Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics

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.

Figure 1

The probability distribution function (PDF) $f(\mathbf{x},t)$ is advected by the flow $\mathbf{v}(\mathbf{x},t)$, as described by the Liouville equation [Eq. (8)].

Figure 2

The constrained Hamiltonian $\mathcal{H}$ [Eq. (35)] is defined on an extended phase space that includes the original phase-space coordinates $\mathbf{x}=(x^1,x^2,\cdots )$, as well as the canonically conjugate momenta $\mathbf{P}=(P_1,P_2,\cdots )$, which act as Lagrange multipliers.

Figure 3

A numerical discretization of phase space implies finite numerical widths $\mathrm{\Delta }x$ and $\mathrm{\Delta }P$. However, the phase-space area $\mathrm{\Delta }x\mathrm{\Delta }P=h/L$ due to the discretization is much smaller than than the Heisenberg limit allows. Squeezed states can be used to reduce the uncertainty to the numerical discretization limit, so that ${\sigma }_x=\mathrm{\Delta }x$.