Koopman linearization

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

Stated for a first-order nonlinear ODE $dx/dt = F(x)$, $x(0) = x_0$, with $x(t) \in \mathbb{C}^n$ and $F : \mathbb{C}^n \to \mathbb{C}^n$, and for a (Banach) space $G$ of observables $\psi : \mathbb{C}^n \to \mathbb{C}$. What has to hold is a property of $G$ rather than of the dynamics, and there are two parts to it. The observable of interest must lie in $G$, because the readout is its expansion $g = d \cdot \Psi = \sum_j d_j \Psi_j$ in a basis $\Psi$ of $G$; and $G$ must be closed under the evolution — Katz, Muraleedharan and Alase write that in defining the Koopman operator they "assumed that the image of $K_t$ is in $G$, which can be ensured in practice by carefully selecting a sufficiently large $G$". Truncation is by projection onto an $N$-dimensional subspace $G_N$, and the paper is explicit that projecting does not commute with the dynamics: $\Psi^{(N)}$ and $\Pi\Psi$ agree at $t = 0$, but "in general $\Psi^{(N)}(t) \neq \Pi\Psi(t)$ for $t > 0$". Convergence of the readout is stated conditionally at this generality and not proved — "As $N \to \infty$, if $g_N(t) \to g(t)$, then the value of the observable obtained using the truncated lifted dynamics $g_N(t)[x_0] = g_N(x(t)) \to g(t)[x_0] = g(x(t))$ converges to the desired value." Every quantitative hypothesis in the paper — the dissipativity condition, the finite-time bound, the truncation-error bounds — is derived for the Fourier basis and belongs to the record below rather than here.

Requires

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

Example

given  a first-order nonlinear ODE  dx/dt = F(x),  x(0) = x_0,  x(t) in C^n,
       an observable g : C^n -> C wanted at the final time,
       a truncation level N

# the parameter of this method is the pair (G, Psi), not a step size
choose  a space of observables G containing the quantity of interest g
choose  a basis Psi = {Psi_j} of G
        # G fixes which observables the lifted dynamics can report
        # Psi fixes the structure of the generator
        # so this is a family of lifts parameterised by that choice,
        # not a single lift

# the Koopman generator, written in that basis
        # (K_t psi)(x(0)) = psi(x(t)),  and K_t is linear on G
        # L(psi) = lim_{t -> 0+} (psi(t) - psi(0)) / t
form    L = [L_jk]   from   L Psi_j = sum_k L_kj Psi_k
write   dPsi(t)/dt = L^T Psi
        # transposed: what evolves is the basis of G, not one vector's
        # representation in it
evaluate at the initial configuration:
        d(Psi(t)[x_0])/dt = L^T (Psi(t)[x_0])
        # the lifted state is now an array of complex numbers

# truncate by projecting onto N basis functions
project onto  G_N = span{Psi_1, ..., Psi_N}  by  Pi : G -> G_N
form    L_N,  whose j-th column is  Pi L Psi_j  in the basis {Psi_j}_{j=1..N}
        # if Pi keeps Psi_j for j <= N and sends it to 0 for j > N,
        # L_N^T is simply the top-left N x N block of L
solve   dPsi^(N)/dt = L_N^T Psi^(N),   initial data Psi^(N)(x_0)
        # not the projected exact equation: Psi^(N)(t) != Pi Psi(t) in
        # general for t > 0, though the two agree at t = 0

# decode
expand  g ~ g_N = sum_{j=1..N} d_j Psi_j      # the closest approximation in G_N
        # obtaining Psi(t)[x_0] at time t also gives g(x(t)) = d . Psi(t)[x_0]

hand the truncated linear ODE, its initial data Psi^(N)(x_0) and the decoding d
     to the layer below

Cost, as the source states it

Read in full: Katz, Muraleedharan and Alase (arXiv:2512.06488) price nothing at the generality this record covers — an arbitrary observable space $G$ and basis $\Psi$, truncated to $N$ basis functions. Their 3.2 and 3.3 establish only that the nonlinear ODE "can be approximately represented as an $N$-dimensional linear ODE", with the proofs placed in "sections 1.4.1.1 and 1.4.1.2 of [31]" — a reference this record does not cite — rather than derived here; no query count, gate count or error bound accompanies them. Everything the paper does price is scoped to the Fourier instance it calls Carleman-Fourier linearization, basis $\Psi_j(x) = [e^{ix}]^{\otimes j}$: its problem statement is already a Fourier ODE, and both complexity theorems are titled for the Fourier algorithm with and without dissipative conditions. Those numbers belong to that record. The conclusion offers trigonometric, Chebyshev and Hermite bases as directions, not results.

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

• Koopman-von Neumann lift to phase-space densities

  Represent nonlinear non-Hamiltonian classical dynamics by the Liouville equation for the phase-space density; the generalized Koopman-von Neumann formulation recasts that as a Schrödinger equation with a Hermitian Hamiltonian operator and a unitary propagator. The lift is exact, and its cost is dimensional rather than an approximation error.

• 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.

Narrower versions of this one

• 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 $n$.

• 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.

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

• Efficient quantum algorithm for solving differential equations with Fourier nonlinearity via Koopman linearization↗Judd Katz, Gopikrishnan Muraleedharan, Abhijeet Alase · 2025theory yes · simulation ? · hardware no