Embed a nonlinear system into a linear one

Why this is a layer

The lifts here are not interchangeable, and they are not all siblings either: Carleman is the monomial-basis instance of Koopman linearization and the Fourier basis is a second instance, so choosing between those two is choosing a basis on a space of observables. Katz et al. list Koopman-von Neumann linearization as its own entry beside Carleman and do not say whether it too is a basis choice, and nothing cited here places level sets (Jin-Liu) or homotopy series terms (Xue et al.) inside that framework either, so all three stay separate on this map. What divides all of them is admissible nonlinearity, whether the truncation converges at all, and what the lifted state physically means — and choosing wrongly here, not downstream, is what usually breaks an end-to-end claim.

Ways to do this

6 methods recorded

• 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.

• 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.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

• Quantum Carleman linearization algorithm

  Carleman-linearize the quadratic ODE, discretize with forward Euler, assemble the whole history into one large sparse linear system, and solve that system with a quantum linear system algorithm. This is the route that made dissipative nonlinear ODEs tractable in evolution time.

• Quantum simulation of the KvN representation

  Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

• Level-set method for observables of nonlinear PDEs

  Use the exact level-set mapping to a linear PDE, solve the linear problem quantumly, and compute physical observables from it. For $M$ sets of initial data the cost does not grow with $M$.

• Homotopy-perturbation series, embedded as a linear ODE

  Embed the homotopy-perturbation series into a finite-dimensional linear ODE system and solve that with a quantum linear-ODE algorithm, obtaining a state $\varepsilon$-close to the normalized exact solution with $Ω(1)$ success probability.

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.