Recently I have read two articles about the Digitized Adiabatic Quantum Computing(DAQC), and tried to factorize $35=5\times7$ and $2479=67\times37$. But some problems came when trying to solve the coefficient of the CD term.
problems for 35
In section IV of the paper "Shortcuts to Adiabaticity in Digitized Adiabatic Quantum Computing", there is a result about the variational coefficient $\alpha_1$, which is $$\alpha_1=\frac{1}{4}\frac{h^2_z+J^2_0}{\lambda^2(h_z^4J_0^4+3h_z^2J^2_0)+(1-\lambda)^2h_x^2(h_z^2+4J_0^2)},$$ with $H_{init}=\sum h_x^j\sigma_x^j, H_{final}=\sum (h_z^j\sigma_z^j+J_0\sigma_z^j\sigma_z^{j+1}), \sigma^{N+1}=\sigma^{0}$.
However, for the simplest example with $N=2, H_{final}=\frac{1}{4}(\sigma_z^0\sigma_z^1+\sigma_z^1\sigma_z^0), h_z=0,J_0=\frac{1}{4}$, I get $$\alpha_1=-\frac{1}{\lambda^2+16(1-\lambda)^2h_x^2}.$$It works for my code to factorize $35$, but it doesn't fit the expression above.
problems for 2479
Moreover, in the paragrah after equation (10) of the paper "Digitized-Adiabatic Quantum Factorization", it provides $$\alpha_1=\frac{0.0830}{h_x^2(1-\lambda)^2+5.0112\lambda^2},$$while $H_{final}$ defined in equation (8).
With $$H_0=(1-\lambda)H_{init}+\lambda H_{final}, A\approx i\alpha_1[H_0,\partial_\lambda H_0]=i\alpha_1[H_{init},H_{final}],\\G=\partial_\lambda H_0+i[A,H_0],\alpha_1=\text{argmin}\text{Tr}(G^2)$$ I get a similar but different $\alpha_1^*\approx -\alpha_1$ $$\alpha^*_1=-\frac{185/2228}{h_x^2(\lambda-1)+5.0112\lambda^2}\approx -\alpha_1.$$
question
It looks like that there are some terrible mistakes about $\alpha_1$, I tried checking my results of $\alpha_1$ with the symbolic operation of Octave but got nothing.
Maybe I did something wrong before I solve the equations $\alpha_1=\text{argmin}\text{Tr}(G^2)$?
Or is there any general expressions of $\alpha_1$ so that I can easily find my mistake?