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.

翻译：变分量子算法是量子计算的重要组成部分，它融合了量子与经典方法以解决机器学习、化学和组合优化中的难题。然而随着算法规模扩大，这些算法无法避免贫瘠高原现象：系统扩展时梯度可能迅速消失，导致深度或随机初始化电路的训练几乎无法进行。为克服贫瘠高原问题，我们提出了一种两阶段优化框架。首先是凸初始化阶段：通过将量子能量景观（希尔伯特景观）塑造成平滑的低能态盆地，使梯度更易识别并避免噪声干扰过程。获得稳定梯度流后进入第二阶段——非凸优化阶段：允许算法探索不同能量极小值，从而增强模型表达能力。最后，我们应用该两阶段方案对量子密钥分发协议（即BB84）进行量子密码分析以确定最优克隆策略。仿真结果表明，所提出的两阶段方案显著优于随机初始化方法。