of the $(n-1)$-th order LPF.

Shortcut to adiabaticity for KPO {#sec-shortcut}
================================

Here we derive the approximate counterdiabatic Hamiltonian given by Eqs. (11) and (12) in the main text and provide numerical evidence for its validity.

First, using the completeness relation $\displaystyle \sum_n |\phi_n \rangle \langle \phi_n |=I$ ($I$ is the identity operator), the counterdiabatic Hamiltonian in Eq. (10) in the main text is rewritten as follows: $$\begin{aligned}
H_{\mathrm{counter}} = 
\frac{i}{2} \sum_n \left(
|\dot{\phi}_n \rangle \langle \phi_n| - |\phi_n \rangle \langle \dot{\phi}_n|
\right).
\label{eq-Hcounter0}\end{aligned}$$

Among $\{ |\phi_n \rangle \}$, we are interested only in the following even cat state: $$\begin{aligned}
|{C_+ (p(t))} \rangle = 
\frac{\left| {\sqrt{p(t)/K}} \right\rangle + \left| {-\sqrt{p(t)/K}} \right\rangle}{\sqrt{2(1+e^{-2p(t)/K})}}.\end{aligned}$$ Note that $\displaystyle H_{\scriptsize{\mbox{KPO}}}(t) |C_+ (p(t)) \rangle = \frac{p(t)^2}{2K} |C_+ (p(t)) \rangle$, and therefore the even cat state is one of the energy eigenstates. Disregarding the energy eigenstates other than $|C_+ \rangle$, the counterdiabatic Hamiltonian in Eq. (\[eq-Hcounter0\]) is approximately given by $$\begin{aligned}
H_{\mathrm{counter}} \approx 
\frac{i}{2}  \left(
|\dot{C}_+ \rangle \langle C_+| - |C_+ \rangle \langle \dot{C}_+|
\right).
\label{eq-Hcounter}\end{aligned}$$

Using the odd cat state $$\begin{aligned}
|C_- (p(t)) \rangle = 
\frac{\left| {\sqrt{p(t)/K}} \right\rangle - \left| {-\sqrt{p(t)/K}} \right\rangle}{\sqrt{2(1-e^{-2p(t)/K})}},\end{aligned}$$ $|\dot{C}_+ \rangle$ becomes $$\begin{aligned}
|\dot{C}_+ \rangle = 
-\frac{\dot{p}}{2K} \tanh \! \frac{p}{K} \, |C_+ \rangle
