[Paper Review] Quantum-inspired annealers as Boltzmann generators for machine learning and statistical physics

Alexander E. Ulanov, Egor Tiunov+1 more|arXiv (Cornell University)|Dec 18, 2019

Quantum Computing Algorithms and Architecture38 references4 citations

TL;DR

This paper introduces a quantum-inspired digital annealer, SimCIM, as a high-quality Boltzmann generator for sampling from complex, high-dimensional Ising distributions. It demonstrates successful training of a fully visible Boltzmann machine on MNIST digits using SimCIM samples, achieving 86.9% classification accuracy and effective partition function estimation, outperforming mean-field methods and offering a plug-and-play alternative to quantum annealers and deep learning models.

ABSTRACT

Quantum simulators and processors are rapidly improving nowadays, but they are still not able to solve complex and multidimensional tasks of practical value. However, certain numerical algorithms inspired by the physics of real quantum devices prove to be efficient in application to specific problems, related, for example, to combinatorial optimization. Here we implement a numerical annealer based on simulating the coherent Ising machine as a tool to sample from a high-dimensional Boltzmann probability distribution with the energy functional defined by the classical Ising Hamiltonian. Samples provided by such a generator are then utilized for the partition function estimation of this distribution and for the training of a general Boltzmann machine. Our study opens up a door to practical application of numerical quantum-inspired annealers.

Motivation & Objective

• To develop a practical, efficient, and noise-resilient method for generating unbiased samples from high-dimensional Boltzmann distributions.
• To apply these samples to training a general Boltzmann machine and estimating the intractable partition function in statistical physics.
• To overcome limitations of existing quantum annealers—such as parameter noise, restricted architectures, and decoherence—by using a classical simulation of the coherent Ising machine.
• To demonstrate that quantum-inspired numerical algorithms can serve as plug-and-play Boltzmann generators, outperforming mean-field approximations and enabling direct use in machine learning and physics applications.

Proposed method

• The study employs the SimCIM algorithm, a numerical simulation of the coherent Ising machine, to generate one-shot samples from the Boltzmann distribution defined by the classical Ising Hamiltonian.
• Samples from SimCIM are used to estimate the partition function via direct summation and annealed importance sampling (AIS), with comparisons to exact enumeration in low-dimensional cases.
• A fully visible, fully connected Boltzmann machine is trained using stochastic gradient ascent on the log-likelihood objective, with gradients computed from SimCIM-generated samples.
• Effective temperature adjustment is applied during training to improve convergence and performance, enhancing both log-likelihood and classification accuracy.
• The method leverages the ability of SimCIM to handle arbitrary real-valued coupling and bias matrices, enabling accurate modeling of complex energy landscapes.
• The approach avoids the need for iterative training like deep neural networks and sidesteps hardware limitations of physical quantum annealers such as D-Wave.

Experimental results

Research questions

• RQ1 Can a quantum-inspired numerical annealer like SimCIM generate high-quality, unbiased samples from high-dimensional Boltzmann distributions?
• RQ2 Can these samples effectively train a general Boltzmann machine for classification tasks, particularly on benchmark datasets like MNIST?
• RQ3 How accurately can the partition function be estimated using SimCIM-generated samples compared to exact or mean-field methods?
• RQ4 Does SimCIM outperform traditional mean-field approximations in capturing strong correlations in spin systems?
• RQ5 Can SimCIM serve as a plug-and-play Boltzmann generator that bypasses the hardware limitations of physical quantum annealers?

Key findings

• The SimCIM-based Boltzmann generator successfully trained a fully visible Boltzmann machine on the MNIST dataset, achieving a classification accuracy of 86.9%.
• The estimated partition function values from SimCIM samples showed strong agreement with exact values computed via exhaustive search in low-dimensional settings.
• The use of effective temperature adjustment during training significantly improved both the average log-likelihood and classification accuracy of the Boltzmann machine.
• SimCIM-generated samples enabled meaningful image reconstruction of handwritten digits, with visual quality improving over training epochs.
• The method outperformed mean-field approximations in capturing long-range spin correlations and provided a more accurate estimation of intractable statistical quantities.
• SimCIM offers a practical, noise-resilient, and hardware-agnostic alternative to physical quantum annealers, enabling direct application in machine learning and statistical physics without iterative training.