提出安全路径算法，解决量子变分算法训练难与表达力差的矛盾。
Adaptive H-EFT-VA: A Provably Safe Trajectory Through the Trainability-Expressibility Landscape of Variational Quantum Algorithms
- 通过自适应扩展希尔伯特空间，实现安全轨迹演化
- 在14量子比特上保真度达0.54，是静态方法的两倍
- 首次严格保证梯度方差下限，适合高精度量子模拟
H-EFT-VA通过层级有效场理论紫外截断解决了巴伦高原问题，确保梯度方差为Ω(1/多项式(N))。但局域性限制其仅能覆盖多项式子空间，对远离|0>^N的状态存在参考态间隙。本文提出自适应H-EFT-VA（A-H-EFT），通过安全轨迹拓展可到达的希尔伯特空间，以平衡训练能力与表达力。定理1表明：若σ(t) ≤ 0.5/√(LN)，则梯度方差保持在Ω(1/多项式(N))。安全扩展引理与单调增长引理证明扩张过程无突变。在16组实验（最大N=14）中，A-H-EFT实现保真度F=0.54，较静态H-EFT-VA的F=0.27提升一倍，显著优于HEA（F≈0.01），且全程梯度方差≥0.5。在Δ_ref=1的海森堡XXZ模型中，成功识别负基态，而静态方法失败。结果统计显著（p < 10^-37）。对三数量级超参数具有鲁棒性，无需调参即可部署。这是首个在变分量子算法景观中严格界定的可行路径。
原文摘要 · Abstract (English)
H-EFT-VA established a physics-informed solution to the Barren Plateau (BP) problem via a hierarchical EFT UV-cutoff, guaranteeing gradient variance in Omega(1/poly(N)). However, localization restricts the ansatz to a polynomial subspace, creating a reference-state gap for states distant from |0>^N. We introduce Adaptive H-EFT-VA (A-H-EFT) to navigate the trainability-expressibility tradeoff by expanding the reachable Hilbert space along a safe trajectory. Gradient variance is maintained in Omega(1/poly(N)) if sigma(t) <= 0.5/sqrt(LN) (Theorem 1). A Safe Expansion Corollary and Monotone Growth Lemma confirm expansion without discontinuous jumps. Benchmarking across 16 experiments (up to N=14) shows A-H-EFT achieves fidelity F=0.54, doubling static H-EFT-VA (F=0.27) and outperforming HEA (F~0.01), with gradient variance >= 0.5 throughout. For Heisenberg XXZ (Delta_ref=1), A-H-EFT identifies the negative ground state while static methods fail. Results are statistically significant (p < 10^-37). Robustness over three decades of hyperparameters enables deployment without search. This is the first rigorously bounded trajectory through the VQA landscape.
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