Overcoming Barren Plateaus in Variational Quantum Circuits using a Two-Step Least Squares Approach

Variational Quantum Algorithms are a vital part of quantum computing. It is a blend of quantum and classical methods for tackling tough problems in machine learning, chemistry, and combinatorial optimization. Yet as these algorithms scale up, they cannot escape the barren-plateau phenomenon. As systems grow, gradients can vanish so quickly that training deep or randomly initialized circuits becomes nearly impossible. To overcome the barren plateau problem, we introduce a two-stage optimization framework. First comes the convex initialization stage. Here, we shape the quantum energy landscape, the Hilmaton landscape, into a smooth, low-energy basin. This step makes gradients easier to spot and keeps noise from derailing the process. Once we have gotten a stable gradient flow, we move to the second stage: nonconvex refinement. In this phase, we allow the algorithm to explore different energy minima, thereby making the model more expressive. Finally, we used our two-stage solution to perform quantum cryptanalysis of the quantum key distribution protocol (i.e., BB84) to determine the optimal cloning strategies. The simulation results showed that our proposed two-stage solution outperforms its random initialization counterpart.