Table 1 Summary of variable names and conventions used throughout the manuscript

x, t, T

Spatial coordinate, time, and final time

u

Solution vector for the nonlinear ODE

y

Carleman state vector: y = [u, u^⊗2, …, u^⊗N]

F₁, F_M

Linear and nonlinear terms of the ODE

N

Carleman truncation order

\({\mathcal{A}}\), \({{\mathcal{A}}}_{N}\)

Full and truncated Carleman system matrix

\({A}_{j}^{(i)}\)

(j, i)-block component of Carleman matrix

ε

Solution error

η

Error in Carleman vector (also due to linearisation)

Δ

Laplace operator

L, L_k

Finite-difference Laplacians of stencil order k

D, c, b

PDE coefficients: diffusion, decay, and nonlinearity

n, d

Number of gridpoints and dimension of the PDE

\({\mathbb{I}}\)

Identity matrix

λ_j( ⋅ )

jth eigenvalue of matrix argument

λ_(⋅)

Subnormalization constant for block encoding

γ

Rescaling factor

\(\widetilde{(\cdot )}\)

Rescaled quantity

R

Nonlinearity-to-decay strength ratio

f_j,k,M

Bound function for Carleman truncation error

Ω_k

Interval defining Carleman error regime

s( ⋅ )

Sparsity of matrix or operator

[N]

Index set {1, 2, …, N}