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Hamiltonian Monte Carlo (HMC) is an efficient Bayesian sampling method thatcan make distant proposals in the parameter space by simulating a Hamiltoniandynamical system. Despite its popularity in machine learning and data science,HMC is inefficient to sample from spiky and multimodal distributions. Motivatedby the energy-time uncertainty relation from quantum mechanics, we propose aQuantum-Inspired Hamiltonian Monte Carlo algorithm (QHMC). This algorithmallows a particle to have a random mass matrix with a probability distributionrather than a fixed mass. We prove the convergence property of QHMC and furthershow why such a random mass can improve the performance when we sample a broadclass of distributions. In order to handle the big training data sets inlarge-scale machine learning, we develop a stochastic gradient version of QHMCusing Nos{\'e}-Hoover thermostat called QSGNHT, and we also provide theoreticaljustifications about its steady-state distributions. Finally in theexperiments, we demonstrate the effectiveness of QHMC and QSGNHT on syntheticexamples, bridge regression, image denoising and neural network pruning. Theproposed QHMC and QSGNHT can indeed achieve much more stable and accuratesampling results on the test cases.

Quantum-Inspired Hamiltonian Monte Carlo for Bayesian Sampling