双曲受限玻尔兹曼机神经量子态
Hyperbolic Restricted Boltzmann Machine Neural Quantum State
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中文总结 AI 辅助
本文提出双曲受限玻尔兹曼机神经量子态，在量子SK模型中优于欧几里得版本，更精确重构纠缠谱，证明双曲几何更适于表示体积律量子系统。
中文摘要 AI 辅助
我们构建了第一种非欧几里得非自回归神经量子态（NQS），其形式为双曲受限玻尔兹曼机（HRBM），并在量子谢林顿-柯克帕特里克（QSK）模型的变分蒙特卡洛（VMC）设置中进行了研究，该模型的基态表现出体积律纠缠。在希尔伯特空间维度增加512倍（对应系统规模从$N=14$增加到$N=24$）的情况下，HRBM NQS在基态能量优化方面以及更低的Renyi-2 $S_2$和von Neumann $S_{vN}$绝对纠缠熵重构误差方面，稳健地优于其欧几里得版本RBM NQS。更重要的是，对于所有QSK系统规模，HRBM NQS在忠实再现QSK模型整个纠缠谱方面表现出优越的表达能力，从顶部特征值一直到跨越15个数量级的尾部，而RBM NQS始终高估次主导模式。这项工作提供了一个概念验证，表明双曲非自回归NQS拟设，由于其构造所依据的双曲几何的指数体积，可能比传统欧几里得NQS更自然地表示体积律量子系统。此外，这项工作的一个有趣副产品是，在QSK体积律系统中，随着希尔伯特空间指数增长，RBM型NQS拟设呈现出多项式缩放结果。
英文摘要
We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from $N=14$ to $N=24$, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2 $S_2$ and von Neumann $S_{vN}$ absolute entanglement entropy reconstruction errors. More importantly, for all tested QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatze, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatze in the QSK volume-law system as the Hilbert space dimension increases exponentially.
