# Question about the counter diabatic(CD) term in Digitized Adiabatic Quantum Computing

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?

Narendra advices me to do some other examples like $$2893=11*263$$ and point out that we may use similar but different $$\alpha$$, Yongcheng tells me the same thing and suggests me to check the process of applying the matrice to the circuit construction.
For any $$H_{init}=\sum h_x\sigma_x$$ with the ground state $$|+\rangle^n$$, here is $$h_x<0$$. Thus I can directly deduce the CD term with the form of $$H_{final}$$. However I make a mistake about $$\sigma_x\sigma_z=-i\sigma_y=\sigma_z\sigma_x$$, and create a wrong local quantum circuit about CD term.