**Smoothness of the solution, and the rate is a function of how much of it there is.** Lemma 1, quoted by Childs and Liu from Gheorghiu, covers a solution $\hat{x}(t) \in C^{r+1}(-1,1)$: $\max_{t}\|\hat{x}(t) - x(t)\| \leq C\max_{t}\|\hat{x}^{(n+1)}(t)\|/n^{r-2}$ for a constant $C$ independent of $n$. Lemma 2 covers $\hat{x}(t) \in C^{\infty}(-1,1)$ and is the one the exponential precision needs: $\max_{t}\|\hat{x}(t) - x(t)\| \leq \sqrt{2/\pi}\max_{t}\|\hat{x}^{(n+1)}(t)\|(e/2n)^{n}$. In Childs and Liu's summary, "the convergence behavior of the spectral method is related to the smoothness of the solution". Chebyshev rather than Fourier is a stated choice with a stated reason: "a Fourier series provides an appropriate basis for periodic problems, whereas Chebyshev polynomials can be applied more generally", and "since general linear ODEs are non-periodic, and interpolation facilitates constructing a straightforward linear system, we develop a quantum algorithm based on the Chebyshev pseudo-spectral method". The Chebyshev-Gauss-Lobatto nodes are likewise chosen rather than assumed — they "achieve the highest convergence rate among all schemes with the same number of nodes", and they have the convenient property $T_k(t_l) = \cos(kl\pi/n)$.

The generator $A(t)$, the interval $[0,T]$, an error tolerance $\varepsilon$, and a target algebraic form.

1. Linear ODE system → Linear system Ax = bsolve for the Chebyshev coefficients

   One system for a whole interval. The unknowns are the Chebyshev coefficients of Eq. (2.1), $x_i(t) = \sum_{k=0}^{n}c_{i,k}T_k(t)$, and what pins them down is Eq. (2.2) — $dx(t_l)/dt = A(t_l)x(t_l) + f(t_l)$ required to hold at each node — together with the initial condition $x_i(t_0) = \gamma_i$ of Eq. (2.3). The derivative never leaves the basis: the Chebyshev identity $2T_k(t) = T'_{k+1}(t)/(k+1) - T'_{k-1}(t)/(k-1)$ of Eq. (2.6) gives the upper-triangular differentiation matrix of Eq. (2.8), $[D_n]_{kj} = 2j/\sigma_k$ for $k+j$ odd and $j > k$ with $\sigma_0 = 2$ and $\sigma_k = 1$ otherwise, so that $c'_{i,k} = \sum_j [D_n]_{kj}c_{i,j}$ and collocation becomes Eq. (2.10), $\sum_k T_k(t_l)c'_{i,k} = \sum_j A_{ij}(t_l)\sum_k T_k(t_l)c_{j,k} + f(t_l)_i$. approximation: The exact solution is replaced by a degree-$n$ Chebyshev polynomial, and the error is governed by how smooth the solution is rather than by any step size. Lemma 2, for $\hat{x} \in C^{\infty}$: $\max_{t\in[-1,1]}\|\hat{x}(t)-x(t)\| \leq \sqrt{2/\pi}\max_{t}\|\hat{x}^{(n+1)}(t)\|(e/2n)^{n}$. Lemma 1, for $\hat{x} \in C^{r+1}$: $\max_{t}\|\hat{x}(t)-x(t)\| \leq C\max_{t}\|\hat{x}^{(n+1)}(t)\|/n^{r-2}$, with $C$ independent of $n$ — the same construction, and the exponential rate is gone. assumption: The solution is smooth. Childs and Liu keep this in the foreground rather than the fine print: "our algorithm must assume that the solution is smooth", listed in the Discussion as an open problem. **Why these nodes and not any others.** The interpolation points are the Chebyshev-Gauss-Lobatto quadrature nodes $t_l = \cos(l\pi/n)$, "since these nodes achieve the highest convergence rate among all schemes with the same number of nodes"; and because $T_k(t_l) = \cos(kl\pi/n)$, evaluating the basis at them costs nothing. Their ordering $1 = t_0 > t_1 > \cdots > t_n = -1$ is load-bearing too — $x(t_0)$ and $x(t_n)$ "are the initial and final states", so both the condition that goes in and the answer that comes out sit exactly on a node.

   approximationassumption

The discrete object, its truncation-error bound, and its conditioning bound.

Read in full: arXiv:1901.00961, the only paper this node cites, carries no numerics anywhere — no figure, no table, no computed value, no dataset and no simulator. Its Appendix B is titled "An example of the quantum spectral method" and is a structural display rather than a run: it takes $d = 1$, $m = 3$, $n = 2$, $p = 1$ and writes out the shape of the linear system with $A(t)$, $f(t)$ and $\gamma$ left symbolic. The register row for 1901.00961 records that as a full-text read.

given  A(t), f(t), initial value gamma, interval [-1, 1], series length n

take the n+1 Chebyshev-Gauss-Lobatto nodes
    t_l = cos(l*pi/n)   for l = 0 ... n,   so  1 = t_0 > t_1 > ... > t_n = -1
    # highest convergence rate of any scheme with this many nodes;
    # t_0 and t_n are the initial and final times, which is what lets the
    # initial condition and the answer both be read off a node

write each component as an undetermined truncated Chebyshev series
    x_i(t) = sum_{k=0..n} c_{i,k} T_k(t)

express the derivative in the same basis
    dx_i/dt = sum_{k=0..n} c'_{i,k} T_k(t)
    c'_{i,k} = sum_{j=0..n} [D_n]_{kj} c_{i,j}
    [D_n]_{kj} = 2j/sigma_k   for k+j odd and j > k,  else 0
    sigma_0 = 2,  sigma_k = 1 for k >= 1
    # D_n is upper triangular; it comes from 2T_k = T'_{k+1}/(k+1) - T'_{k-1}/(k-1)

collocate: demand the ODE hold at every node
    for l = 0 ... n, for i = 0 ... d-1:
        sum_k T_k(t_l) c'_{i,k} = sum_j A_ij(t_l) sum_k T_k(t_l) c_{j,k} + f_i(t_l)
    # T_k(t_l) = cos(k*l*pi/n), so no evaluation of T_k is needed

add the initial condition as further rows
    x_i(1) = gamma_i        for i = 0 ... d-1

return the linear system in the unknowns c_{i,k}
    # one global system for the whole interval, not one system per step

What Childs and Liu state for the discretization by itself is **the number of terms**, not an end-to-end complexity — the latter belongs to the route that calls a solver on the assembled system. Under Lemma 2's $C^{\infty}$ hypothesis: "if the solution is in $C^{\infty}$, the spectral method approximates the solution to within $\varepsilon$ using only $n = \mathrm{poly}(\log(1/\varepsilon))$ terms in the Chebyshev series". Under Lemma 1's weaker hypothesis the same construction costs "$n = \mathrm{poly}(1/\varepsilon)$". That gap between the two is the whole reason a spectral discretization is worth building, and it is also the reason the smoothness hypothesis cannot be dropped quietly.