Algorithm 2: Short-step IF-QIPM

1:Choose $ϵ>0$, $\delta \in (0,1)$, $\theta \in (0,{\theta }_0)$, $\beta \in (0,1)$ and $\sigma =(1-\frac{\beta }{\sqrt{n}})$. 2:$k\leftarrow 0$ 3:Choose initial feasible interior solution $(x^0,y^0,s^0)\in \mathcal{N}\left(\theta \right)$ 4:while$(x^k,y^k,s^k)\notin {\mathcal{PD}}_{ϵ}$do 5:   ${\mu }^k\leftarrow \frac{{\left(x^k\right)}^T{s}^k}{n}$ 6:   ${ϵ}_{OSS}^k\leftarrow \delta {\parallel r_c^k\parallel }_2/\left(\sqrt{2}{∥\left[\begin{array}{cc}{S}^{k}V+X^{k}Q{V}^k& -X^k{A}^T\end{array}\right]∥}_2\right)$ 7:   $({\lambda }^k,\mathsf{\Delta }{y}^k)\leftarrow$solve system (6) with accuracy ${ϵ}_{OSS}^k$ quantumly 8:   $\mathsf{\Delta }{x}^k=V{\lambda }^k$ and $\mathsf{\Delta }{s}^k=-A^T\mathsf{\Delta }{y}^k$ 9:   $(x^{k+1},y^{k+1},s^{k+1})\leftarrow (x^k,y^k,s^k)+(\mathsf{\Delta }{x}^k,\mathsf{\Delta }{y}^k,\mathsf{\Delta }{s}^k)$ 10:   $k\leftarrow k+1$ 11:end while 12:return$(x^k,y^k,s^k)$