# Quantum Regularized Least Squares

arXiv: https://arxiv.org/abs/2206.13143 · HTML: https://mlsanity.com/paper/2206.13143/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2022.

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

- [Towards efficient quantum algorithms for diffusion probabilistic models](https://mlsanity.com/paper/2502.14252/) (2025)
- [Quantum Next Generation Reservoir Computing: An Efficient Quantum Algorithm for Forecasting Quantum Dynamics](https://mlsanity.com/paper/2308.14239/) (2023)
- [Evaluating the Potential of Quantum Machine Learning in Cybersecurity: A Case-Study on PCA-based Intrusion Detection Systems](https://mlsanity.com/paper/2502.11173/) (2025)
- [Quantum Algorithms for the Pathwise Lasso](https://mlsanity.com/paper/2312.14141/) (2023)
- [Quantum Speedup for Spectral Approximation of Kronecker Products](https://mlsanity.com/paper/2402.07027/) (2024)
- [Accelerating Inference for Multilayer Neural Networks with Quantum Computers](https://mlsanity.com/paper/2510.07195/) (2025)
- [Quantum Sparse Recovery and Quantum Orthogonal Matching Pursuit](https://mlsanity.com/paper/2510.06925/) (2025)
- [Block-encodings as programming abstractions: The Eclipse Qrisp BlockEncoding Interface](https://mlsanity.com/paper/2604.18276/) (2026)

## Abstract (arXiv)

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general ℓ2-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result. To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.