Simulation of realistic classical mechanical systems is of great importance to many areas of engineering such as robotics, dynamics of rotating machinery and control theory. In this work, we develop quantum algorithms to estimate quantities of interest such as the kinetic energy in a given classical mechanical system in the presence of friction or damping as well as forcing or source terms, which makes the algorithm of practical interest. We show that for such systems, the quantum algorithm scales polynomially with the logarithm of the dimension of the system. We cast this problem in terms of Hamilton's equations of motion (equivalent to the first variation of the Lagrangian) and solve them using quantum algorithms for differential equations. We then consider the hardness of estimating the kinetic energy of a damped coupled oscillator system. We show that estimating the kinetic energy at a given time of this system to within additive precision is BQP hard when the strength of the damping term is bounded by an inverse polynomial in the number of qubits. We then consider the problem of designing optimal control of classical systems, which can be cast as the second variation of the Lagrangian. In this direction, we first consider the Riccati equation, which is a nonlinear differential equation ubiquitous in control theory. We give an efficient quantum algorithm to solve the Riccati differential equation well into the nonlinear regime. To our knowledge, this is the first example of any nonlinear differential equation that can be solved when the strength of the nonlinearity is asymptotically greater than the amount of dissipation. We then show how to use this algorithm to solve the linear quadratic regulator problem, which is an example of the Hamilton-Jacobi-Bellman equation.

翻译：经典力学系统的实际模拟在机器人学、旋转机械动力学及控制理论等工程领域具有重要应用。本文提出量子算法，用于存在摩擦/阻尼及外力/源项时估计经典力学系统中的动能等关键物理量，从而赋予算法实际应用价值。我们证明：对此类系统，量子算法复杂度随系统维度的对数呈多项式增长。通过将问题转化为哈密顿运动方程（等价于拉格朗日量的第一变分），并利用微分方程量子算法求解。进一步研究阻尼耦合振子系统的动能估计难度：当阻尼项强度受量子比特数逆多项式约束时，在给定时间内实现该动能加法精度估计属于BQP困难问题。针对经典系统最优控制设计（可表述为拉格朗日量的第二变分），我们首先考虑控制理论中普遍存在的非线性微分方程——里卡蒂方程，提出高效量子算法以深入非线性区域求解该方程。据我们所知，这是首个能以渐近大于耗散量的非线性强度求解非线性微分方程的范例，并进一步展示如何将此算法应用于哈密顿-雅可比-贝尔曼方程的典型实例——线性二次型调节器问题。