of exponential speedup for some of these QML algorithms: the data-loading problem and, more recently, the stunning "dequantization" results of Tang et al. A third issue, namely the fault-tolerance requirements of most QML algorithms, has further hindered their practical realization. The quantum topological data analysis (QTDA) algorithm of Lloyd, Garnerone and Zanardi was one of the first QML algorithms that convincingly offered an expected exponential speedup. From the outset, it did not suffer from the dataloading problem. A recent result has also shown that the generalized problem solved by this algorithm is likely classically intractable, and would therefore be immune to any dequantization efforts. However, the QTDA algorithm of Lloyd et al. has a time complexity of O(n 4 /(ǫ 2 δ √ ζ)) (where n is the number of data points, ǫ is the error tolerance, δ is the smallest nonzero eigenvalue of the restricted Laplacian, and ζ is the fraction of all simplices in the complex) and requires fault-tolerant quantum computing, which has not yet been achieved. In this paper, we completely overhaul the QTDA algorithm to achieve an improved exponential speedup and depth complexity of O(n log(1/(δǫ))). The latter depth complexity opens the door for an implementation on near-term quantum hardware, potentially making it the first useful algorithm to achieve quantum advantage on general classical data. Our approach includes three key innovations: (a) an efficient realization of the combinatorial Laplacian as a sum of Pauli operators; (b) a quantum rejection sampling and projection approach to restrict the superposition to the simplices of the desired order in the complex (replacing Grover's search of Lloyd et al.); and (c) a stochastic rank estimation method to estimate the Betti numbers (replacing quantum phase estimation of Lloyd et al.). We present a theoretical error analysis for the proposed algorithm, and present the circuit and computational time and depth complexities for Betti number estimation up to the error tolerance ǫ. The techniques presented herein have wider potential applications than QTDA or even rank estimation.

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“…n and D is sufficiently large. Knowles and Rosenthal [35] derived a more detailed result on the distribution of the spectrum when np(1 − p) ≫ log 4 n.…”

Section: 𝜆(Y) ≤ H(y)mentioning

confidence: 99%

Algebraic and combinatorial expansion in random simplicial complexes

Fountoulakis

Przykucki

2021

Random Struct Algorithms

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In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a d-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal, and Tessler. We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum co-degree of among all (𝑑 − 1)-faces. Furthermore, we consider a random walk on such a complex, which generalizes the standard random walk on a graph. We show that the associated conductance is with high probability bounded away from 0, resulting in a bound on the mixing time that is logarithmic in the number of vertices of the complex.

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Eigenvalue confinement and spectral gap for random simplicial complexes

Cited by 13 publications

References 62 publications

Random Simplicial Complexes: Models and Phenomena

Random Simplicial Complexes: Models and Phenomena

Quantum Topological Data Analysis with Linear Depth and Exponential Speedup

Algebraic and combinatorial expansion in random simplicial complexes

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