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

Question

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?

Answer 1

With help from Narendra N. Hegade and Yongcheng Ding I succeed in finding out 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.

Finally I got some results different from those in the referred articles, but work.