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The recent developments of quantum computing present potential novel pathwaysfor quantum chemistry, as the increased computational power of quantumcomputers could be harnessed to naturally encode and solve electronic structureproblems. Theoretically exact quantum algorithms for chemistry have beenproposed (e.g. Quantum Phase Estimation) but the limited capabilities ofcurrent noisy intermediate scale quantum devices (NISQ) motivated thedevelopment of less demanding hybrid algorithms. In this context, theVariational Quantum Eigensolver (VQE) algorithm was successfully introduced asan effective method to compute the ground state energy of small molecules. Thecurrent study investigates the Folded Spectrum (FS) method as an extension tothe VQE algorithm for the computation of molecular excited states. It providesthe possibility of directly computing excited states around a selected targetenergy, using the same ansatz as for the ground state calculation. Inspired bythe variance-based methods from the Quantum Monte Carlo literature, the FSmethod minimizes the energy variance, thus requiring a computationallyexpensive squared Hamiltonian. We alleviate this potentially poor scaling byemploying a Pauli grouping procedure, identifying sets of commuting Paulistrings that can be evaluated simultaneously. This allows for a significantreduction of the computational cost. We apply the FS-VQE method to smallmolecules (H$_2$,LiH), obtaining all electronic excited states with chemicalaccuracy on ideal quantum simulators.

Folded Spectrum VQE : A quantum computing method for the calculation of molecular excited states