弱相互作用费米子动力学的高效经典模拟
Efficient Classical Simulation of Weakly Interacting Fermion Dynamics
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中文总结 AI 辅助
该研究针对弱相互作用费米子系统实时动力学模拟问题，提出结合连续时间QMC等思想的多项式时间经典算法，可在特定条件下高效估计局域可观测量期望值，明确了经典可处理的宽范围区域。
中文摘要 AI 辅助
我们考虑弱相互作用费米子系统的实时动力学模拟任务，尤其聚焦于计算时间t处局域可观测量A的期望值。通过分析海森堡绘景下可观测量关于相互作用强度λ的微扰展开收敛性，当哈密顿量在D维晶格上呈几何局域性、且满足弱相互作用区域条件λ|t|^(2D+1)=O(1)时，我们提出了一种多项式时间算法来估计该期望值，重要的是，该条件与系统规模无关。若目标是近似归一化弗罗贝尼乌斯范数下的时间演化可观测量，我们将收敛区域扩展至λ|t|=O(1)，运行时间为拟多项式。当非相互作用部分呈现安德森局域性时，我们的多项式时间算法可扩展至λ|t|=O(1)，仅需多对数因子的修正。该算法整合了连续时间QMC、图示QMC及Majorana Propagation的思想，但采用了新的海森堡绘景算符增长分析，使采样复杂度可严格控制，从而在相互作用足够弱、采样方差与系统规模无关的区域，得到可证高效的经典算法。这些结果明确了可利用弱相互作用、局域性和局域化使实时费米子动力学在经典层面可处理的广泛区域。
英文摘要
We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $λ$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $λ|t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $λ|t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $λ|t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.
