---
title: Truncated Carleman Linearization
url: https://www.emergentmind.com/topics/truncated-carleman-linearization
type: topic
---

# Truncated Carleman Linearization

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 [2410.13688][1711.02552].

## 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 $F_j(t,p)$ in each tensor position. This is the canonical block-tridiagonal lifting pattern for quadratic nonlinearities [2410.13688].

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 $D$, and the Lie derivative operator splits as $A=A_1+A_2$, where $A_1$ is degree-preserving and $A_2$ is degree-raising. Homogeneous-degree formulations likewise write
\[
\dot z_k(t)=\sum_{l=k}^{\infty}A_{k,l}(t)z_l(t),
\]
so the infinite lift is naturally upper-triangular across degrees [2606.24473][2207.07755].

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 $N$,
\[
w(t)=\bigl(w_1(t),w_2(t),\ldots,w_N(t)\bigr)^\top \in \mathbb{R}^{N_c},
\qquad
N_c=\frac{n^{N+1}-n}{n-1}.
\]
In the quadratic case, the truncated operator $A_N(t,p)$ has upper off-diagonal blocks $A^i_{i+1}$ from quadratic interaction, diagonal blocks $A^i_i$ from linear dynamics, and lower off-diagonal blocks $A^i_{i-1}$ from inhomogeneous forcing; truncation removes the $A^N_{N+1}w_{N+1}$ term and thereby closes the finite chain [2410.13688].

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 $p$ is
\[
M(n,p)=\sum_{k=1}^{p}\binom{n+k-1}{k}
      =\binom{n+p}{p}-1,
\]
in contrast to the tensor-ordering count
\[
\sum_{k=1}^{p} n^k=\frac{n^{p+1}-n}{n-1}.
\]
This reduction changes the combinatorics of assembly without changing the underlying finite-section idea [2605.05901].

In discrete-time stochastic polynomial systems, truncation appears after expectation rather than before time integration. If $y(s,t)$ stacks monomials up to degree $s$, then the expected moment vector obeys a deterministic lower-triangular recursion
\[
M^{(s)}(t+1)=E(s,s)\,M^{(s)}(t),
\]
and truncation at degree $s$ yields an exact computation of the $j_0$th moment at time $t$ whenever $j_0^t\le s$ [1911.12683].

## 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
\[
E_2(t)=\frac{\|x_0\|\,e^{\|F_1\|t}}{(1+\beta_0)-\beta_0 e^{\|F_1\|t}}
\Big[\beta_0(e^{\|F_1\|t}-1)\Big]^N,
\qquad
\beta_0=\frac{\|x_0\|\|F_2\|}{\|F_1\|},
\]
together with the local convergence horizon
\[
T^\ast=\frac{1}{\|F_1\|}\ln\Bigl(1+\frac{1}{\beta_0}\Bigr)
\]
[1711.02552].

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
\[
\rho=\frac{(D_0+R\mu_0)\|x_0\|_2}{R^2\mu_0}<1,
\]
so the truncation error decays like $\rho^N$ for all $t\ge t_0$ [2207.07755].

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 $k_D$ 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 [2606.24473].

The dissipative regime is not the only one admitting rigorous truncation control. A resonance-based theory replaces classical dissipativity with a spectral nonresonance gap
\[
\Delta=\inf_{k\in[m]}\inf_{\substack{m_j\ge 0\\ \sum m_j\ge 2}}
\left|\lambda_k-\sum_{j=1}^m m_j\lambda_j\right|,
\]
leading to the global-in-time bound
\[
\|x(T)-y_1(T)\|_1 \le N C T \,\mathfrak{B}_r^{\,N-1},
\qquad
\mathfrak{B}_r=\frac{4 e \mu \kappa_1(W)\|F_2\|_1}{\Delta},
\]
when $\mathfrak{B}_r<1$ [2405.12714]. A different route is pivot-shifted Carleman linearization, which first re-centers the dynamics at a pivot $s$, 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 [2605.20071].

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 [2510.00722].

## 4. Structural variants of truncated Carleman linearization

Several modern variants alter the lift rather than the truncation principle. Pivot-shifted Carleman linearization replaces $x$ by $u=x-s$, so that the shifted coefficients become
\[
F_{2,s}=F_2,\qquad
F_{1,s}=F_1+F_2(s\otimes I+I\otimes s),\qquad
F_{0,s}=F_2s^{\otimes 2}+F_1s+F_0.
\]
Combined with a Lyapunov transform $v=Qu$, 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 [2605.20071].

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,
\[
\kappa(\gamma)=\sum_{j=1}^{n}\gamma_j\,2^{(j-1)b},
\qquad
b=\left\lceil\log_2(2Q+1)\right\rceil.
\]
This permits sparse triplet coalescing of repeated target monomials and preserves the truncated affine dynamics exactly, up to floating-point summation order [2605.05901].

For periodic vector fields, Carleman-Fourier linearization replaces monomials by Fourier basis functions
\[
w_{\alpha}=e^{i\alpha\cdot x},
\]
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 [2411.11598].

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 [2410.13688].

For quantum data loading, one line of work develops an efficient decomposition of the Carleman-linearized one-dimensional Burgers equation. After truncation at order $\alpha$, the resulting linear system is embedded into a larger system whose matrix can be decomposed into
\[
\mathcal{O}(\log n_t+\alpha^2\log n_x)
\]
terms, with block encodings suitable for VQLS and two-qubit gate depth bounded by
\[
\mathcal{O}\bigl(\alpha(\log n_x)^2\bigr)
\]
[2505.00285]. 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 [2602.07097].

Fluid dynamics has become a major testing ground. For steady-state incompressible flow, the lowest nontrivial truncation $K=2$ (“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 $10^{-5}$–$10^{-4}$ range [2605.23380]. 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 [2310.17973].

In chemical kinetics, Carleman linearization was applied to a scalar test problem and to zero-dimensional homogeneous ignition for H$_2$/air and CH$_4$/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 [2207.01818]. 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 [2205.15009].

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 [2605.15366].

## 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 [2605.05901].

The dominant structural limitation is dimensional growth. In tensor-product form, the lifted dimension grows like $\sum_{k=1}^{N}n^k$; 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 [2510.00722]. 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 $\|u_0\|$ may require higher $N$ [2410.13688].

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 [2410.13688]. In pivot-shifted Carleman methods, optimal pivot selection remains open, as do extensions beyond quadratic nonlinearities and efficient implementations of multiple pivot switches [2605.20071]. In lattice-Boltzmann quantum algorithms, the multi-step version remains an open topic for future research [2310.17973].

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.

Source: https://www.emergentmind.com/topics/truncated-carleman-linearization