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Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms

François Le Gall

TL;DR

The paper tackles the robustness of quantum dequantization results for linear-algebra-based quantum machine learning under a weakened classical access model, introducing epsilon-approximate sampling-and-query access. It develops three core tools—robust inner-product estimation, robust matrix multiplication via approximate importance sketches, and robust rejection sampling—to translate oversampling-based techniques to the approximate setting. The authors prove that robust dequantization extends to low-rank QSVT, sparse QSVT, and a range of ML tasks (supervised clustering, recommendation systems, and low-rank inversion), with performance matching quantum algorithms up to polynomial factors under suitably small epsilon. This work strengthens the case that quantum advantages in the QRAM model may be fragile even when classical methods receive near-ideal, yet imperfect, access to input data. Overall, it provides a comprehensive framework showing that approximate sampling access suffices to replicate the performance of several quantum algorithms, highlighting limitations on exponential quantum speedups for ML tasks in practical data-access regimes.

Abstract

Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.

Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms

TL;DR

The paper tackles the robustness of quantum dequantization results for linear-algebra-based quantum machine learning under a weakened classical access model, introducing epsilon-approximate sampling-and-query access. It develops three core tools—robust inner-product estimation, robust matrix multiplication via approximate importance sketches, and robust rejection sampling—to translate oversampling-based techniques to the approximate setting. The authors prove that robust dequantization extends to low-rank QSVT, sparse QSVT, and a range of ML tasks (supervised clustering, recommendation systems, and low-rank inversion), with performance matching quantum algorithms up to polynomial factors under suitably small epsilon. This work strengthens the case that quantum advantages in the QRAM model may be fragile even when classical methods receive near-ideal, yet imperfect, access to input data. Overall, it provides a comprehensive framework showing that approximate sampling access suffices to replicate the performance of several quantum algorithms, highlighting limitations on exponential quantum speedups for ML tasks in practical data-access regimes.

Abstract

Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.
Paper Structure (47 sections, 26 theorems, 164 equations, 2 figures, 1 table)

This paper contains 47 sections, 26 theorems, 164 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation: Approximate sampling-and-query access
  4. Our results
  5. Approximate oversampling and closeness properties.
  6. Matrix multiplication.
  7. Applications.
  8. More recent quantum-inspired algorithms.
  9. Overview of our techniques
  10. Robust estimation of the inner product.
  11. Extension to robust estimation of matrix multiplication.
  12. Robust rejection sampling.
  13. Preliminaries
  14. Notations and general definitions
  15. General notations.
  16. ...and 32 more sections

Key Result

Theorem 1

Given two matrices $X\in \mathbb{C}^{m\times n}$ and $Y\in\mathbb{C}^{m\times n'}$, assume that we have $\varepsilon$-approximate sampling-and-query access to $X$, for any $\varepsilon\in[0,1]$. For any $\xi>0$ and any $\delta\in(0,1]$, we can construct efficiently a matrix sketch $\Sigma\in \mathbb

Figures (2)

  • Figure 1: Relation between the notions of sampling access to a vector defined in Section \ref{['sub:def1']}. A plain arrow shows a trivial generalization. The dotted arrow refers to the implementation proved in Proposition \ref{['prop:oversampling']} (which modifies the parameter $\varepsilon$).
  • Figure 2: Procedure $\mathsf{Sample}$.

Theorems & Definitions (61)

  • Theorem 1: Informal version
  • Theorem 2: Informal version
  • Theorem 3: Informal version
  • Lemma 1: Hoffman-Wielandt theorem HW53
  • Lemma 2: Corollary 2.3 in Gil10
  • Lemma 3: MacDiarmid's bounded difference inequality McDiarmid89
  • Lemma 4
  • proof
  • Definition 1
  • Definition 2
  • ...and 51 more