Evolved Quantum Boltzmann Machines Enable Efficient Generative Modeling of Complex Probability Distributions for Challenging Simulations

Born-rule generative modelling tackles the challenge of creating probability distributions that complex systems can efficiently sample, a crucial task with implications for fields ranging from materials science to machine learning. Mark M. Wilde from Cornell University addresses a longstanding obstacle in this field, demonstrating a practical method for training quantum Boltzmann machines for generative modelling. The research leverages the Donsker-Varadhan representation of relative entropy and a Boltzmann gradient estimator to achieve this, and extends the approach to a more general framework known as an evolved Boltzmann machine. This work not only provides a pathway to efficiently capture complex probability distributions, but also establishes theoretical convergence guarantees for several hybrid quantum-classical training algorithms, representing a significant advance in the development of quantum generative models.

Classical methods alone prove insufficient for this task, yet quantum Boltzmann machines were proposed approximately one decade ago as a potential solution, though efficient training methods have remained elusive. This paper overcomes this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modelling, extending to a more general model, known as an evolved quantum Boltzmann machine, which combines parameterized real and imaginary components. Two key ingredients underpin this proposal, namely a mathematical representation of relative entropy and a quantum Boltzmann gradient estimator detailed in previous work.

Quantum Gradient Estimation for Born-Rule Models

Scientists have achieved a breakthrough in training evolved quantum Boltzmann machines for Born-rule generative modeling, overcoming a longstanding challenge in quantum machine learning. The work proposes a practical solution by combining a quantum gradient estimator with a mathematical formula for analyzing relative entropy, enabling efficient training of these complex models. The core of the method lies in accurately calculating the gradients needed to optimize the model parameters, a process previously hindered by computational complexity. Researchers developed a quantum algorithm to estimate these gradients, leveraging quantum circuits designed to efficiently measure the necessary quantities.

Specifically, the algorithm estimates partial derivatives by preparing quantum states and measuring observables, utilizing efficient Hamiltonian simulation and thermal state preparation techniques. The team demonstrated that the algorithm can accurately calculate the gradients required for training, even when the model’s Hamiltonians are both Hermitian and unitary, incorporating standard quantum gates to facilitate these measurements with time values randomly selected to optimize the estimation process. Measurements confirm that the proposed method allows for the efficient estimation of all partial derivatives needed for training, a critical step in minimizing the difference between a target probability distribution and the model’s output. This breakthrough delivers a significant advancement in the field of quantum machine learning, paving the way for new applications in areas such as data generation and pattern recognition.

Efficient Training of Quantum Generative Models

This work presents a practical solution for training quantum Boltzmann machines and their more general form, evolved quantum Boltzmann machines, for Born-rule generative modeling, demonstrating how to efficiently train these models to accurately simulate probability distributions. The core achievement lies in combining a mathematical representation of relative entropy with a Boltzmann gradient estimator, enabling effective parameter tuning and accurate sampling by minimizing the difference between a target probability distribution and one generated by the quantum model. Furthermore, they developed and analyzed four distinct hybrid quantum-classical algorithms, extragradient, two-timescale gradient descent-ascent, follow-the-ridge, and HessianFR, to optimise the training process, providing theoretical convergence guarantees for each. While acknowledging the computational complexity inherent in quantum-classical hybrid algorithms, this research establishes a viable pathway for leveraging quantum computers in generative modeling tasks, with future work potentially focusing on exploring performance on larger datasets and investigating hardware implementations to enhance efficiency.