Transition dipole moments in the Subspace Search Variational Quantum Eigensolver method.

In this article, as a continuation of "The lecture of Subspace-Search Variational Quantum Eigensolver method for beginners by Hikaru Wakaura | blueqat", we will calculate the transition dipole moments using the Subspace-Search Variational Quantum Eigensolver (SSVQE) method. In the SSVQE method, we look for an operator that transforms any ground set into a set of ground and excited states with a common operator. This makes it easy to create superpositions between the states sought. For the non-diagonal terms, we can use only the diagonal states

$\langle \Phi_j \mid H \mid \Phi_k \rangle = a + ib$

$a= \langle + \mid H \mid + \rangle - \frac{1}{2}(\langle \Phi_j \mid H \mid \Phi_j \rangle + \langle \Phi_k \mid H \mid \Phi_k \rangle)$

$b= \langle y+ \mid H \mid y+ \rangle - \frac{1}{2}(\langle \Phi_j \mid H \mid \Phi_j \rangle + \langle \Phi_k \mid H \mid \Phi_k \rangle)$

This is expressed as where $\mid + \rangle$ and $\mid y+ \rangle$ are the equal superposition of state j and state k, respectively, and the equal superposition including the phase $\pi/2$ . These are achieved by superposition of the initial states. This time, as before, the cluster and Hamiltonian depths are set to 2, and the optimization method is BFGS. Figure 1 shows the normal logarithm (normal logarithmic error) of the difference between the energy levels in the ground state, triplet state, singlet state, and two-electron excited state and the levels calculated for the exact solution in the STO-3G basis. Although there are no levels that exceed the chemical accuracy, the overall logarithmic error is only about -1.5. Table 2 shows the transition dipole moments between them. The transition dipole moment in molecular hydrogen should be zero due to its shape, but it is too large to be ignored. However, the accuracy of VQE was worse than that of VQE, and it did not reach the exact solution.

Table 1 Energy levels(Hartree) and logarithmic error.

Table 2 Transition dipole moments(Debye) between levels.

Deriving dipole moments using Subspace Search Variational Quantum Eigensolver method.

Hikaru Wakaura