Table 3 Jordan–Wigner transformation for the main operators appearing in the Hamiltonian and in our ADAPT-VQE operator pool.
From: Nuclear shell-model simulation in digital quantum computers
Fermion operators | Qubit operators | |
|---|---|---|
\(n_p\) | \(a_p^{\dag }a_p\) | \(\dfrac{1}{2} (1-Z_p)\) |
\(h_{pqrs}\) | \(\begin{array}{l} a_p^{\dag }a_q^{\dag }a_r a_s \\ \quad + a_r^{\dag }a_s^{\dag }a_p a_q\end{array}\) | \(\begin{array}{l} \dfrac{1}{8} P_{rs}^{pq}\, ( -X_p X_q X_r X_s + X_p X_q Y_r Y_s \\ \quad \qquad - X_p Y_q X_r Y_s- X_p Y_q Y_r X_s \\ \quad \qquad - Y_p Y_q Y_r Y_s + Y_p Y_q X_r X_s \\ \quad \qquad - Y_p X_q Y_r X_s - Y_p X_q X_r Y_s)\end{array}\) |
\(T_{rs}^{pq}\) | \(\begin{array}{l} i(a_p^{\dag }a_q^{\dag }a_r a_s \\ \quad - a_r^{\dag }a_s^{\dag }a_p a_q)\end{array}\) | \(\begin{array}{l} \dfrac{1}{8} P_{rs}^{pq}\, ( -X_p Y_q Y_r Y_s - Y_p X_q Y_r Y_s \\ \quad \qquad + Y_p Y_q X_r Y_s + Y_p Y_q Y_r X_s \\ \quad \qquad +Y_p X_q X_r X_s + X_p Y_q X_r X_s \\ \quad \qquad - X_p X_q Y_r X_s - X_p X_q X_r Y_s)\end{array}\) |
\(h_{pq}\) | \(a_p^{\dag }a_q + a_q^{\dag }a_p\) | \(\dfrac{1}{2}\left( \prod _{n=p+1}^{q-1}Z_n\right) \left( X_p X_q+Y_q Y_p\right)\) |
\(T_{pq}\) | \(i(a_p^{\dag }a_q - a_q^{\dag } a_p)\) | \(\dfrac{1}{2}\left( \prod _{n=p+1}^{q-1}Z_n\right) \left( Y_p X_q-X_q Y_p\right)\) |
