Quantum Subspace Methods for Quantum chemistry*

The Variational Quantum Eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with problem size. Increased depth can both degrade the accuracy of the results and reduce trainability due to the noises. Besides, near-term quantum devices have a limited number of qubits. This work proposes a novel approach to reduce ansatz circuit depth and the number of qubits by mutual-information-based permutation of qubits and iterative folding of ansatz into an effective Hamiltonian. This breaks the original VQE algorithm into a series of VQE simulations with shallower circuit depth and fewer qubits. Besides, to calculate excited states properties of molecules, we developed an efficient quantum Krylov subspace algorithm by leveraging the iterative growth of Krylov subspace. Such subspace methods remove the optimization problems suffered in the VQE-type of methods. The newly proposed quantum Krylov subspace algorithm is employed to study the excited-state properties of various systems on both simulators and quantum devices. The complexity of the quantum Krylov subspace method is also analyzed. We believe the newly developed quantum Krylov subspace algorithm provides a feasible tool for studying excited-state phenomena on quantum computers.