Quantum Boltzmann machine learning of ground-state energies

Abstract: Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an $\varepsilon$-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a novel quantum circuit construction that combines classical sampling, Hamiltonian simulation, and the Hadamard test, thus overcoming a key obstacle to quantum Boltzmann machine learning that has been left open since [Amin et al., Phys. Rev. X 8, 021050 (2018)]. Additionally supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.

• The paper introduces a novel hybrid algorithm employing QBMs to estimate ground-state energies while avoiding barren-plateau issues seen in traditional methods.
• It rigorously proves convergence to an ϵ-approximate stationary point with sample complexity polynomial in key parameters.
• A new quantum circuit design efficiently computes energy gradients through classical sampling, Hamiltonian simulation, and the Hadamard test.

Quantum Boltzmann Machine Learning of Ground-State Energies

The paper “Quantum Boltzmann Machine Learning of Ground-State Energies” presents a rigorous analysis and novel approach using Quantum Boltzmann Machines (QBMs) to estimate ground-state energies of Hamiltonians, a problem with significant implications in quantum physics and material sciences. The research addresses the limitations of traditional quantum algorithms and establishes the efficiency of QBMs in overcoming such challenges.

Key Contributions

1. Algorithm Development: The paper introduces a hybrid quantum-classical algorithm employing QBMs, which avoids the notorious barren-plateau problem often encountered in Variational Quantum Eigensolver (VQE) methods. Unlike VQE, which relies on parameterized quantum circuits, the QBM approach uses parameterized thermal states.
2. Theoretical Insights: The authors rigorously prove that their method converges to an $\varepsilon$-approximate stationary point of the energy function. This convergence is achieved with a sample complexity polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian.
3. Quantum Circuit Construction: They present a novel quantum circuit construction for efficiently estimating the gradient of the energy function. This involves a combination of classical sampling, Hamiltonian simulation, and the Hadamard test.
4. Analytical Calculations: The study includes comprehensive calculations of the gradient and Hessian of the energy function. These insights are crucial for understanding the landscape of the optimization problem.

Numerical Results and Claims

The paper substantiates its claims with numerical evidence, showing that QBMs do not suffer from barren-plateau problems. Although QBMs have mostly been explored for generative modeling or Hamiltonian learning, this research applies them to ground-state energy estimation, marking a pivotal advancement in their application.

Implications

The findings have profound implications for both theoretical and practical aspects:

• Theoretical Implications: The results provide an analytical framework that justifies the use of QBMs in more complex quantum optimization tasks, challenging long-standing complexity barriers related to preparing thermal states.
• Practical Implications: Practically, the work indicates a more resource-efficient pathway for quantum optimization problems, especially when dealing with Hamiltonians that are computationally challenging for classical algorithms.

Future Developments

The research opens avenues for exploring QBMs beyond ground-state energy estimation. This includes extending their application to other optimization and learning problems in quantum computing, such as semi-definite programming and constrained Hamiltonian optimization.

A critical open question remains regarding the absence of barren plateaus in all contexts involving QBMs, which is numerically evident but lacks comprehensive theoretical proof. Future work could focus on providing analytical guarantees in this domain, which would further establish QBMs as a robust tool in quantum computing.

Conclusion

The paper successfully challenges conventional approaches by utilizing Quantum Boltzmann Machines for efficient ground-state energy estimation, providing a theoretical foundation and practical framework for further exploration in the field of quantum computing. This represents a significant step toward broadening the applicability of quantum algorithms and enhancing their operational efficiency in the near term.