标题： 量子杂质问题的复杂性 显示英文标题

标题： Complexity of quantum impurity problems

摘要： 我们给出了一个拟多项式时间的经典算法，用于估算量子杂质模型的基态能量并计算低能态。 此类模型描述了一组自由费米子与一个小的相互作用子系统（称为杂质）耦合。完整系统由 $n$ 个费米模式组成，哈密顿量为 $H=H_0+H_{imp}$，其中 $H_0$ 是产生-湮灭算符的二次形式，而 $H_{imp}$ 是作用于某子集上 $O(1)$ 个模式上的任意哈密顿量。 我们证明了 $H$ 的基态能量可以用加性误差 $2^{-b}$ 在时间 $n^3 \exp{[O(b^3)]}$ 内近似。 我们的算法还找到了一个达到该近似的低能态。 这个低能态被表示为 $\exp{[O(b^3)]}$ 个费米子Gaussian态的叠加。 为了得到这一结果，我们证明了关于杂质模型精确基态的若干定理。 特别是，我们证明了基态协方差矩阵的特征值以指数形式衰减，且衰减速率非常轻微地依赖于 $H_0$ 的谱隙。 我们的证明中的关键部分是Zolotarev对 $\sqrt{x}$ 函数的有理逼近。 我们预计，我们的算法可能用于基于动力学平均场理论的强关联材料的混合量子-经典模拟。 我们实现了一个简化实用版本的算法，并使用单杂质Anderson模型对其进行了基准测试。 显示英文摘要

摘要： We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of $n$ fermionic modes and has a Hamiltonian $H=H_0+H_{imp}$, where $H_0$ is quadratic in creation-annihilation operators and $H_{imp}$ is an arbitrary Hamiltonian acting on a subset of $O(1)$ modes. We show that the ground energy of $H$ can be approximated with an additive error $2^{-b}$ in time $n^3 \exp{[O(b^3)]}$. Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of $\exp{[O(b^3)]}$ fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of $H_0$. A key ingredient of our proof is Zolotarev's rational approximation to the $\sqrt{x}$ function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.