We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, having a good trial wavefunction to guide the random walk is crucial for avoiding sign problems. Typically, this trial wavefunction is fixed throughout the simulation. Our proposed method iterates between determining a new trial wavefunction in the form of a tensor train, derived from the current walkers, and using this updated trial wavefunction to anchor the next phase of AFQMC. Numerical results demonstrate that our algorithm is highly accurate for large spin systems, achieving a relative error of \(10^{-5}\) in estimating ground-state energies. Additionally, the overlap between our estimated trial wavefunction and the ground-state wavefunction achieves a high-fidelity. We provide a convergence proof, highlighting how an effective trial wavefunction can reduce the variance in the AFQMC energy estimate.

翻译：我们提出了一种新颖算法，通过将辅助场量子蒙特卡洛（AFQMC）与张量链草图技术相结合，用于计算量子多体系统的基态能量。在AFQMC中，拥有一个良好的试探波函数来引导随机行走对于避免符号问题至关重要。通常，该试探波函数在整个模拟过程中是固定的。我们提出的方法在以下两个步骤之间迭代：首先，从当前行走者中推导出张量链形式的新的试探波函数；然后，使用这个更新后的试探波函数来锚定下一阶段的AFQMC。数值结果表明，我们的算法对于大型自旋系统具有很高的精度，在估计基态能量时实现了 \(10^{-5}\) 的相对误差。此外，我们估计的试探波函数与基态波函数之间的重叠度达到了很高的保真度。我们提供了收敛性证明，阐明了有效的试探波函数如何能够降低AFQMC能量估计的方差。