Quantum simulation of excited states with the transcorrrelated Hamiltonian: higher accuracy with fewer qubits

In this work, we use the canonical transcorrelated theory to construct effective and compact molecular Hamiltonians, which can make accurate quantum simulation of ground and excited states on Noisy Intermediate-Scaling Quantum (NISQ) devices much cheaper. This approach involves a unitary transformation of the Hamiltonian with a geminal F12 operator which is a function of the inter-electronic distances. The coulomb singularities appearing in the regular Hamiltonian is removed after this procedure, as a result of which, any many-body method (UCC, FCI) with this effective Hamiltonian requires a fewer number of qubits to obtain accurate energies. We tested this approach for simulating ground states of small molecules earlier (Phys. Chem. Chem. Phys., 2020, 22, 24270-24281) and got very encouraging results which motivated us to consider excited states as well. Since, excited states are usually much more entangled than ground states, the Hamiltonian needs to be dressed with generalized geminal excitation operators unlike quasi-double geminal excitations that we used for the ground state simulation. This approach enables us to drastically reduce the number of qubits and circuit depth while maintaining the desired accuracy for both ground and excited-state properties.