  During the noisy intermediate-scale quantum (NISQ) era, quantum computational
approaches refined to overcome the challenge of limited quantum resources are
highly valuable. However, the accuracy of the molecular properties predicted by
most of the quantum computations nowadays is still far off (not within chemical
accuracy) compared to their corresponding experimental data. Here, we propose a
promising qubit-efficient quantum computational approach to calculate the
harmonic vibrational frequencies of a large set of neutral closed-shell
diatomic molecules with results in great agreement with their experimental
data. To this end, we construct the accurate Hamiltonian using molecular
orbitals, derived from density functional theory to account for the electron
correlation and expanded in the Daubechies wavelet basis set to allow an
accurate representation in real space grid points, where an optimized compact
active space is further selected so that only a reduced small number of qubits
is sufficient to yield an accurate result. To justify the approach, we
benchmark the performance of the Hamiltonians spanned by the selected molecular
orbitals by first transforming the molecular Hamiltonians into qubit
Hamiltonians and then using the exact diagonalization method to calculate the
results, regarded as the best results achievable by quantum computation.
Furthermore, we show that the variational quantum circuit with the
chemistry-inspired UCCSD ansatz can achieve the same accuracy as the exact
diagonalization method except for systems whose Mayer bond order indices are
larger than 2. For those systems, we demonstrate that the heuristic
hardware-efficient RealAmplitudes ansatz, even with a shorter circuit depth,
can provide a significant improvement over the UCCSD ansatz, verifying that the
harmonic vibrational frequencies could be calculated accurately by quantum
computation in the NISQ era.
