Koopman-von Neumann lift to phase-space densities

What it fills

• Embed a nonlinear system into a linear one

  Given a nonlinear vector field $F$, produce a (truncated) linear generator on a lifted space, a lift of the initial condition into that space, and a decoding of the target quantity, such that linear evolution reproduces the nonlinear dynamics to accuracy $\varepsilon$. The truncation or lift parameter fixes both the accuracy and the dimension.

When it applies

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.

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

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

Cost, as the source states it

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.

Implementations

Nobody has written one up yet. That is a gap in this record, not a statement that the method has never been run — the paper register already records, per paper, which sources report numerics or a hardware run.

What it needs

Nothing below this — it bottoms out here.

Other ways to fill the same slot

Different approaches

• Carleman linearization a narrower version of Koopman linearization

  Lift the quadratic ODE onto the tower $y, y⊗y, y⊗y⊗y, \ldots$ , on which the dynamics is exactly linear and each level couples only to its neighbours, then truncate at level $N$. The lift itself is exact; all of the error comes from the truncation. Katz, Muraleedharan and Alase derive it as one instance of Koopman linearization: taking the space of observables to be the polynomials and the basis functions to be the monomials reproduces exactly this tower, in one variable and in $n$.

• Koopman linearization

  Pick a space of observables $G$ containing the quantity of interest and a basis $Ψ$ for it; the Koopman generator acting on $Ψ$ gives an infinite-dimensional linear ODE, truncated by projecting onto $N$ basis functions. $G$ fixes which observables the lifted dynamics can report and $Ψ$ fixes the structure of the generator, so this is a family of lifts parameterised by that choice rather than a single lift. Only basis choices a cited paper has carried through are recorded here — Katz, Muraleedharan and Alase name Chebyshev and Hermite bases as directions rather than results — so the narrower versions recorded under it are a sample of the framework and not an enumeration of it.

• Carleman-Fourier linearization a narrower version of Koopman linearization

  Lift the rescaled ODE $dx/dt = F_0 + F_1 e^{ix}$ — the problem as posed is $du/dt = G_0 + G_1 e^{iu}$, rescaled so that $F_0 = G_0$ and $F_1 = νG_1$ — onto the Fourier tower $e^{ix}, (e^{ix})^{\otimes2}, \ldots$ instead of the monomial tower, then truncate at level $N$. Katz, Muraleedharan and Alase give the reason for the choice: expanding the same equation in monomials leaves the coefficient matrix non-sparse, whereas in the Fourier basis the coefficient matrix of their single-variable illustration has only two non-zero entries in each row.

• Level-set exact linearization

  Map a nonlinear PDE exactly onto a linear one using the level-set method, with no truncation and therefore no convergence parameter. The price is a higher-dimensional linear problem.

• Homotopy perturbation embedding

  Convert the original nonlinear ODE into another nonlinear system whose homotopy-perturbation terms embed into a single finite-dimensional linear ODE system, truncated at a chosen homotopy order. The embedding is finite-dimensional by construction.

In the Atlas

No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.

Sources

• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics↗Ilon Joseph · 2020theory yes · simulation ? · hardware no