A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

• URL: http://arxiv.org/abs/2209.12887v1
• Date: Mon, 26 Sep 2022 17:56:11 GMT
• Title: A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits
• Authors: Sam McArdle, Andr\'as Gily\'en, Mario Berta
• Abstract summary: We present an improved quantum algorithm for computing persistent Betti numbers. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest.
• Score: 5.478764356647437
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyse and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over rigorous state-of-the-art classical algorithms for calculating the persistent Betti numbers to constant additive error - the salient task for applications. However, this may be reduced to closer to quadratic when compared against heuristic classical methods and observed scalings. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence that this is the case.

Related papers

• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [63.733312560668274]
  Given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties? We prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. We devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to scaling in many practical settings.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms [0.0]
  We show how many techniques from randomized linear algebra can be adapted to work under this weaker assumption. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms.
  arXiv  Detail & Related papers  (2023-04-11T02:09:13Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis [59.545114016224254]
  Quantum algorithms for topological data analysis seem to provide an exponential advantage over the best classical approach. We show that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. We argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices.
  arXiv  Detail & Related papers  (2022-09-28T17:53:25Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Quantifying Grover speed-ups beyond asymptotic analysis [0.0]
  We consider an approach that combines classical emulation with detailed complexity bounds that include all constants. We apply our method to some simple quantum speedups of classical algorithms for solving the well-studied MAX-$k$-SAT optimization problem. This requires rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines.
  arXiv  Detail & Related papers  (2022-03-09T19:00:00Z)
• 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)
• Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
  We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression. We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
  arXiv  Detail & Related papers  (2020-11-09T01:13:07Z)
• Towards quantum advantage via topological data analysis [0.0]
  We study the quantum-algorithmic methods behind the algorithm for topological data analysis of Lloyd, Garnerone and Zanardi. We provide a number of new quantum algorithms for problems such as rank estimation and complex network analysis.
  arXiv  Detail & Related papers  (2020-05-06T06:31:24Z)
• Statistical Limits of Supervised Quantum Learning [90.0289160657379]
  We show that if the bound on the accuracy is taken into account, quantum machine learning algorithms for supervised learning cannot achieve polylogarithmic runtimes in the input dimension. We conclude that, when no further assumptions on the problem are made, quantum machine learning algorithms for supervised learning can have at most speedups over efficient classical algorithms.
  arXiv  Detail & Related papers  (2020-01-28T17:35:32Z)

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.