论文标题
Koopman-von Neumann方法的量子模拟非线性经典动力学
Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
论文作者
论文摘要
量子计算机可用于通过使用经典力学的广义Koopman-von Neumann公式在相空间上模拟非线性非汉顿经典动力学。 Koopman-von Neumann配方意味着，如Liouville方程所表达的相位空间上概率分布函数的保存可以重塑为Hilbert Space上的等效schrödinger方程，并带有Hermitian Hamiltonian hamiltonian操作员和单位宣传者。该schrödinger方程在该动量上是线性的，因为它源自具有经典相位空间维度的两倍的受约束的哈密顿系统。具有有限资源的量子计算机可用于模拟该单一进化运算符的有限维近似。如果Koopman-von Neumann Hamiltonian稀疏，经典动力学的量子模拟比确定性的Eulerian离散化更为有效。利用量子步行技术进行状态制备和振幅估计进行计算，可观察到，导致对经典概率蒙特卡洛算法的二次改善。
Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space by using the generalized Koopman-von Neumann formulation of classical mechanics. The Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space, as expressed by the Liouville equation, can be recast as an equivalent Schrödinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schrödinger equation is linear in the momenta because it derives from a constrained Hamiltonian system with twice the classical phase space dimension. A quantum computer with finite resources can be used to simulate a finite-dimensional approximation of this unitary evolution operator. Quantum simulation of classical dynamics is exponentially more efficient than a deterministic Eulerian discretization of the Liouville equation if the Koopman-von Neumann Hamiltonian is sparse. Utilizing quantum walk techniques for state preparation and amplitude estimation for the calculation of observables leads to a quadratic improvement over classical probabilistic Monte Carlo algorithms.
