# Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra

arXiv: https://arxiv.org/abs/2011.04125 · HTML: https://mlsanity.com/paper/2011.04125/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2020.

## Built on by (most cited first, 8 of 8)

- [Low-Rank Approximation with 1/ε1/3 Matrix-Vector Products](https://mlsanity.com/paper/2202.05120/) (2022)
- [Quantum Topological Data Analysis with Linear Depth and Exponential Speedup](https://mlsanity.com/paper/2108.02811/) (2021)
- [Topological data analysis on noisy quantum computers](https://mlsanity.com/paper/2209.09371/) (2022)
- [Quantum algorithms for SVD-based data representation and analysis](https://mlsanity.com/paper/2104.08987/) (2021)
- [Krylov Methods are (nearly) Optimal for Low-Rank Approximation](https://mlsanity.com/paper/2304.03191/) (2023)
- [Multivariate trace estimation using quantum state space linear algebra](https://mlsanity.com/paper/2405.01098/) (2024)
- [Scalable Community Detection Using Quantum Hamiltonian Descent and QUBO Formulation](https://mlsanity.com/paper/2411.14696/) (2024)
- [Fast Length-Squared Sampling for Positive-Semidefinite Matrices](https://mlsanity.com/paper/2608.12503/) (2026)

## Abstract (arXiv)

We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of attention in recent years; we obtain sharper bounds for these problems. More significantly, we achieve these improvements by arguing that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise; these are powerful and standard techniques in randomized numerical linear algebra. With this recognition, we are able to employ the large body of work in numerical linear algebra to obtain algorithms for these problems that are simpler or faster (or both) than existing approaches. Our experiments demonstrate that the proposed data structures also work well on real-world datasets.