Warm-Starting the VQE with Approximate Complex Amplitude Encoding

The Variational Quantum Eigensolver (VQE) is a Variational Quantum Algorithm (VQA) to determine the ground state of quantum-mechanical systems. As a VQA, it makes use of a classical computer to optimize parameter values for its quantum circuit. However, each iteration of the VQE requires a multitude of measurements, and the optimization is subject to obstructions, such as barren plateaus, local minima, and subsequently slow convergence. We propose a warm-starting technique, that utilizes an approximation to generate beneficial initial parameter values for the VQE aiming to mitigate these effects. The warm-start is based on Approximate Complex Amplitude Encoding, a VQA using fidelity estimations from classical shadows to encode complex amplitude vectors into quantum states. Such warm-starts open the path to fruitful combinations of classical approximation algorithms and quantum algorithms. In particular, the evaluation of our approach shows that the warm-started VQE reaches higher quality solutions earlier than the original VQE.