Stated for the rescaled ODE $dx/dt = F_0 + F_1 e^{ix}$ with time-independent coefficient matrices. Katz et al. give two truncation regimes. The dissipative one requires $µ̃_0 := \min_j Im${$(F_0)_j$}$\geq 0$ and $R_p := ||F_1||_row,q ||\Psi_1(0)||_p / µ̃_0 < 1$, under which the $k$-th truncation error component is bounded by $(||\Psi_1(0)||_p)^{N+1} (||F_1||_row,q / µ̃_0)^{N+1−k}$. The second drops dissipativity and holds only on a finite interval $[0, T_max]$ with $T_max = \min${$T_r, \ln r / (||F_0||_∞ + ||F_1||_row,q)$}, where $r$ is the rescaling parameter and $T_r$ is the horizon their Lemma 4.3 supplies. Chen, Motee and Sun state the linearization for periodic vector fields with several fundamental frequencies and prove exponential convergence in the truncation length, achieved across the whole time horizon only for particular classes of system.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, and a truncation or lift parameter (Carleman truncation level $N$, a phase-space grid, the level-set dimension, the homotopy order).

1. Nonlinear initial-value problem → Linear ODE systemlift on the Fourier basis, not monomials

   The problem as posed is $du/dt = G_0 + G_1 e^{iu}$; the substitution $x_j := u_j + i\ln(\nu)$, $j = 1,\ldots,n$, rescales the exponent of the initial state, $e^{ix_0} = e^{iu_0}/\nu$, and carries the equation to $dx/dt = F_1 e^{ix} + F_0$ with $F_1 = \nu G_1$ and $F_0 = G_0$. assumption: $\nu$ is taken large enough that $\gamma := \lVert e^{ix_0}\rVert < 1$, which is what the readout needs; Katz, Muraleedharan and Alase observe that $\tilde\mu_0$ and $R_p$ are invariant under this rescaling, and rescale the readout coefficients with it as $c_j = \nu^{\lvert j\rvert} d_j$ so that $f(x) = \sum_j c_j e^{ix \cdot j}$ satisfies $f(x) = g(u)$. The lift then takes the Fourier basis $\Psi_j(x) = [e^{ix}]^{\otimes j}$ in place of monomials. Differentiating the first block gives $de^{ix}/dt = i\tilde{F}_1\Psi_2 + i\tilde{F}_0\Psi_1$, where $\tilde{F}_0 = \mathrm{diag}(F_0) \in \mathbb{C}^{n \times n}$ and $\tilde{F}_1 \in \mathbb{C}^{n \times n^2}$ lays out the rows of $F_1$ blockwise; differentiating $\Psi_j$ by the product rule and splitting the sum in two gives the recurrence $d\Psi_j/dt = B^{(1)}_{j+1}\Psi_{j+1} + B^{(0)}_j\Psi_j$ with $B^{(1)}_{j+1} = i\tilde{F}_1 \otimes I^{\otimes(j-1)} + I \otimes i\tilde{F}_1 \otimes I^{\otimes(j-2)} + \cdots + I^{\otimes(j-1)} \otimes i\tilde{F}_1$ and $B^{(0)}_j$ the same expression with $\tilde{F}_0$, so the infinite lifted system $d\Psi/dt = L^{T}\Psi$ is block upper bidiagonal. That is the reason for the basis: in the single-variable illustration it reads $d(e^{ix})/dt = iF_0e^{ix} + iF_1e^{2ix}$, "only two non-zero entries in each row", whereas expanding the same equation in monomials leaves the coefficient matrix not sparse. approximation: The tower is cut at level $N$, $d\Psi^{(N)}/dt = L_N^{T}\Psi^{(N)}$ with $L_N^{T}$ the top-left block of $L^{T}$, whose last block row is $(0,\ldots,0,B^{(0)}_N)$ — the coupling $B^{(1)}_{N+1}$ to level $N+1$ is dropped, and that is where the error $\eta_k = \Psi_k - \Psi^{(N)}_k$ enters. Blocks of unequal size are then padded, $\Psi^{(N)}(t) = \bigoplus_{j=1}^{N} e_1^{\otimes(N-j)} \otimes \Psi_j(t) \in \mathbb{C}^{Nn^{N}}$, to simplify the circuit. assumption: Which bound that error obeys depends on the regime. Under dissipativity — $\tilde\mu_0 := \min_j \mathrm{Im}\{(F_0)_j\} \ge 0$ and $R_p := \lVert F_1\rVert_{\mathrm{row},q}\lVert\Psi_1(0)\rVert_p/\tilde\mu_0 < 1$, with $1/p + 1/q = 1$ — Theorem 4.2 gives $\lVert\eta_k\rVert_p \le (\lVert\Psi_1(0)\rVert_p)^{N+1}(\lVert F_1\rVert_{\mathrm{row},q}/\tilde\mu_0)^{N+1-k}$ for all time. Without it, Theorem 4.4 requires $\lVert\Psi_1(0)\rVert_p < 1/r$ for some $r > 1$ and holds only on $[0,T_{\max}]$, $T_{\max} = \min\{T_r,\ \ln r/(\lVert F_0\rVert_{\infty} + \lVert F_1\rVert_{\mathrm{row},q})\}$ with $T_r$ from Lemma 4.3, bounding $\lVert\eta(t)\rVert_p \le (1/r)(e^{(\lVert F_0\rVert_{\infty} + \lVert F_1\rVert_{\mathrm{row},q})t}/r)^{N}$.

   approximationassumption

A linear generator with any inhomogeneity, a lift map, a readout map, and an error bound as a function of the truncation parameter.

given  the Fourier ODE  du/dt = G_0 + G_1 e^{iu},  u(0) = u_0,  u(t) in C^n,
       G_0 in C^n,  G_1 in C^{n x n},  e^{iu} the vector with entries {e^{iu_j}},
       the Fourier coefficients d_j of the readout g,
       a truncation level N

# rescale first
choose  nu > 0 large enough that  gamma := ||e^{ix_0}|| = ||e^{iu_0}|| / nu  <  1
set     x_j = u_j + i ln(nu),   j = 1..n
        # the rescaled ODE is  dx/dt = F_1 e^{ix} + F_0,
        # with  F_1 = nu G_1  and  F_0 = G_0
        # mu~_0 and R_p are invariant under this rescaling
set     c_j = nu^{|j|} d_j
        # f(x) = sum_j c_j e^{i x . j}  then satisfies  f(x) = g(u)

# lift onto the Fourier tower, not the monomial tower
set     Psi_j(x) = [e^{ix}]^{tensor j},   j = 1, 2, ...
        # expanding the same equation in monomials leaves the coefficient
        # matrix non-sparse; in the single-variable illustration
        #   d(e^{ix})/dt = i F_0 e^{ix} + i F_1 e^{2ix}
        # each row of the Fourier coefficient matrix has two non-zero entries
form    F~_0 = diag(F_0)  in C^{n x n}
        F~_1 in C^{n x n^2}, the rows of F_1 laid out blockwise
        # d(e^{ix})/dt = i F~_1 Psi_2 + i F~_0 Psi_1
form    B^(1)_{j+1} = i F~_1 (x) I^(x)(j-1) + I (x) i F~_1 (x) I^(x)(j-2)
                      + ... + I^(x)(j-1) (x) i F~_1
        B^(0)_j     = the same expression with F~_0 in place of F~_1
        # recurrence:  dPsi_j/dt = B^(1)_{j+1} Psi_{j+1} + B^(0)_j Psi_j
        # so  dPsi/dt = L^T Psi  is block upper bidiagonal

# truncate at level N
keep    the first N blocks:  dPsi^(N)/dt = L_N^T Psi^(N),  initial data Psi^(N)(x_0)
        # L_N^T is the top-left block of L^T; its last block row is
        #   ( 0 ... 0  B^(0)_N ),  i.e. the coupling B^(1)_{N+1} is dropped,
        # and that is where  eta_k = Psi_k - Psi^(N)_k  enters
pad     Psi^(N)(t) = direct sum over j = 1..N of  e_1^(x)(N-j) (x) Psi_j(t),
        in C^{N n^N}
        # blocks of unequal size complicate the circuit

hand the truncated linear ODE, its initial data Psi^(N)(x_0) and the rescaled
     readout coefficients c to the layer below
     # which bound eta obeys depends on the regime: Theorem 4.2 for all time
     # under mu~_0 >= 0 and R_p < 1, Theorem 4.4 on [0, Tmax] without them

Chen, Motee and Sun price nothing algorithmic: their Theorem 4.1 bounds each entry of the order-$N$ truncation's first block classically by $C_1 N^{-3/2}e^{D_1(t-\tilde{T}_{CF})N}$ up to a horizon $\tilde{T}_{CF}$, with $C_1$, $D_1$, $\tilde{T}_{CF}$ fixed by the field's frequencies and initial state; no query or gate count appears. The query count is Katz, Muraleedharan and Alase's: their informal Theorem 6.3 gives, under dissipativity, $\tilde{O}\left(\frac{T^{3/2}\lVert G_0\rVert_\infty\lVert d\rVert}{\varepsilon}\left(1+\frac{\lVert G_1\rVert_2}{\lVert G_1\rVert_{\mathrm{row},q}}\right)\left(1+\frac{\tilde\mu_0}{\lVert G_1\rVert_{\mathrm{row},q}}\right)^K\right)$ queries to oracles for the ODE's $n$-dimensional coefficients $G_0,G_1$: $T$ the final time, $\varepsilon$ the accuracy, $K$ the Fourier degree of the readout and $d$ its coefficient vector, $\tilde\mu_0$ the dissipativity parameter, $p$ and $q$ Hölder conjugates. That is $\mathrm{polylog}(n)$, hence exponentially faster than classical, only once all six of $T$, $\lVert G_0\rVert_\infty$, $\lVert d\rVert$, $1/\varepsilon$, $\lVert G_1\rVert_2/\lVert G_1\rVert_{\mathrm{row},q}$ and $\tilde\mu_0/\lVert G_1\rVert_{\mathrm{row},q}$ are themselves $\mathrm{polylog}(n)$; they call the $T^{3/2}$ scaling suboptimal. Without dissipativity (Theorem 6.7) the count loses $T$, $\lVert G_0\rVert_\infty$ and $\lVert G_1\rVert_{\mathrm{row},q}$, growing instead with the rescaling parameter $r$, $\lVert d\rVert_q$, $\lVert e^{iu_0}\rVert_p$ and $1/\varepsilon$.