Second order at the same stability class as backward Euler, so it is more accurate at equal step size, but the implicit solve does not disappear on a quantum computer — it is the layer below, and what this one hands down to it is the assembled system. Second order buys a larger $h$ at the same accuracy, so written out step by step the loop turns fewer times than backward Euler's; it is the same loop, one linear solve per turn with the previous turn's state as its right-hand side. Being A-stable removes the step-size restriction, not the repetition. Second order still leaves a polynomial dependence on $1/\varepsilon$; only propagator-series or spectral discretizations reach $\log(1/\varepsilon)$. Dong, Li and Xue encode diagonal Padé approximations of the matrix exponential into a large, block-sparse linear system solved via a quantum linear system algorithm, but state no complexity for the $(1,1)$ case specifically.

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

1. Linear ODE system → Linear system Ax = baverage the generator across the step

   The step averages the generator at the two ends, $(I - \frac{h}{2}A)u_{k+1} = (I + \frac{h}{2}A)u_k + \frac{h}{2}(b_k + b_{k+1})$, which as a rational approximation of $e^{hA}$ is the $(1,1)$ diagonal Padé approximant $R_{11}(Ah) = [D_{11}(Ah)]^{-1}N_{11}(Ah)$ of Dong, Li and Xue's Definition 3.1. approximation: The propagator $e^{hA}$ is replaced by a ratio of degree-one polynomials in $Ah$, whose denominator $D_{11}(Ah) = I - \frac{h}{2}A$ is the matrix each implicit step solves against; second order, so the precision dependence stays polynomial in $1/\varepsilon$. For the autonomous case they treat, a step of this form, $\hat{x}(sh) = R(Ah)\hat{x}((s-1)h) + (R(Ah) - I)A^{-1}b$, is encoded as block rows of one large block-sparse linear system, and that assembled system is what this layer hands down. assumption: The denominator must be non-singular, which the source assures only when $p$ and $q$ are large enough or when the eigenvalues of $A$ are all negative.

   approximationassumption

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

Crank-Nicolson is the (1,1) member of the family Dong, Li and Xue (arXiv:2504.06948) do encode, but k = 1 is the one order their results never reach: Theorem 3.6 is stated for k ≥ 3, Table 1 begins at k = 5, and every experiment in Section 5.2 runs at k = 9 or higher. The words Crank-Nicolson and trapezoidal appear nowhere in the paper.

given  A, b, u_0, step size h, horizon T

for k = 0 ... T/h - 1:
    # the generator averaged at the two ends of the step
    # one linear solve per turn, the previous turn's state as its right-hand side
    solve (I - (h/2)A) u_{k+1} = (I + (h/2)A) u_k + (h/2)(b_k + b_{k+1})

return u_{T/h}

# second order at backward Euler's stability class, so a larger h at the same
# accuracy and fewer turns of this loop -- the same loop, not a shorter one.
# A-stability removes the step-size restriction, not the repetition.

Dong, Li and Xue's complexity theorems require diagonal Padé order $k \ge 3$, and Crank–Nicolson is the $(1,1)$ approximant — so no verified source states its end-to-end cost, and the field says so. What second order buys — a larger $h$, hence fewer turns of the same solve-per-step loop — is an accuracy statement, recorded in conditions, not a stated complexity.