A State Distillation Method Using Quantum Imaginary Time Control for Solving the Generalized Eigenproblems

Finding eigenvalues of Hamiltonians has many important applications in generalized eigenproblems, such as calculating chemical reaction rates and approximating spectral densities of large matrices. While variational quantum methods (VQEs) and quantum Krylov subspace methods (QKS) have been used to estimate the ground and excited state energies of quantum many-body systems, different problems exist when computing excited states. VQEs face optimization problems, such as barren plateaus induced by high expressibility ansatz. QKS, whether it is QLanczos or QDavidson, faces the problem of long circuit depth and the requirement of extra ancilla qubits. Moreover, the number of measurements of the QLanczos increases exponentially with the expansion of the system. In this work, we proposed a state distillation method using imaginary time control which requires the quadratic form of the problem Hamiltonian and controls to adjust the order in which states vanish in imaginary time evolution. Using this state distillation, we can obtain states for constructing and solving a generalized eigenvalue problem to get the approximate eigenvalue and eigenvector of unknown excited states. The algorithm only needs an approximate energy of one of the eigenstates as the initial value to obtain excited states with energy higher than it. As the size of the system expands, it computes excited states with a quadratic increase in the number of measurements and little extra resources using the variational-ansatz based imaginary time method. In principle, using the state distillation method, our method is robust to control errors while solving the eigenspectrum of the n-local Hamiltonian matrix compared to the methods mentioned above, therefore more achievable for NISQ-era quantum computers.