Lie-Algebraic Subspace Quantization for Zero-Shot Quantum Learning and Barren-Plateau Mitigation

Research Approach

This paper addresses the trainability bottlenecks in parameterized quantum circuits (PQCs), specifically the barren plateau phenomenon where gradients vanish exponentially with system size. The authors propose a zero-shot, analytical parameter transfer map that converts classical neural network weights into quantum evolutions. This process involves three main steps: extracting a low-dimensional subspace using singular value decomposition (SVD), projecting the retained block onto the nearest unitary manifold via polar decomposition, and mapping the resulting unitary to a Hermitian generator using the principal matrix logarithm.

Key Findings

• Analytical Warm-Start: By initializing quantum circuits with generators derived from identity-centered classical architectures (like residual networks), the system avoids the Haar-random regime that typically triggers barren plateaus. This ensures that initial gradients are non-zero and scale polynomially with the subspace dimension k, rather than exponentially with the total number of qubits.
• Subspace Quantization Theorem: The authors provide a rigorous error bound for this mapping, showing that reconstruction error is governed by geometric truncation and the deviation of the classical weight from the unitary manifold. In the near-unitary regime, these errors are shown to be second-order.
• Hardware Validation: Experiments on the 156-qubit IBM Heron processor (ibm_kobe) demonstrate that the compiled circuits achieve high Hellinger fidelity (0.987 at k=8) and maintain resolvable gradients up to 128 physical qubits, significantly outperforming standard hardware-efficient ansatzes.

This work provides a bridge between classical deep learning and quantum hardware, enabling the deployment of complex quantum models without the prohibitive cost of iterative gradient-based optimization on noisy quantum devices. By treating the Lie-algebraic representation as a structural communication protocol, the framework allows for the efficient synthesis of quantum models through purely classical algebraic manipulation, offering a scalable path for quantum machine learning.