Recursive quantum algorithm to find the lowest eigenstate of a general Hamiltonian

We propose a quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to $\simeq 1$. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called "Demon-like algorithmic cooling (DLAC)", recently proposed in [Nature Photonics 8, 113 (2014)]. The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best "cooling" with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number $n_c$ of iterations is proportional to $\simeq {\cal O}(D^{-1}\epsilon^{-0.19})$, where $D$ is the difference between the two lowest eigenvalues, and $\epsilon$ is an error defined as the probability that the finally obtained system state is in an unexpected (i.e. not the lowest) eigenstate.