# Quantum-Inspired Hamiltonian Monte Carlo for Bayesian Sampling

arXiv: https://arxiv.org/abs/1912.01937 · HTML: https://mlsanity.com/paper/1912.01937/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2019.

## Built on by (most cited first, 4 of 4)

- [Particle-based Online Bayesian Sampling](https://mlsanity.com/paper/2302.14796/) (2023)
- [Antithetic Riemannian Manifold And Quantum-Inspired Hamiltonian Monte Carlo](https://mlsanity.com/paper/2107.02070/) (2021)
- [Repelling-Attracting Hamiltonian Monte Carlo](https://mlsanity.com/paper/2403.04607/) (2024)
- [Gaussian Process Regression with Soft Inequality and Monotonicity Constraints](https://mlsanity.com/paper/2404.02873/) (2024)

## Abstract (arXiv)

Hamiltonian Monte Carlo (HMC) is an efficient Bayesian sampling method that can make distant proposals in the parameter space by simulating a Hamiltonian dynamical system. Despite its popularity in machine learning and data science, HMC is inefficient to sample from spiky and multimodal distributions. Motivated by the energy-time uncertainty relation from quantum mechanics, we propose a Quantum-Inspired Hamiltonian Monte Carlo algorithm (QHMC). This algorithm allows a particle to have a random mass matrix with a probability distribution rather than a fixed mass. We prove the convergence property of QHMC and further show why such a random mass can improve the performance when we sample a broad class of distributions. In order to handle the big training data sets in large-scale machine learning, we develop a stochastic gradient version of QHMC using Nos'e-Hoover thermostat called QSGNHT, and we also provide theoretical justifications about its steady-state distributions. Finally in the experiments, we demonstrate the effectiveness of QHMC and QSGNHT on synthetic examples, bridge regression, image denoising and neural network pruning. The proposed QHMC and QSGNHT can indeed achieve much more stable and accurate sampling results on the test cases.