Quantum-enhanced Markov chain Monte Carlo (2203.12497v1)

Abstract: Sampling from complicated probability distributions is a hard computational problem arising in many fields, including statistical physics, optimization, and machine learning. Quantum computers have recently been used to sample from complicated distributions that are hard to sample from classically, but which seldom arise in applications. Here we introduce a quantum algorithm to sample from distributions that pose a bottleneck in several applications, which we implement on a superconducting quantum processor. The algorithm performs Markov chain Monte Carlo (MCMC), a popular iterative sampling technique, to sample from the Boltzmann distribution of classical Ising models. In each step, the quantum processor explores the model in superposition to propose a random move, which is then accepted or rejected by a classical computer and returned to the quantum processor, ensuring convergence to the desired Boltzmann distribution. We find that this quantum algorithm converges in fewer iterations than common classical MCMC alternatives on relevant problem instances, both in simulations and experiments. It therefore opens a new path for quantum computers to solve useful--not merely difficult--problems in the near term.

• The paper presents a novel quantum algorithm to enhance Markov chain Monte Carlo (MCMC) sampling from complex distributions, particularly in low-temperature Ising models.
• Combining quantum proposal generation with classical acceptance rules, the method demonstrates improved convergence rates numerically and on a quantum processor, showing potential speedup over classical methods.
• This quantum-enhanced MCMC offers practical potential for optimization and machine learning problems, highlighting the integration of quantum techniques into existing computational frameworks.

Overview of Quantum-Enhanced Markov Chain Monte Carlo

The paper "Quantum-enhanced Markov chain Monte Carlo" by Layden et al. presents a novel approach to improve the convergence rate of Markov chain Monte Carlo (MCMC) algorithms using quantum computing. Traditional MCMC approaches frequently encounter limitations when sampling from complex probability distributions, particularly in low-temperature Boltzmann distributions of classical Ising models. These distributions are crucial in various applications within statistical physics, machine learning, and combinatorial optimization, yet they pose significant computational challenges for classical algorithms.

Quantum Algorithm Details

The key contribution of this work is a quantum algorithm specifically designed to perform MCMC sampling from the Boltzmann distribution of classical Ising models, exploiting quantum superposition to enhance exploration of the energy landscape. The algorithm combines quantum computing to propose candidate moves and classical computing for the acceptance or rejection of these moves, based on the Metropolis-Hastings (M-H) framework. A quantum processor, through the application of unitary operators, generates proposals that are hard to efficiently replicate classically yet facilitate a rapid traversal of the energy landscape.

In this algorithm, the quantum step prepares a quantum state corresponding to the current configuration of spins, applies a unitary transformation, and measures the resulting state to propose a new configuration. The transformation $U = e^{-iHt}$ employed involves a Hamiltonian that incorporates the problem Hamiltonian for the Ising model and a mixing Hamiltonian to ensure exploration. Notably, the requirement for symmetric proposal probabilities is satisfied by the chosen unitaries, ensuring convergence to the target distribution.

Numerical and Experimental Results

The results indicate that the quantum algorithm offers a marked improvement in average-case convergence compared to classical counterparts, particularly in the low-temperature spin-glass regime where classical methods struggle. Through extensive numerical simulations, the authors demonstrate an exponential scaling improvement with problem size, indicating a polynomial enhancement in spectral gap—a key measure of convergence—to the quantum method versus classical strategies.

Furthermore, the experimental implementation of the algorithm on IBM’s superconducting quantum processor confirms this advantage, despite the presence of experimental noise. The authors report a significant quantum speedup in convergence, with the proposed algorithm largely overcoming the challenges associated with rugged energy landscapes typical of spin glasses.

Implications and Future Directions

The proposed quantum-enhanced MCMC algorithm opens new avenues for practical quantum computing applications, offering potential benefits in areas like optimization and machine learning where sampling complexity remains a bottleneck. This work suggests that quantum computers could soon solve not only intricate problems but also those of genuine practical value within reasonable computational constraints.

Looking forward, possible improvements in the algorithm could focus on optimizing the time evolution parameters and exploring different quantum Hamiltonian formulations. Additionally, integrating such quantum-accelerated MCMC techniques into larger computational frameworks, possibly synergizing with classical and variational approaches, could yield even broader applicability and utility.

The work also raises intriguing questions about the scalability and error resilience of the algorithm as quantum hardware continues to advance. Understanding how the observed quantum speedup scales with the problem size and hardware fidelity could shape future research on the utilization of quantum resources for probabilistic sampling tasks. Crucially, the integration of this quantum-accelerated sampling approach into current machine learning and optimization pipelines stands as a promising avenue for future exploration, with the potential to significantly enhance the computational capabilities for high-dimensional and complex problem spaces.