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Beyond Sparsity: Quantum Block Encoding for Dense Matrices via Hierarchically Low Rank Compression

Kun Tang, Jun Lai

TL;DR

This work extends quantum linear system methods to dense, kernel-generated matrices by leveraging hierarchically block separable (HBS) structure. It develops two quantum-friendly strategies: an extended sparsification that yields a sparse augmented system and a direct block-encoding scheme that recursively encodes the HBS factors using LCU and QMM. The authors provide rigorous complexity and error analyses for both approaches, including conditioning via regularization and a polynomial-in-N subnormalization bound for the direct encoding. Numerical experiments on 2D Helmholtz integral equations demonstrate linear scalability in preprocessing and effective compression, supporting potential quantum speedups for dense, structured problems. Overall, the results expand the applicability of quantum linear system solvers to a broad class of dense matrices arising in potential theory, covariance modeling, and computational physics.

Abstract

While quantum algorithms for solving large scale systems of linear equations offer potential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.

Beyond Sparsity: Quantum Block Encoding for Dense Matrices via Hierarchically Low Rank Compression

TL;DR

This work extends quantum linear system methods to dense, kernel-generated matrices by leveraging hierarchically block separable (HBS) structure. It develops two quantum-friendly strategies: an extended sparsification that yields a sparse augmented system and a direct block-encoding scheme that recursively encodes the HBS factors using LCU and QMM. The authors provide rigorous complexity and error analyses for both approaches, including conditioning via regularization and a polynomial-in-N subnormalization bound for the direct encoding. Numerical experiments on 2D Helmholtz integral equations demonstrate linear scalability in preprocessing and effective compression, supporting potential quantum speedups for dense, structured problems. Overall, the results expand the applicability of quantum linear system solvers to a broad class of dense matrices arising in potential theory, covariance modeling, and computational physics.

Abstract

While quantum algorithms for solving large scale systems of linear equations offer potential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.
Paper Structure (17 sections, 12 theorems, 53 equations, 9 figures)

This paper contains 17 sections, 12 theorems, 53 equations, 9 figures.

Table of Contents

  1. Introduction
  2. quantum linear system algorithms
  3. Hierarchically block separable matrices
  4. A Sparsification Based Quantum Approach
  5. A Direct Block Encoding Algorithm for Hierarchical Matrices
  6. Numerical experiments
  7. Conclusions
  8. Analysis of the Compression Algorithm
  9. Proxy Approximation
  10. Computational Complexity
  11. Element Bound
  12. Complexity and Error Analysis for Quantum Algorithm
  13. Analysis for the Sparsification Based Qauntum Algorithm
  14. Suppressing Auxiliary Variable Norms
  15. Control of Condition Number
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $A\in\mathbb{C}^{2^n\times 2^n}$ be a matrix that is $s_r$-row-sparse and $s_c$-column-sparse. Suppose we are given access to the following $(n+1)$-qubit oracles: where $r_{ik}$ is the column index of the $k$-th nonzero entry in the $i$-th row of $A$, and $c_{lj}$ is the row index of the $l$-th nonzero entry in the $j$-th column of $A$. If there are fewer than $k$ nonzero entries, the oracle

Figures (9)

  • Figure 1: This figure illustrates the algorithmic process of the recursive HBS factorization. At level 1, (a) the matrix $A$ is decomposed into its near-field blocks ($D_1$) and far-field remainder ($B_1$). (b) The far-field part is approximated as $B_1 \approx L_1 S_1 R_1$ using ID. (c) The process is recursively applied to $S_1$, and the indices of the resulting far-field part $B_2$ are reordered for the subsequent level.
  • Figure 2: An index tree for the interval $[0,1]$. The interval is first partitioned into $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$, and the two resulting nodes are then partitioned further. Each node at each level corresponds to an interval.
  • Figure 3: Construction of HBS Factors via Spatial Partitioning
  • Figure 4: Quantum circuit for LCU.
  • Figure 5: Quantum circuit for QMM.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Definition 2.1: QLSP
  • Definition 2.2: Block Encoding
  • Lemma 2.1: Block encoding nguyenBlockencodingDenseFullrank2022
  • Definition 3.1
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Definition 5.1: State preparation pair
  • Lemma 5.1: LCU gilyenQuantumSingularValue2019
  • Lemma 5.2: QMM gilyenQuantumSingularValue2019
  • ...and 7 more