An Incremental Span-Program-Based Algorithm and the Fine Print of Quantum Topological Data Analysis

• URL: http://arxiv.org/abs/2307.07073v1
• Date: Thu, 13 Jul 2023 21:46:45 GMT
• Title: An Incremental Span-Program-Based Algorithm and the Fine Print of Quantum Topological Data Analysis
• Authors: Mitchell Black and William Maxwell and Amir Nayyeri
• Abstract summary: 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.
• Score: 1.2246649738388387
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove upper and lower bounds on the effective resistance and capacitance, showing both quantities can be exponentially large with respect to the size of the complex, implying that our algorithm would have to run for exponential time to exactly compute Betti numbers. However, as a corollary to these bounds, we show that the spectral gap of the combinatorial Laplacian can be exponentially small. As the runtime of all previous quantum algorithms for computing Betti numbers are parameterized by the inverse of the spectral gap, our bounds show that all quantum algorithms for computing Betti numbers must run for exponentially long to exactly compute Betti numbers. Finally, we prove some novel formulas for effective resistance and effective capacitance to give intuition for these quantities.

Related papers

• Generalized quantum Arimoto-Blahut algorithm and its application to quantum information bottleneck [55.22418739014892]
  We generalize the quantum Arimoto-Blahut algorithm by Ramakrishnan et al. We apply our algorithm to the quantum information bottleneck with three quantum systems. Our numerical analysis shows that our algorithm is better than their algorithm.
  arXiv  Detail & Related papers  (2023-11-19T00:06:11Z)
• A simple quantum algorithm to efficiently prepare sparse states [0.0]
  We show that the gate complexity is linear in the number of non-zero amplitudes in the state and quadratic in the number of qubits. This is competitive with the best known algorithms for sparse state preparation.
  arXiv  Detail & Related papers  (2023-10-30T07:05:15Z)
• Quantum Multiplication Algorithm Based on the Convolution Theorem [0.0]
  We propose a quantum algorithm for integer multiplication with time complexity $O(sqrtnlog2 n)$. Unlike the Harvey algorithm, our algorithm does not have the restriction of being applicable solely to extremely large numbers. The paper also reviews the history and development of classical multiplication algorithms and motivates us to explore how quantum resources can provide new perspectives and possibilities for this fundamental problem.
  arXiv  Detail & Related papers  (2023-06-14T12:40:54Z)
• 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)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• 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)
• A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits [5.478764356647437]
  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.
  arXiv  Detail & Related papers  (2022-09-26T17:56:11Z)
• 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)
• Pauli String Partitioning Algorithm with the Ising Model for Simultaneous Measurement [0.0]
  We propose an efficient algorithm for partitioning Pauli strings into subgroups, which can be simultaneously measured in a single quantum circuit. Our partitioning algorithm drastically reduces the total number of measurements in a variational quantum eigensolver for a quantum chemistry.
  arXiv  Detail & Related papers  (2022-05-09T01:49:21Z)
• Improved Quantum Algorithms for Fidelity Estimation [77.34726150561087]
  We develop new and efficient quantum algorithms for fidelity estimation with provable performance guarantees. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation. We prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general.
  arXiv  Detail & Related papers  (2022-03-30T02:02:16Z)
• 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)

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.