Applies to nonlinear non-Hamiltonian classical dynamics on phase space. Joseph's efficiency claim holds when the Koopman-von Neumann Hamiltonian is sparse, and is stated relative to a deterministic Eulerian discretization of the Liouville equation. The lifted object is a distribution over phase space, so recovering a single trajectory or a pointwise value is a separate readout problem.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, and a truncation or lift parameter (Carleman truncation level $N$, a phase-space grid, the level-set dimension, the homotopy order).

1. Nonlinear initial-value problem → Linear ODE systemcarry the density as an amplitude

   The lifted object is the phase-space density, not a trajectory. For $d$ ODEs $\dot{x} = v(x,t)$ with $v$ an arbitrary vector field, conservation of the density is the Liouville equation $\dot{f} + f\nabla\cdot v = \partial_t f + \nabla\cdot(vf) = 0$, whose Liouville operator $-(d/dt)^\dagger = \partial_t + \nabla\cdot v$ is the adjoint of the advection operator $d/dt = \partial_t + v\cdot\nabla$ and coincides with it, both anti-Hermitian, only when $\nabla\cdot v = 0$ — the general case is exactly what this lift has to handle. assumption: The lift's postulate is that the density is the inner product of a complex probability amplitude with its adjoint, $f = \psi^\dagger\psi$, so that $\psi = f^{1/2}e^{i\varphi}$, together with an assumed equation of motion for the phase, $\partial_t\varphi + v\cdot\nabla\varphi = -W(x,t)/\hbar$; the phase has no effect on the classical dynamics and is not measurable within it, so the choice of $W$ is a choice of gauge, the trivial choice $W = 0$ being the constrained classical action. assumption: Expectations are read back as $\langle O\rangle = \int O\psi^\dagger\psi\,d^dx = \int Of\,d^dx$, which requires the density normalized to yield unit probability after integration over all of phase space. Differentiating $\psi$ then gives $\partial_t\psi + \frac{1}{2}(v\cdot\nabla + \nabla\cdot v)\psi = -iW\psi/\hbar$, and multiplying by $i\hbar$ gives the generalized Koopman-von Neumann equation $i\hbar\partial_t\psi = -i\hbar\frac{1}{2}(v\cdot\nabla + \nabla\cdot v)\psi + W\psi$, which has the form of a Schrödinger equation $i\hbar\partial_t\psi = \hat{H}\psi$: promoting $\hat{x} = x$, $\hat{P} = -i\hbar\nabla$ and every function of the coordinates to an operator by its formal Taylor series gives $\hat{H} = \frac{1}{2}(\hat{P}\cdot\hat{v} + \hat{v}\cdot\hat{P}) + \hat{W}$. Integration by parts over $\langle\phi|\psi\rangle = \int\phi^\dagger(x,t)\psi(x,t)\,d^dx$ proves $\hat{H}$ Hermitian, so the propagator $\hat{U}$ of $i\hbar\partial_t\hat{U} = \hat{H}\hat{U}$ is unitary for any set of ODEs, not only Hamiltonian ones; solved by characteristics along $x = \xi(x_0,t)$ it is explicit, $\langle x|\hat{U}_{t,t_0}|x_0\rangle = J_0^{-1/2}(x_0,t)\,\delta^d(x - \xi(x_0,t))\,e^{-i\int_{t_0}^{t}W_0(x_0,t)dt/\hbar}$ with $J_0(x_0,t) = \det(\partial x_0^j/\partial x^k)$. The branch of $J_0^{1/2}$ is irrelevant to the classical system, but Joseph records that the phase shift $\Delta\varphi = -\nu\pi/2$ set by the Maslov index $\nu$, which counts the zeros of the Jacobian along the trajectory, must be accounted for to obtain the correct semiclassical phase factor — a framework he leaves for future work. Because $\hat{H}$ is linear in $\hat{P}$, Heisenberg's equations for the coordinates are exactly the classical ones, $d\hat{x}/dt = [\hat{x},\hat{H}]/i\hbar = v(\hat{x},t)$, and the uncertainty principle binds each $x^j$ only to its own conjugate momentum and never one original variable to another, so there is "complete fidelity to the classical phase space evolution". Nothing is truncated and there is no convergence parameter; the price is dimensional, $\hat{H}$ being the quantization of the constrained Hamiltonian $H(x,P,t) = P\cdot v(x,t) + W(x,t)$ on twice the classical phase-space dimension, whose momenta are Lagrange multipliers enforcing the equations of motion.

   assumption

A linear generator with any inhomogeneity, a lift map, a readout map, and an error bound as a function of the truncation parameter.

given  a classical system presented as first order in time,
           dx/dt = v(x,t),   x in R^d,   v an arbitrary vector field
       # nonlinear, and not required to be Hamiltonian; a PDE reaches this form
       # through the method of lines

# the object lifted is the phase-space density, not a trajectory
write  the Liouville equation for conservation of the PDF f on phase space:
           d f/dt + f div v  =  d_t f + div( v f ) = 0

# the KvN postulate
set    f = psi^dagger psi,     i.e.   psi = f^(1/2) e^(i phi)
choose W(x,t) and let the phase obey
           d_t phi + v . grad phi = -W(x,t)/hbar
       # within classical dynamics phi is not measurable, so W is a gauge
       # choice; W = 0 is the constrained classical action

differentiate psi and multiply by i hbar:
           i hbar d_t psi = -i hbar (1/2)( v . grad + div v ) psi + W psi
       # a Schroedinger equation, i hbar d_t psi = H_hat psi

promote x -> x_hat, P -> P_hat = -i hbar grad, and every function of the
coordinates to an operator by its formal Taylor series:
           H_hat = (1/2)( P_hat . v_hat + v_hat . P_hat ) + W_hat

# Hermitian over <phi|psi> = int phi^dagger(x,t) psi(x,t) d^d x, by integration
# by parts, for ANY set of ODEs -- Hamiltonian or not, dissipative or not --
# so the propagator U_hat of  i hbar d_t U_hat = H_hat U_hat  is unitary
# no truncation, hence no convergence parameter: the lift is exact

return H_hat (Hermitian), U_hat (unitary), the lift f -> psi, and the readout
           f = psi^dagger psi,   <O> = int O f d^d x
       # the lifted object is a distribution over phase space, so a single
       # trajectory or a pointwise value is a separate readout problem

# the price is dimensional, not an approximation error: H_hat is the
# quantization of the constrained Hamiltonian H = P . v(x,t) + W(x,t) on twice
# the classical phase space dimension, the momenta acting as Lagrange
# multipliers that enforce the equations of motion

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 (abstract). No unconditional end-to-end query count for a general nonlinear system is stated.