Truncated Carleman Linearization

• Truncated Carleman linearization is a method that lifts nonlinear ODEs and PDEs into an infinite-dimensional linear system and truncates it to obtain a finite surrogate.
• It employs algebraic constructions using tensor-power or symmetry-reduced monomial bases to generate block-tridiagonal or upper-triangular linear operators with explicit error bounds.
• The approach finds applications in fluid dynamics, quantum algorithms, and nonlinear optimization, though its efficiency depends on convergence, conditioning, and computational cost.

Searching arXiv for recent and foundational work on truncated Carleman linearization to ground the article in cited literature. arXiv search query: "Truncated Carleman linearization nonlinear ODE PDE convergence quantum" Truncated Carleman linearization is the finite-section form of Carleman linearization: a nonlinear polynomial dynamical system is lifted to an infinite-dimensional linear system on monomials or tensor powers, and that hierarchy is then cut at a finite order so that linear-system techniques can be applied. In quadratic systems, the lifted derivative of the degree-$i$ block couples linearly only to degree $i-1$, $i$, and $i+1$ blocks; truncation closes the chain by discarding couplings to degrees above the retained order, producing a finite linear surrogate whose utility depends on convergence, conditioning, and closure error (Gnanasekaran et al., 2024, Forets et al., 2017).

1. Algebraic construction of the lift

A standard starting point is a polynomial ODE

$$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$

with $u^{[i]}$ the Kronecker power of $u$. Carleman lifting introduces one variable for each monomial degree,

$w_i(t)=u(t)^{[i]}, \qquad i\in\mathbb{N},$

and differentiation of these monomials yields an infinite linear system. In the quadratic case,

$$\dot{w}_i=\sum_{j=0}^{2}A^{i}_{i+j-1}(t,p)\,w_{i+j-1},$$

where the blocks $A^{i}_{i+j-1}(t,p)$ are Kronecker sums obtained by placing $i-1$0 in each tensor position. This is the canonical block-tridiagonal lifting pattern for quadratic nonlinearities (Gnanasekaran et al., 2024).

Equivalent formulations work directly in a monomial basis ordered by total degree. In that setting, the lifted coordinate vector collects all nonzero monomials up to some degree $i-1$1, and the Lie derivative operator splits as $i-1$2, where $i-1$3 is degree-preserving and $i-1$4 is degree-raising. Homogeneous-degree formulations likewise write

$i-1$5

so the infinite lift is naturally upper-triangular across degrees (Boreale et al., 23 Jun 2026, Amini et al., 2022).

This algebraic mechanism is the essential content of Carleman linearization: nonlinear multiplication in the original coordinates becomes linear coupling between blocks of different degree. The price is dimensional inflation from a finite nonlinear system to an infinite linear one.

2. Truncation, closure, and basis choice

Truncation selects a finite lifted state and drops all couplings to higher monomials. For tensor-power lifting up to order $i-1$6,

$i-1$7

In the quadratic case, the truncated operator $i-1$8 has upper off-diagonal blocks $i-1$9 from quadratic interaction, diagonal blocks $i$0 from linear dynamics, and lower off-diagonal blocks $i$1 from inhomogeneous forcing; truncation removes the $i$2 term and thereby closes the finite chain (Gnanasekaran et al., 2024).

Because the tensor-product basis contains many repeated monomial contributions, several later formulations replace it with symmetry-reduced monomial bases. In that representation, the number of retained monomials up to order $i$3 is

$i$4

in contrast to the tensor-ordering count

$i$5

This reduction changes the combinatorics of assembly without changing the underlying finite-section idea (Akiba et al., 7 May 2026).

In discrete-time stochastic polynomial systems, truncation appears after expectation rather than before time integration. If $i$6 stacks monomials up to degree $i$7, then the expected moment vector obeys a deterministic lower-triangular recursion

$i$8

and truncation at degree $i$9 yields an exact computation of the $i+1$0th moment at time $i+1$1 whenever $i+1$2 (Pruekprasert et al., 2019).

3. Error bounds and convergence regimes

A central issue is whether the finite-section approximation converges to the original nonlinear dynamics as the truncation order grows. Explicit local error bounds were given for polynomial ODEs through backward integration and a generating-function construction. In the quadratic case, the latter yields a computable local error bound

$i+1$3

together with the local convergence horizon

$i+1$4

(Forets et al., 2017).

For analytic systems with exponentially decaying Maclaurin coefficients, finite-section approximations were later shown to converge exponentially in truncation order, both locally in time and, under stronger stability assumptions, uniformly on the entire half-line. In the globally stable case, the approximation error satisfies a geometric bound with ratio

$i+1$5

so the truncation error decays like $i+1$6 for all $i+1$7 (Amini et al., 2022).

More recent analysis works directly in the original monomial basis and focuses on observables such as state coordinates rather than the full lifted state. Using a Dyson–Duhamel expansion, the degree-preserving linear part is separated from the degree-raising nonlinear part, which yields degree-aware bounds with a support gap $i+1$8 and geometric convergence on certified horizons. This makes explicit how truncation error propagates back to the observable and retains logarithmic-norm information from the original linear dynamics (Boreale et al., 23 Jun 2026).

The dissipative regime is not the only one admitting rigorous truncation control. A resonance-based theory replaces classical dissipativity with a spectral nonresonance gap

$i+1$9

leading to the global-in-time bound

$$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$0

when $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$1 (Wu et al., 2024). A different route is pivot-shifted Carleman linearization, which first re-centers the dynamics at a pivot $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$2, then applies a Lyapunov transform and rescaling. Under stability after pivot shift and a weighted nonlinear stability condition, the truncation order scales logarithmically with simulation time and target precision, and the conventional lower-bound requirement on the initial condition is removed (Wang et al., 19 May 2026).

For parabolic semilinear PDEs with quadratic nonlinearities, the truncated lift is formulated directly in infinite-dimensional Hilbert spaces. There, the total approximation error decomposes into a discretization error and a linearization error, and the convergence radius and convergence rate of the linearization are independent of the spatial discretization (Heinzelreiter et al., 1 Oct 2025).

4. Structural variants of truncated Carleman linearization

Several modern variants alter the lift rather than the truncation principle. Pivot-shifted Carleman linearization replaces $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$3 by $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$4, so that the shifted coefficients become

$$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$5

Combined with a Lyapunov transform $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$6, this enlarges the class of systems for which truncated Carleman embeddings converge efficiently, especially when the pivot is chosen near a locally exponentially stable equilibrium (Wang et al., 19 May 2026).

A different structural refinement is duplicate-aware shift-and-lift. Here the vector field is first expanded around a moving center, and the lifted operator is assembled in a symmetry-reduced basis with packed exponent-key indexing,

$$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$7

This permits sparse triplet coalescing of repeated target monomials and preserves the truncated affine dynamics exactly, up to floating-point summation order (Akiba et al., 7 May 2026).

For periodic vector fields, Carleman-Fourier linearization replaces monomials by Fourier basis functions

$$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$8

leading to a block upper-triangular linear system in harmonic order rather than polynomial degree. Finite-section approximations in this basis admit explicit error bounds and, for suitable classes of systems, exponential convergence over larger regions and longer horizons than monomial-based Carleman linearization (Chen et al., 2024).

These variants show that truncated Carleman linearization is not a single fixed algorithmic object. It is a family of finite-section approximations whose basis, centering, and symmetry conventions are chosen to exploit specific structure in the nonlinear dynamics.

5. Numerical methods and application domains

In nonlinear PDE-constrained optimization, truncated Carleman linearization is used after spatial semi-discretization of the PDE. The lifted linear ODE is discretized in time, embedded into a stacked linear system, and solved inside a bi-level variational quantum PDE-constrained optimization framework using a Variational Quantum Linear Solver (VQLS); the paper demonstrates this pipeline on an inverse Burgers problem and gives explicit truncation, Euler, and condition-number bounds (Gnanasekaran et al., 2024).

For quantum data loading, one line of work develops an efficient decomposition of the Carleman-linearized one-dimensional Burgers equation. After truncation at order $$\dot{u}(t)=F_0(t,p)+F_1(t,p)u+F_2(t,p)u^{[2]}+\cdots+F_k(t,p)u^{[k]},$$9, the resulting linear system is embedded into a larger system whose matrix can be decomposed into

$u^{[i]}$0

terms, with block encodings suitable for VQLS and two-qubit gate depth bounded by

$u^{[i]}$1

(Demirdjian et al., 1 May 2025). A related construction replaces Pauli decompositions by a Sigma-basis linear combination of nonunitaries, with the explicit claim of an exponential reduction in the number of decomposition terms compared to traditional Pauli-based decompositions (Ali, 6 Feb 2026).

Fluid dynamics has become a major testing ground. For steady-state incompressible flow, the lowest nontrivial truncation $u^{[i]}$2 (“C2”) was shown to recover not only the initial transient but also the late-time steady state, with two-dimensional tests at moderate Reynolds number reporting mean L2 relative errors in the $u^{[i]}$3–$u^{[i]}$4 range (Cappelli et al., 22 May 2026). In a lattice-Boltzmann formulation, second-order Carleman truncation was reported to be sufficient for Kolmogorov-like flows at moderate Reynolds numbers between 10 and 100; the quantum circuit for the single time-step collision operator has a fixed depth, regardless of the number of lattice sites, but such depth is of the order of ten thousands quantum gates (Sanavio et al., 2023).

In chemical kinetics, Carleman linearization was applied to a scalar test problem and to zero-dimensional homogeneous ignition for H$u^{[i]}$5/air and CH$u^{[i]}$6/air mixtures. The reported outcome was that the method accurately reproduced reference data, and that increasing the truncation order improved accuracy even with a large time-step size (Akiba et al., 2022). In nonlinear system identification, truncated Carleman lifting is used as a model class: the truncation order is selected from a prescribed error bound, then the finite lifted linear system is estimated from trajectory data, with demonstration on the Van der Pol oscillator (Abudia et al., 2022).

Hybrid quantum-classical Duffing pipelines further illustrate the current algorithmic state of the art. There, Carleman-linearized and time-discretized block-banded systems are solved with VQLS using symmetry-grouped Hadamard Test evaluations under global and local costs; across IBM and Xanadu implementations, the reported behavior is near-unity fidelity and vanishing relative residuals for the test cases considered (Liu et al., 14 May 2026).

6. Limitations, misconceptions, and open problems

A common misconception is that higher-order truncation is automatically superior to classical local linearization. Comparative evaluation against Jacobian linearization does not support that claim uniformly: the reported outcome is regime-dependent accuracy gains for duplicate-aware shift-and-lift rather than universal superiority (Akiba et al., 7 May 2026).

The dominant structural limitation is dimensional growth. In tensor-product form, the lifted dimension grows like $u^{[i]}$7; in PDE settings the discretized lifted space can still be prohibitively large even when the linearization error behaves well, which is why sparse grids, tensor formats, and structured decompositions are emphasized (Heinzelreiter et al., 1 Oct 2025). Strong nonlinearities, large initial conditions, or conservative dissipativity conditions can force larger truncation orders and thereby increase cost substantially; this is stated explicitly for the nonlinear PDE-constrained optimization setting, where very strong nonlinearities or large $u^{[i]}$8 may require higher $u^{[i]}$9 (Gnanasekaran et al., 2024).

Quantum-advantage claims are also conditional. Some complexity bounds rely on empirical VQLS scaling assumptions rather than proven worst-case guarantees, and practical performance depends heavily on structured matrices, efficient block encodings, and favorable conditioning (Gnanasekaran et al., 2024). In pivot-shifted Carleman methods, optimal pivot selection remains open, as do extensions beyond quadratic nonlinearities and efficient implementations of multiple pivot switches (Wang et al., 19 May 2026). In lattice-Boltzmann quantum algorithms, the multi-step version remains an open topic for future research (Sanavio et al., 2023).

Truncated Carleman linearization is therefore best understood as a rigorous but regime-sensitive finite-section methodology. Its success depends on how well the chosen lift, truncation order, basis, and numerical realization align with the structure of the underlying nonlinear system.