论文标题
哈密顿量的光谱估计：经典假想时间演化与量子实时演变之间的比较
Spectral estimation for Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution
论文作者
论文摘要
我们提出了一个经典的蒙特卡洛（MC）方案，该方案有效地估计了一个假想时间，腐烂的信号（即无标志 - 问题）当地的哈密顿量。该信号中的衰减速率对应于汉密尔顿特征值（带有输入态的相关本征），并且可以使用ESPRIT（例如ESPRIT）的经典信号处理方法进行经典提取。我们将该MC方案的效率与其量子对应物进行了比较，在该方案中，人们再次使用ESPRIT方法从实时的，通过量子相估计电路获得的实时的振荡信号提取了局部汉密尔顿一般的特征值。我们证明，ESPRIT方法可以通过量子相估计回路，使用poly（n）量子和经典的努力来解决s = poly（poly（n）特征值，假设它们有效地制备了输入状态。我们证明，我们的蒙特卡洛方案加ESPRIT方法可以解决S = O（1）特征值，假设它们之间存在1/poly（poly（n）），而poly（n）纯粹是经典的努力，则需要用于输入状态的某些访问结构。但是，我们还表明，在这些假设（即s = o（1）特征值下，假设它们之间存在1/poly（poly（n））差距，并且对输入状态的某些访问结构可以实现这一目标，则可以用poly（n）纯经典的当地汉密尔顿人来实现这一点。因此，这些结果量化了古典蒙特卡洛方法的一些机会和局限性，以估计哈密顿量的光谱估计。我们通过数值来比较MC特征值估计方案（对于杂种汉密尔顿人）和QPE特征值估计方案，通过为原型的型号汉密尔顿系统实施它们：横向式磁场。
We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be classically extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits, assuming efficient preparation of the input state. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of classical Monte Carlo methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the QPE eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain.
