Stated for $n$-dimensional nonlinear dissipative ODEs. Krovi describes Xue et al., alongside Liu et al., as additionally requiring normality of the matrix that models dissipation.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, 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 systemembed the chain of homotopy orders

   Homotopy perturbation replaces Eq. (1) by the homotopy $H(\nu,p) = \frac{d\nu}{dt} - F_1\nu - pF_2\nu^{\otimes 2} = 0$ with $\nu(0,p) = u_{in}$, writes $\nu = \nu_0 + p\nu_1 + p^2\nu_2 + \cdots + p^c\nu_c$, and equates the terms with identical powers of $p$, which gives the chain of Eq. (6): $\frac{d\nu_0}{dt} = F_1\nu_0$ with $\nu_0(0) = u_{in}$, and $\frac{d\nu_i}{dt} = F_1\nu_i + F_2\sum_{j=0}^{i-1}\nu_j \otimes \nu_{i-1-j}$ with $\nu_i(0) = 0$; at $p = 1$ the solution is $\tilde{u} = \nu_0 + \nu_1 + \cdots + \nu_c$. That chain is what embeds. Write $\vec{y} = [\vec{y}_0,\vec{y}_1,\ldots,\vec{y}_c]$ with $\vec{y}_0 = [\nu_0 + \nu_1 + \cdots + \nu_c]$ and, for $1 \le i \le c$, $\vec{y}_{i,j} = \bigotimes_{k=0}^{i}\nu_{a_{i,j,k}}$ over the index vectors satisfying $a_{i,j,k} \ge 0$ and $i+1 \le \sum_{k=0}^{i}(a_{i,j,k}+1) \le c+1$, so that $\beta_0 = 1$ and $\beta_i = \sum_{k=i}^{c}\binom{k}{i}$ blocks sit at level $i$ and the whole space has dimension $N = \sum_{i=0}^{c}n^{i+1}\beta_i = (n+1)^{c+1} - 1 - cn$. Differentiating $\vec{y}_{i,j}$ by Eq. (16) gives a linear part inside level $i$ and, through $F_2$, terms every one of which lies in $\vec{y}_{i+1}$ — that closure is what makes the generator block bidiagonal, $A_{i,i} = I_{\beta_i}\otimes(\sum_{j=0}^{i}I_{n^j}\otimes F_1\otimes I_n^{\otimes i-j})$ with $A_{i,i+1}$ read off the $F_2$ terms and nothing below the diagonal, and level $c$ carries $A_{c,c}$ with no block above it. Setting $\vec{y}_{i,0} = \nu_0^{\otimes i+1}$ makes the lift of the initial condition Eq. (12), $y_{in} = [[u_{in}],[u_{in}^{\otimes 2},0,\ldots,0],\ldots,[u_{in}^{\otimes c+1}]]$. approximation: The embedding is finite only because the expansion stops at order $c$; the exact solution is $u(t) = \sum_{i=0}^{\infty}\nu_i(t)$. What is dropped is bounded through the Catalan recursion $\alpha_{i+1} = \sum_{j=0}^{i}\alpha_j\alpha_{i-j}$ with $\alpha_0 = 1$ and $\alpha_i = \frac{1}{i+1}\binom{2i}{i} < 4^i$, giving $\lVert \nu_i(t)\rVert < (4K_1)^i\lVert u_{in}\rVert \le K^{i+1}$ with $K_1 = \lVert u_{in}\rVert\lVert F_2\rVert/\lvert \mathrm{Re}(\lambda_1)\rvert$ and $K = 4K_1$, so the truncation error is the tail $\sum_{i=c+1}^{\infty}K^{i+1}$ and Lemma 9 reads $\lVert u(t) - \tilde{u}(t)\rVert \le \varepsilon$ once $c > \log_{1/K}\frac{1}{\varepsilon(1-K)}$. assumption: $F_1$ and $F_2$ are time-independent and $s$-sparse, $F_1$ is normal, and its eigenvalues satisfy $\mathrm{Re}(\lambda_n) \le \cdots \le \mathrm{Re}(\lambda_1) < 0$ — the dissipation the whole bound rests on, entering the induction as $\lVert \nu_0(t)\rVert \le \lVert e^{F_1t}\rVert \lVert u_{in}\rVert \le \lVert u_{in}\rVert$ and $\int_0^t \lVert e^{\mathrm{Re}(\lambda_1)(t-\tau)}\rVert\,d\tau \le (1 - e^{\mathrm{Re}(\lambda_1)t})/\lvert \mathrm{Re}(\lambda_1)\rvert$. The tail converges only for $K < 1$, and the step $\lVert \nu_i\rVert \le K^{i+1}$ additionally needs $K \ge \lVert u_{in}\rVert$, which the paper arranges by rescaling $u$ to $\zeta u$ with a constant that leaves $4\lVert u_{in}\rVert\lVert F_2\rVert/\lvert \mathrm{Re}(\lambda_1)\rvert$ unchanged.

   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  du/dt = F_1 u + F_2 u^{(x)2},  u(0) = u_in            # the quadratic ODEs, Eq.(1)
       F_1, F_2 time-independent and s-sparse
       F_1 normal, eigenvalues Re(lambda_n) <= ... <= Re(lambda_1) < 0
       homotopy truncation order c

# --- 1. homotopy perturbation: one nonlinear system becomes a chain -----------
#     construct the homotopy nu(t,p), Eq.(4):
#         H(nu,p) = dnu/dt - F_1 nu - p F_2 nu^{(x)2} = 0,   nu(0,p) = u_in
#     write nu = nu_0 + p nu_1 + p^2 nu_2 + ... + p^c nu_c and equate the terms
#     with identical powers of p, giving Eq.(6):
    dnu_0/dt = F_1 nu_0,                                       nu_0(0) = u_in
    dnu_i/dt = F_1 nu_i + F_2 sum_{j=0..i-1} nu_j (x) nu_{i-1-j},  nu_i(0) = 0
#     at p = 1:   utilde = nu_0 + nu_1 + ... + nu_c

# --- 2. embed that chain in linear ODEs, Eq.(8)-(12) -------------------------
    y_0     = [nu_0 + nu_1 + ... + nu_c]
    y_{i,j} = nu_{a_{i,j,0}} (x) nu_{a_{i,j,1}} (x) ... (x) nu_{a_{i,j,i}},   1 <= i <= c
              over the index vectors a_{i,j} with
                  a_{i,j,k} >= 0   and   i+1 <= sum_k (a_{i,j,k} + 1) <= c+1
    beta_0  = 1,   beta_i = sum_{k=i..c} binom(k,i)         # blocks at level i
    N       = sum_{i=0..c} n^{i+1} beta_i = (n+1)^{c+1} - 1 - cn

#     differentiating y_{i,j} gives a linear part inside level i and, through
#     F_2, terms that all lie in y_{i+1} -- that is the closure, and it is why A
#     is block bidiagonal with nothing below the diagonal:
    A_{i,i}   = I_{beta_i} (x) sum_{j=0..i} I_{n^j} (x) F_1 (x) I_n^{(x) i-j}
    A_{i,i+1} = the F_2 terms of that derivative,  i = 0 ... c-1
#     level c has A_{c,c} and nothing above it -- the truncation closes the system

#     fix y_{i,0} = nu_0^{(x) i+1}, so the lift of the initial condition is
    y_in = [[u_in], [u_in^{(x)2}, 0, ..., 0], ..., [u_in^{(x)c+1}, 0, ..., 0]]

# --- 3. what this layer hands down -------------------------------------------
return  dy/dt = A y,  y(0) = y_in       # finite-dimensional by construction
        the readout: the first block of y is utilde, so the answer is on |0,0>
        sparsity O(s c^2);  ||A|| <= (c+1)(||F_1|| + ||F_2||);  Re(gamma_i) < 0
        truncation error as a function of c:
            ||nu_i(t)|| < (4 K_1)^i ||u_in|| <= K^{i+1},
                K_1 = ||u_in|| ||F_2|| / |Re(lambda_1)|,   K = 4 K_1
            ||u(t) - utilde(t)|| <= eps   when K < 1 and
                c > log_{1/K} ( 1 / (eps (1-K)) )

# O_A is built from O(c) queries to O_F1 and O(1) queries to O_F2   (Lemma 6)
# solving dy/dt = A y with a quantum linear-ODE algorithm is the layer below

Xue, Wu and Guo state the algorithm's complexity as $O(g\eta T\,\mathrm{poly}(\log(nT/\varepsilon)))$, where $T$ is the evolution time, $n$ the dimension, $\varepsilon$ the allowed error, and $\eta$ and $g$ are quantities measuring the decay of the solution; the returned state is $\varepsilon$-close to the normalized exact solution with success probability $\Omega(1)$. As with the Carleman route, this is a complexity for the complete algorithm — the embedding together with the quantum linear-ODE solver it feeds — not a standalone cost for the lift. The abstract claims exponential improvement over the best classical algorithms or previous quantum algorithms in $n$ or $\varepsilon$; how the homotopy truncation order enters the bound is in the paper's full text and is not quoted here.

Xue and co-authors later report that applying quantum simulation to each step of the related homotopy analysis method makes complexity grow exponentially with the truncation order, and introduce a quantum-compatible linearization that maps the whole process into one system of linear PDEs so that complexity grows only polynomially with that order.