# Quantum Machine Learning For Classical Data

arXiv: https://arxiv.org/abs/2105.03684 · HTML: https://mlsanity.com/paper/2105.03684/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 183 references that are in the map.
- First posted on arXiv 8 May 2021.

## Builds on (most cited first, 27 of 27)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Faster Kernel Ridge Regression Using Sketching and Preconditioning](https://mlsanity.com/paper/1611.03220/) (2016)
- [Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions](https://mlsanity.com/paper/1811.00414/) (2018)
- [Quantum Deep Learning](https://mlsanity.com/paper/1412.3489/) (2014)
- [Quantum SDP-Solvers: Better upper and lower bounds](https://mlsanity.com/paper/1705.01843/) (2017)
- [Reinforcement Learning Using Quantum Boltzmann Machines](https://mlsanity.com/paper/1612.05695/) (2016)
- [Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension](https://mlsanity.com/paper/1811.04909/) (2018)
- [Quantum-inspired algorithms in practice](https://mlsanity.com/paper/1905.10415/) (2019)
- [Quantum assisted Gaussian process regression](https://mlsanity.com/paper/1512.03929/) (2015)
- [Learning Thermodynamics with Boltzmann Machines](https://mlsanity.com/paper/1606.02718/) (2016)
- [Universal discriminative quantum neural networks](https://mlsanity.com/paper/1805.08654/) (2018)
- [Quantum linear systems algorithms: a primer](https://mlsanity.com/paper/1802.08227/) (2018)
- [Generative training of quantum Boltzmann machines with hidden units](https://mlsanity.com/paper/1905.09902/) (2019)
- [Quantum-inspired sublinear classical algorithms for solving low-rank linear systems](https://mlsanity.com/paper/1811.04852/) (2018)
- [Lecture Notes on Randomized Linear Algebra](https://mlsanity.com/paper/1608.04481/) (2016)
- [Learning DNFs under product distributions via μ-biased quantum Fourier sampling](https://mlsanity.com/paper/1802.05690/) (2018)
- [A quantum algorithm for simulating non-sparse Hamiltonians](https://mlsanity.com/paper/1803.08273/) (2018)
- [Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming](https://mlsanity.com/paper/1901.03254/) (2019)
- [Statistical Limits of Supervised Quantum Learning](https://mlsanity.com/paper/2001.10477/) (2020)
- [Weighted sampling of outer products](https://mlsanity.com/paper/1410.4429/) (2014)
- [Quantum Inspired Training for Boltzmann Machines](https://mlsanity.com/paper/1507.02642/) (2015)
- [Nonlinear regression based on a hybrid quantum computer](https://mlsanity.com/paper/1808.09607/) (2018)

## Built on by (most cited first, 2 of 2)

- [Hyperparameter Importance of Quantum Neural Networks Across Small Datasets](https://mlsanity.com/paper/2206.09992/) (2022)
- [QCA-MolGAN: Quantum Circuit Associative Molecular GAN with Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2509.05051/) (2025)

## Abstract (arXiv)

In this dissertation, we study the intersection of quantum computing and supervised machine learning algorithms, which means that we investigate quantum algorithms for supervised machine learning that operate on classical data. This area of research falls under the umbrella of quantum machine learning, a research area of computer science which has recently received wide attention. In particular, we investigate to what extent quantum computers can be used to accelerate supervised machine learning algorithms. The aim of this is to develop a clear understanding of the promises and limitations of the current state of the art of quantum algorithms for supervised machine learning, but also to define directions for future research in this exciting field. We start by looking at supervised quantum machine learning (QML) algorithms through the lens of statistical learning theory. In this framework, we derive novel bounds on the computational complexities of a large set of supervised QML algorithms under the requirement of optimal learning rates. Next, we give a new bound for Hamiltonian simulation of dense Hamiltonians, a major subroutine of most known supervised QML algorithms, and then derive a classical algorithm with nearly the same complexity. We then draw the parallels to recent "quantum-inspired" results, and will explain the implications of these results for quantum machine learning applications. Looking for areas which might bear larger advantages for QML algorithms, we finally propose a novel algorithm for Quantum Boltzmann machines, and argue that quantum algorithms for quantum data are one of the most promising applications for QML with potentially exponential advantage over classical approaches.