On the natural gradient for variational quantum eigensolver (1909.05074v1)

Abstract: The variational quantum eigensolver is a hybrid algorithm composed of quantum state driving and classical parameter optimization, for finding the ground state of a given Hamiltonian. The natural gradient method is an optimization method taking into account the geometric structure of the parameter space. Very recently, Stokes et al. developed the general method for employing the natural gradient for the variational quantum eigensolver. This paper gives some simple case-studies of this optimization method, to see in detail how the natural gradient optimizer makes use of the geometric property to change and improve the ordinary gradient method.

Exploring Optimization in Variational Quantum Eigensolvers Using Natural Gradient Methods

The paper "On the Natural Gradient for Variational Quantum Eigensolver" by Naoki Yamamoto investigates the application of natural gradient optimization in improving the performance of the Variational Quantum Eigensolver (VQE) for finding the ground state of quantum systems. By addressing how geometric properties of parameter spaces are utilized, this study provides critical insights into the efficiency of hybrid quantum-classical algorithms.

Core Concepts and Methodologies

The VQE, a prominent algorithm in the domain of quantum computing, leverages parametrized quantum circuits to approximate the ground state of Hamiltonians. It operates as a quantum-classical hybrid system, where quantum computations generate state vectors, and classical computation utilizes optimization techniques to adjust parameters iteratively. Traditionally, optimization within VQE employs methods like gradient descent. Nevertheless, this research emphasizes the potential of natural gradient descent, which incorporates the geometric nature of the parameter space via the Fubini-Study metric.

The natural gradient is applicable when optimizing complex systems with non-Euclidean parameter spaces. It dampens local minimums and plateaus in parameter landscapes through a tailored approach to parameter update, thereby sidestepping convergence issues prevalent in conventional methods. This approach contrasts with the ordinary gradient, which assumes a flat Euclidean space and often struggles around singular regions.

Numerical Experiments

The paper presents two key case studies to illustrate the advantages of using natural gradients within VQE frameworks: a single-qubit system and the problem of finding the ground state of the ${\rm H}_2$ molecule.

Single Qubit Case Study

For a single qubit, the natural gradient is shown to effectively navigate the parameter landscape, especially near singular points where parameters become indistinguishable under certain metrics. When compared with ordinary gradient descent, natural gradient methods demonstrate a superior capability to avoid local traps and optimize more efficiently, showcasing faster energy convergence in certain instances.

${\rm H}_2$ Molecule Case Study

In modeling a simplified ${\rm H}_2$ molecule, the Hamiltonian reveals complex entanglement properties when reduced to a two-qubit form. The research demonstrates that natural gradient techniques can more effectively handle the optimization paths, particularly across plateaus in energy landscapes of VQE iterations. The inherent ability of natural gradients to adapt dynamically to the local geometric structure allows for the efficient exploration of solution spaces, speeding up convergence towards the desired ground state.

Implications for Quantum Computing and Future Directions

The findings of this paper suggest that natural gradient methods provide a sophisticated optimization tool capable of enhancing the performance of VQE in finding the ground states of quantum systems. The inherent adaptability of these methods to the geometry of parameter spaces can lead to more efficient explorations of highly non-Euclidean landscapes typical in many quantum systems. However, care must be taken when singular points lie near target states, indicating a need for dynamic strategy adjustments in practical applications.

Future research could focus on comparing various metrics, such as classical Fisher information, and their impact on parameter space topology. Additionally, extending these insights to more complex quantum systems with broader applications, such as material science and chemistry, could further establish the utility of natural gradients in quantum optimization.

Conclusion

The exploration of natural gradients for VQE optimizations marks a valuable contribution to quantum computing methodologies, enhancing the efficiency and accuracy of hybrid optimization algorithms. By leveraging the geometric peculiarities of quantum parameter spaces, these techniques promise to advance computational strategies in quantum systems, thereby bolstering the potential of quantum computational approaches across various scientific domains.