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Table 8 Phase parameters required to simulate the time evolution of the vertical hopping \(h_{3}\) and \(h_{8}\), see Eq. (108c)

From: Superconducting circuit architecture for digital-analog quantum computing

Vertical Hopping
Operator \(\tilde{\varphi }^{(j)}_{1}\) \(\tilde{\varphi }^{(j)}_{2}\)
\(U_{(9,11)}^{{y,y}^{\dagger }}U_{(4,6)}^{{y,y}^{\dagger }}\) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{1}= \pi \) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=\pi \)
\(U_{10}^{x}U_{5}^{x}\) \(\tilde{\varphi }^{(5),(10)}_{1}= 2\pi \) \(\tilde{\varphi }^{(5),(10)}_{2}=\pi \)
\(\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{4}^{x}\sigma _{5}^{y}+\sigma _{9}^{x}\sigma _{10}^{y})\frac{t}{n} )\) \(\tilde{\varphi }^{(4)}_{1}= 1/2\pi \)
\(\tilde{\varphi }^{(9)}_{1}= 3/2\pi \)
\(\tilde{\varphi }^{(4)}_{2}=3/2\pi \)
\(\tilde{\varphi }^{(9)}_{2}=3/2\pi \)
\(U_{5}^{x^{\dagger }}U_{10}^{x^{\dagger }} \) \(\tilde{\varphi }^{(5),(10)}_{1}= \pi \) \(\tilde{\varphi }^{(5),(10)}_{2}=2\pi \)
\(U_{(4,6)}^{y,y}U_{(9,11)}^{y,y}\) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{1}={2} \pi \) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{2}={2}\pi \)
\(U_{(9,11)}^{{x,x}^{\dagger }}U_{(4,6)}^{{x,x}^{\dagger }}\) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{1}=\pi \) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=2\pi \)
\(U_{10}^{y^{\dagger }}U_{5}^{y^{\dagger }}\) \(\tilde{\varphi }^{(5),(10)}_{1}=\pi \) \(\tilde{\varphi }^{(5),(10)}_{2}=\pi \)
\(\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{4}^{y}\sigma _{5}^{x}+\sigma _{9}^{y}\sigma _{10}^{x})\frac{t}{n} )\) \(\tilde{\varphi }^{(4)}_{1}=3/2\pi \)
\(\tilde{\varphi }^{(9)}_{1}=1/2\pi \)
\(\tilde{\varphi }^{(4)}_{2}=3/2\pi \)
\(\tilde{\varphi }^{(9)}_{2}=3/2\pi \)
\(U_{5}^{y}U_{10}^{y} \) \(\tilde{\varphi }^{(5),(10)}_{1}=2\pi \) \(\tilde{\varphi }^{(5),(10)}_{2}=2\pi \)
\(U_{(4,6)}^{x,x}U_{(9,11)}^{x,x}\) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{1}=2\pi \) \(\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=\pi \)