Neural-network quantum states (NQS) are a leading variational tool for quantum many-body physics, yet their optimization is fragile whenever the ground state carries a non-trivial sign or complex phase structure, a situation generic to gauge fields, broken time-reversal symmetry, and fermionic statistics. We trace this fragility to the stochastic estimator of the phase gradient rather than to network expressiveness. The phase sector of the Monte Carlo energy gradient is a noisy score-function estimator; differentiating the local energy instead yields a direct estimator that is unbiased for the same phase force, has far lower variance, and requires only a separated amplitude--phase ansatz. Demonstrated on a 100-site flux ladder, a small network trained this way reaches $0.89\%$ median error, where tuned standard baselines plateau at $1.8\%$ and wider or deeper standard-gradient networks degrade from $8.4\%$ to $24.6\%$. The advantage carries over to chiral XXX chains: the direct estimator again converges to a markedly lower error than the standard one, across $α$ and size; it grows with flux and vanishes in zero-flux controls. An adaptive-mixture of the two estimators is provably never worse in variance than the better endpoint at the optimal mixing coefficient, with seed-resolved diagnostics tracing much of the gain to eliminating failed runs. Estimator design thus emerges as a first-class lever for complex-valued neural quantum states.

翻译：神经网络量子态（NQS）是量子多体物理中领先的变分工具，然而当基态携带非平凡符号或复杂相位结构时（这在规范场、时间反演对称性破缺及费米子统计中普遍存在），其优化过程十分脆弱。我们发现这种脆弱性源于相位梯度的随机估计量，而非网络表达能力。蒙特卡洛能量梯度中的相位扇区是一个含噪声的得分函数估计量；而对局域能量求导可得到一种直接估计量，该估计量对相同相位力无偏、方差显著更低，且仅需振幅-相位分离的假设形式。在100格点通量梯子模型上的实验表明，采用此方法训练的小型网络达到0.89%的中位误差，而经过调优的标准基线停滞于1.8%，更宽或更深的梯度标准网络误差从8.4%退化至24.6%。该优势延续至手性XXX链：直接估计量在不同α和系统尺寸下均收敛到显著低于标准方法的误差；该优势随通量增强，并在零通量对照实验中消失。两种估计量的自适应混合在最优混合系数下，其方差可证明从不劣于两者中更优的端点，而种子解析诊断表明大部分收益源于消除失败运行。因此，估计量设计成为复值神经量子态的一级杠杆。