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

TL;DR

This paper introduces a two-stage optimization method for variational quantum algorithms that effectively overcomes barren plateaus, improving training stability and performance in quantum machine learning and cryptanalysis.

Contribution

A novel two-step optimization framework combining convex initialization and nonconvex refinement to mitigate barren-plateau issues in variational quantum circuits.

Findings

The method reduces dependence on the condition number of the quantum least squares matrix.

Simulation results show improved performance over random initialization.

Successfully applied to quantum cryptanalysis of BB84 protocol.

Abstract

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 let the algorithm wander through different energy minima, making the model more expressive. We show that our proposed algorithm theoretically reduces the dependence on the condition number of the underlying quantum least squares approximate matrix via Riemannian manifold optimization. Finally, we used our two-stage solution to perform quantum cryptanalysis of 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.