A tree-approach Pauli decomposition algorithm with application to quantum computing

• URL: http://arxiv.org/abs/2403.11644v1
• Date: Mon, 18 Mar 2024 10:38:06 GMT
• Title: A tree-approach Pauli decomposition algorithm with application to quantum computing
• Authors: Océane Koska, Marc Baboulin, Arnaud Gazda,
• Abstract summary: We propose an algorithm with a parallel implementation that optimize this decomposition using a tree approach. We also explain how some particular matrix structures can be exploited to reduce the number of operations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Pauli matrices are 2-by-2 matrices that are very useful in quantum computing. They can be used as elementary gates in quantum circuits but also to decompose any matrix of $\mathbb{C}^{2^n \times 2^n}$ as a linear combination of tensor products of the Pauli matrices. However, the computational cost of this decomposition is potentially very expensive since it can be exponential in $n$. In this paper, we propose an algorithm with a parallel implementation that optimizes this decomposition using a tree approach to avoid redundancy in the computation while using a limited memory footprint. We also explain how some particular matrix structures can be exploited to reduce the number of operations. We provide numerical experiments to evaluate the sequential and parallel performance of our decomposition algorithm and we illustrate how this algorithm can be applied to encode matrices in a quantum memory.

Related papers

• Pauli Decomposition via the Fast Walsh-Hadamard Transform [0.0]
  We present a new exact and explicit formula for the Pauli string coefficients. We show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. A numerical implementation of our equation outperforms currently available solutions.
  arXiv  Detail & Related papers  (2024-08-12T14:56:45Z)
• Decomposing dense matrices into dense Pauli tensors [0.0]
  We derive a fixed-memory, branchless algorithm to compute the inner product between a 2N-by-2N complex matrix and an N-term Pauli tensor in O(2N) time. Our scheme permits the embarrassingly parallel decomposition of a matrix into a weighted sum of Pauli strings in O(8N) time.
  arXiv  Detail & Related papers  (2024-01-29T18:18:11Z)
• 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)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra [3.4137115855910767]
  We propose a class of randomized quantum algorithms for the task of sampling from matrix functions. Our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures.
  arXiv  Detail & Related papers  (2023-02-03T17:22:49Z)
• 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)
• A Quantum Computer Amenable Sparse Matrix Equation Solver [0.0]
  We study problems involving the solution of matrix equations, for which there currently exists no efficient, general quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware.
  arXiv  Detail & Related papers  (2021-12-05T15:42:32Z)
• Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
  We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
  arXiv  Detail & Related papers  (2021-06-16T04:07:48Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Sketching Transformed Matrices with Applications to Natural Language Processing [76.6222695417524]
  We propose a space-efficient sketching algorithm for computing the product of a given small matrix with the transformed matrix. We show that our approach obtains small error and is efficient in both space and time.
  arXiv  Detail & Related papers  (2020-02-23T03:07:31Z)

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.