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

TL;DR

The paper tackles the difficulty of training variational quantum algorithms due to barren plateaus by proposing a two-stage optimization: a convex initialization that stabilizes gradients in a local, well-behaved region of the Hilbert space, followed by a nonconvex refinement that enhances expressivity and exploration. The framework, inspired by two-stage least squares, provides convergence guarantees under explicit assumptions and applies to a quantum cloning application within BB84, achieving higher fidelity than random initializations. The key contributions include formal convergence analysis, a practical convex surrogate loss over local observables, and demonstration that a convex warm start can avoid exponential gradient decay while enabling effective nonconvex optimization. This approach improves trainability and scalability of VQAs, with potential impact on quantum machine learning and quantum cryptographic analyses in near-term devices.

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 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.

Figures (2)

• Figure 1: Average cloning fidelity vs Number of layers with a fixed qubit of 10.
• Figure 2: Average Fidelity vs iteration

Theorems & Definitions (14)

• Remark 1
• Theorem 1
• Corollary 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof