Block encoding of sparse matrices with a periodic diagonal structure

• URL: http://arxiv.org/abs/2602.10589v1
• Date: Wed, 11 Feb 2026 07:24:33 GMT
• Title: Block encoding of sparse matrices with a periodic diagonal structure
• Authors: Alessandro Andrea Zecchi, Claudio Sanavio, Luca Cappelli, Simona Perotto, Alessandro Roggero, Sauro Succi,
• Abstract summary: We provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure.<n>Various applications for the presented methodology are discussed in the context of solving differential problems.
• Score: 67.45502291821956
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.

Related papers

• On Encoding Matrices using Quantum Circuits [5.877573384886684]
  We study encodings matrices in the form of block encodings and state preparation circuits.<n>Two key results we establish are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form, and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits.
  arXiv  Detail & Related papers  (2025-10-22T21:20:08Z)
• Efficient Quantum Access Model for Sparse Structured Matrices using Linear Combination of Things [0.6138671548064355]
  We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices.<n>LCU is a foundational primitive in both variational and fault-tolerant quantum algorithms.<n>We introduce the Sigma basis, a compact set of simple, non-unitary operators that can better capture sparsity and structure.
  arXiv  Detail & Related papers  (2025-07-04T17:05:07Z)
• Matrix encoding method in variational quantum singular value decomposition [49.494595696663524]
  We propose the variational quantum singular value decomposition based on encoding the elements of the considered $Ntimes N$ matrix into the state of a quantum system of appropriate dimension.<n> Controlled measurement is involved to avoid small success in ancilla measurement.
  arXiv  Detail & Related papers  (2025-03-19T07:01:38Z)
• Efficient Variational Quantum Linear Solver for Structured Sparse Matrices [0.6138671548064355]
  We show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix. We employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions.
  arXiv  Detail & Related papers  (2024-04-25T19:22:05Z)
• Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems [43.53835128052666]
  We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an $(N-1)times (N-1)$ matrix.<n>The basic idea is to encode each row of the matrix into a pure state of some quantum system.
  arXiv  Detail & Related papers  (2024-01-29T23:23:27Z)
• An Efficient Algorithm for Clustered Multi-Task Compressive Sensing [60.70532293880842]
  Clustered multi-task compressive sensing is a hierarchical model that solves multiple compressive sensing tasks. The existing inference algorithm for this model is computationally expensive and does not scale well in high dimensions. We propose a new algorithm that substantially accelerates model inference by avoiding the need to explicitly compute these covariance matrices.
  arXiv  Detail & Related papers  (2023-09-30T15:57:14Z)
• Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization [43.74656748515853]
  Non-stationary multi-block bilevel optimization problems involve $mgg 1$ lower level problems and have important applications in machine learning. We aim to achieve three properties for our algorithm: a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ samples for each sampled block per-iteration; and (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator.
  arXiv  Detail & Related papers  (2023-05-30T04:10:11Z)
• Automatic and effective discovery of quantum kernels [41.61572387137452]
  Quantum computing can empower machine learning models by enabling kernel machines to leverage quantum kernels for representing similarity measures between data.<n>We present an approach to this problem, which employs optimization techniques, similar to those used in neural architecture search and AutoML.<n>The results obtained by testing our approach on a high-energy physics problem demonstrate that, in the best-case scenario, we can either match or improve testing accuracy with respect to the manual design approach.
  arXiv  Detail & Related papers  (2022-09-22T16:42:14Z)
• 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)
• High-Dimensional Sparse Bayesian Learning without Covariance Matrices [66.60078365202867]
  We introduce a new inference scheme that avoids explicit construction of the covariance matrix. Our approach couples a little-known diagonal estimation result from numerical linear algebra with the conjugate gradient algorithm. On several simulations, our method scales better than existing approaches in computation time and memory.
  arXiv  Detail & Related papers  (2022-02-25T16:35:26Z)
• 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)
• 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)

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.