A general quantum matrix exponential dimensionality reduction framework based on block-encoding

• URL: http://arxiv.org/abs/2306.09606v1
• Date: Fri, 16 Jun 2023 03:36:03 GMT
• Title: A general quantum matrix exponential dimensionality reduction framework based on block-encoding
• Authors: Yong-Mei Li, Hai-Ling Liu, Shi-Jie Pan, Su-Juan Qin, Fei Gao, Qiao-Yan Wen
• Abstract summary: Matrix Exponential Dimensionality Reduction (MEDR) deals with the small-sample-size problem that appears in linear Dimensionality Reduction (DR) algorithms. High complexity is the bottleneck in this type of DR algorithm because one has to solve a large-scale matrix exponential eigenproblem. We design a general quantum algorithm framework for MEDR based on the block-encoding technique.
• Score: 4.501305807267216
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: As a general framework, Matrix Exponential Dimensionality Reduction (MEDR) deals with the small-sample-size problem that appears in linear Dimensionality Reduction (DR) algorithms. High complexity is the bottleneck in this type of DR algorithm because one has to solve a large-scale matrix exponential eigenproblem. To address it, here we design a general quantum algorithm framework for MEDR based on the block-encoding technique. This framework is configurable, that is, by selecting suitable methods to design the block-encodings of the data matrices, a series of new efficient quantum algorithms can be derived from this framework. Specifically, by constructing the block-encodings of the data matrix exponentials, we solve the eigenproblem and then obtain the digital-encoded quantum state corresponding to the compressed low-dimensional dataset, which can be directly utilized as input state for other quantum machine learning tasks to overcome the curse of dimensionality. As applications, we apply this framework to four linear DR algorithms and design their quantum algorithms, which all achieve a polynomial speedup in the dimension of the sample over their classical counterparts.

Related papers

• Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices [0.0]
  Solving differential equations is one of the most computationally expensive problems in classical computing. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable.
  arXiv  Detail & Related papers  (2024-10-07T17:44:30Z)
• 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)
• 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)
• A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph [4.045204834863644]
  We propose an efficient quantum algorithm to solve the eigenproblem of the Laplacian matrix of a fully connected weighted graph. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (non-symmetric) normalized Laplacian matrix.
  arXiv  Detail & Related papers  (2022-03-28T02:24:08Z)
• Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices [4.2389474761558406]
  We show how efficient quantum circuits can be explicitly constructed for some well-structured matrices. We also provide implementations of these quantum circuits in sparse strategies.
  arXiv  Detail & Related papers  (2022-03-19T03:50:16Z)
• Quantum algorithms for matrix operations and linear systems of equations [65.62256987706128]
  We propose quantum algorithms for matrix operations using the "Sender-Receiver" model. These quantum protocols can be used as subroutines in other quantum schemes.
  arXiv  Detail & Related papers  (2022-02-10T08:12:20Z)
• Adiabatic Quantum Graph Matching with Permutation Matrix Constraints [75.88678895180189]
  Matching problems on 3D shapes and images are frequently formulated as quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. We propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures.
  arXiv  Detail & Related papers  (2021-07-08T17:59:55Z)
• Quantum Dimensionality Reduction by Linear Discriminant Analysis [14.671957651032638]
  Dimensionality reduction (DR) of data is a crucial issue for many machine learning tasks, such as pattern recognition and data classification. We present a quantum algorithm and a quantum circuit to efficiently perform linear discriminant analysis (LDA) for dimensionality reduction.
  arXiv  Detail & Related papers  (2021-03-04T16:06:30Z)
• Joint Deep Reinforcement Learning and Unfolding: Beam Selection and Precoding for mmWave Multiuser MIMO with Lens Arrays [54.43962058166702]
  millimeter wave (mmWave) multiuser multiple-input multiple-output (MU-MIMO) systems with discrete lens arrays have received great attention. In this work, we investigate the joint design of a beam precoding matrix for mmWave MU-MIMO systems with DLA.
  arXiv  Detail & Related papers  (2021-01-05T03:55:04Z)
• 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)
• Block-encoding based quantum algorithm for linear systems with displacement structures [4.145426157018113]
  We present efficient and memory-reduced quantum algorithms for solving linear systems with displacement structures. The proposed block-encodings provide a quadratic speedup with respect to the dimension over classical algorithms. One of the quantum linear system solvers is applied to the linear prediction of time series.
  arXiv  Detail & Related papers  (2019-12-27T16:10:13Z)

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.