Quantum Algorithm for Estimating Betti Numbers Using a Cohomology Approach

• URL: http://arxiv.org/abs/2309.10800v2
• Date: Fri, 20 Oct 2023 12:32:08 GMT
• Title: Quantum Algorithm for Estimating Betti Numbers Using a Cohomology Approach
• Authors: Nhat A. Nghiem, Xianfeng David Gu and Tzu-Chieh Wei
• Abstract summary: Calculating Betti numbers classically is a daunting task due to the massive volume of data. We consider the dual' approach, which is inspired by Hodge theory and de Rham cohomology. Our algorithm can calculate its $r$-th Betti number $beta_r$ up to some multiplicative error.
• Score: 2.2000560351723504
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Topological data analysis has emerged as a powerful tool for analyzing large-scale data. High-dimensional data form an abstract simplicial complex, and by using tools from homology, topological features could be identified. Given a simplex, an important feature is so-called Betti numbers. Calculating Betti numbers classically is a daunting task due to the massive volume of data and its possible high-dimension. While most known quantum algorithms to estimate Betti numbers rely on homology, here we consider the `dual' approach, which is inspired by Hodge theory and de Rham cohomology, combined with recent advanced techniques in quantum algorithms. Our cohomology method offers a relatively simpler, yet more natural framework that requires exponentially less qubits, in comparison with the known homology-based quantum algorithms. Furthermore, our algorithm can calculate its $r$-th Betti number $\beta_r$ up to some multiplicative error $\delta$ with running time $\mathcal{O}\big( \log(c_r) c_r^2 / (c_r - \beta_r)^2 \delta^2 \big)$, where $c_r$ is the number of $r$-simplex. It thus works best when the $r$-th Betti number is considerably smaller than the number of the $r$-simplex in the given triangulated manifold.

Related papers

• Alternative Method for Estimating Betti Numbers [0.0]
  We provide an alternative method for estimating Betti numbers and normalized Betti numbers of given simplicial complex. Our method can be faster than the best-known classical method for finding Betti numbers. Our method could match the running time of best-known quantum method in the case of dense simplices.
  arXiv  Detail & Related papers  (2024-03-07T17:32:42Z)
• Gapped Clique Homology on weighted graphs is $\text{QMA}_1$-hard and contained in $\text{QMA}$ [0.0]
  We study the complexity of a classic problem in computational topology, the homology problem. We find that the complexity is characterized by quantum complexity classes. Our results can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics.
  arXiv  Detail & Related papers  (2023-11-28T21:15:30Z)
• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Higher-order topological kernels via quantum computation [68.8204255655161]
  Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. We propose a quantum approach to defining Betti kernels, which is based on constructing Betti curves with increasing order.
  arXiv  Detail & Related papers  (2023-07-14T14:48:52Z)
• An Incremental Span-Program-Based Algorithm and the Fine Print of Quantum Topological Data Analysis [1.2246649738388387]
  We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. Our algorithm works best when the complex has close to the maximal number of simplices.
  arXiv  Detail & Related papers  (2023-07-13T21:46:45Z)
• Private estimation algorithms for stochastic block models and mixture models [63.07482515700984]
  General tools for designing efficient private estimation algorithms. First efficient $(epsilon, delta)$-differentially private algorithm for both weak recovery and exact recovery.
  arXiv  Detail & Related papers  (2023-01-11T09:12:28Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Quantum Approximate Counting for Markov Chains and Application to Collision Counting [0.0]
  We show how to generalize the quantum approximate counting technique developed by Brassard, Hoyer and Tapp [ICALP 1998] to estimating the number of marked states of a Markov chain. This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha.
  arXiv  Detail & Related papers  (2022-04-06T03:04:42Z)
• Minimax Optimal Quantization of Linear Models: Information-Theoretic Limits and Efficient Algorithms [59.724977092582535]
  We consider the problem of quantizing a linear model learned from measurements. We derive an information-theoretic lower bound for the minimax risk under this setting. We show that our method and upper-bounds can be extended for two-layer ReLU neural networks.
  arXiv  Detail & Related papers  (2022-02-23T02:39:04Z)
• Quantum algorithm for persistent Betti numbers and topological data analysis [0.0]
  This paper shows the first quantum algorithm that can estimate the persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.
  arXiv  Detail & Related papers  (2021-10-31T09:02:01Z)
• Dist2Cycle: A Simplicial Neural Network for Homology Localization [66.15805004725809]
  Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes.
  arXiv  Detail & Related papers  (2021-10-28T14:59:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.