We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions psi(x_1,x_2,...,x_D) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy Theta(1/N^2) is Theta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c) gate operations, where N is the number of points to which each argument is discretized, nu and c are implementation dependent constants of O(1). Optimal classical methods require Theta(N^D) bits and Omega(N^D) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D > 2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.
Quantum Eigensolver for Non-Normal Matrices via Ground State Energy Estimation
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；State Key Laboratory of Mathematical Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）
Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits
量子相位估计是量子算法设计中的一个基石，它允许对指数大的稀疏矩阵的特征值进行推断。这些特征值可以被学习的最大速率（被称为海森堡极限）受到模拟任意哈密顿量所需的电路复杂性的限制。量子相位估计的单控制量子位变体不需要实验之间的相干性，近年来由于较低的电路深度和最小的量子位开销而引起了人们的兴趣。在这项工作中，我们表明，这些方法可以实现海森堡极限，{ \em也}当一个是无法准备本征态的系统。给定一个量子子程序，它提供了一个“相位函数”$g(k)=\sum_j A_j e^{i ϕ_j k}$与未知的本征相位$ϕ_j$和重叠$A_j$在量子成本$O(k)$的样本，我们展示了如何估计相位$\{ϕ_j\}$与（均方根）误差$δ$的总量子成本$T=O(δ^{-1})$。我们的方案结合了海森堡限制的多阶量子相位估计的思想，用于单个特征值相位[Higgins et al（2009）和Kimmel et al（2015）]，具有所谓的密集量子相位估计的子程序，该子程序通过时间序列分析使用经典处理QEEP问题[Somma（2019）]或矩阵束方法。对于我们的算法，自适应固定的选择$k$在$g(k)$，我们证明海森堡限制缩放时，我们使用的时间序列/QEEP子程序。我们提出的数值证据表明，使用矩阵束技术的算法可以实现海森堡有限的缩放以及。
Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation
