Quantum Dynamical Hamiltonian Monte Carlo

Abstract: One of the open challenges in quantum computing is to find meaningful and practical methods to leverage quantum computation to accelerate classical machine learning workflows. A ubiquitous problem in machine learning workflows is sampling from probability distributions that we only have access to via their log probability. To this end, we extend the well-known Hamiltonian Monte Carlo (HMC) method for Markov Chain Monte Carlo (MCMC) sampling to leverage quantum computation in a hybrid manner as a proposal function. Our new algorithm, Quantum Dynamical Hamiltonian Monte Carlo (QD-HMC), replaces the classical symplectic integration proposal step with simulations of quantum-coherent continuous-space dynamics on digital or analogue quantum computers. We show that QD-HMC maintains key characteristics of HMC, such as maintaining the detailed balanced condition with momentum inversion, while also having the potential for polynomial speedups over its classical counterpart in certain scenarios. As sampling is a core subroutine in many forms of probabilistic inference, and MCMC in continuously-parameterized spaces covers a large-class of potential applications, this work widens the areas of applicability of quantum devices.

• The paper introduces QD-HMC, a quantum-enhanced variant of HMC that uses quantum circuits to simulate coherent dynamics while preserving key MCMC properties.
• The paper demonstrates that QD-HMC achieves lower autocorrelation times and higher acceptance rates than classical HMC in low-temperature scenarios.
• The paper validates QD-HMC’s theory by maintaining detailed balance and energy conservation, and outlines future work on practical quantum hardware applications.

Quantum Dynamical Hamiltonian Monte Carlo: A Detailed Exploration

The paper "Quantum Dynamical Hamiltonian Monte Carlo" presents an innovative approach at the intersection of quantum computing and classical machine learning workflows, specifically in the context of Markov Chain Monte Carlo (MCMC) methods. The authors propose a novel algorithm, Quantum Dynamical Hamiltonian Monte Carlo (QD-HMC), which leverages quantum computations to improve upon the classical Hamiltonian Monte Carlo (HMC) method. This work aims to address a key challenge in quantum computing: finding meaningful and practical methods to utilize quantum computation to accelerate various workflows, particularly those involving complex sampling tasks integral to machine learning applications.

Overview of Hamiltonian Monte Carlo

Hamiltonian Monte Carlo is a widely used method in probabilistic inference, enabling efficient exploration of complex, high-dimensional probability distributions. By introducing auxiliary momentum variables, HMC exploits concepts from Hamiltonian dynamics to propose transitions in a state space that respects conservation laws, significantly improving sample efficiency over random walk-based approaches. The leapfrog integrator, commonly employed in HMC, simulates these dynamics by iteratively updating position and momentum variables, ensuring high acceptance rates of proposed samples due to the approximate conservation of the Hamiltonian.

Quantum Approach: Quantum Dynamical HMC

The proposed QD-HMC algorithm extends this classical approach by incorporating quantum dynamics, utilizing quantum computers to perform the dynamical simulation steps. This algorithm replaces the classical symplectic leapfrog integrator with quantum simulations of coherent continuous-space dynamics. The core component, a quantum circuit analogous to a variational Trotterization, facilitates the transformation of initial states into new proposals while ensuring the key HMC properties of volume preservation and reversibility are maintained.

The quantum integrated approach in QD-HMC introduces potential advantages in sampling efficiency, particularly in scenarios where classical simulations become computationally demanding. The authors posit that this method could provide polynomial speedups under certain conditions, particularly in low-temperature regimes where quantum dynamics can mimic a Grover-like search, enhancing the probability of escaping local minima more effectively than classical methods.

Theoretical Implications and Empirical Results

The paper demonstrates the theoretical foundations necessary for QD-HMC to function as a valid MCMC sampler. It effectively meets criteria such as detailed balance and energy conservation, crucial for ensuring convergence to the true target distribution in the asymptotic limit. The symmetry in the proposal distribution functions via quantum operations and the optional momentum flip ensure reversibility, akin to classical HMC's momentum reversal.

Empirically, QD-HMC is evaluated against a series of optimization benchmarks. The results illustrate comparable performance to classical HMC in high-temperature scenarios, with potential advantages manifesting in lower autocorrelation times and higher acceptance rates, particularly at lower temperatures. These advantages suggest that QD-HMC can maintain proposal acceptance probabilities independent of temperature, in contrast to classical HMC's reliance on finely tuned hyperparameters.

Future Directions and Challenges

The introduction of QD-HMC opens several avenues for future research, notably concerning the implementation on practical quantum hardware and further exploration of quantum advantage in specific sampling regimes. Key challenges include the development of adaptable quantum circuits that can efficiently handle problem-specific log-probability landscapes, ensuring scalability and performance consistency across a broader range of applications.

Further integration with variational quantum algorithms and exploration into parameterized quantum circuits could enhance the versatility of QD-HMC, potentially impacting fields such as reinforcement learning, complex statistical modeling, and beyond. Additionally, actual empirical deployment on both digital and analogue quantum processors will be critical in quantifying real-world impacts and the practical utility of this quantum-classical hybrid approach.

In conclusion, the QD-HMC algorithm represents a meaningful stride towards integrating quantum computational techniques within classical machine learning frameworks. By leveraging quantum dynamics, this approach not only extends the applicability of quantum devices in probabilistic inference but also sets a foundation for further exploration and optimization in the complex landscape of quantum-enhanced computational methods.