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We propose a quantum algorithm to obtain the lowest eigenstate of anyHamiltonian simulated by a quantum computer. The proposed algorithm begins withan arbitrary initial state of the simulated system. A finite series oftransforms is iteratively applied to the initial state assisted with anancillary qubit. The fraction of the lowest eigenstate in the initial state isthen amplified up to $\simeq 1$. We prove that our algorithm can faithfullywork for any arbitrary Hamiltonian in the theoretical analysis. Numericalanalyses are also carried out. We firstly provide a numericalproof-of-principle demonstration with a simple Hamiltonian in order to compareour scheme with the so-called "Demon-like algorithmic cooling (DLAC)", recentlyproposed in [Nature Photonics 8, 113 (2014)]. The result shows a good agreementwith our theoretical analysis, exhibiting the comparable behavior to the best"cooling" with the DLAC method. We then consider a random Hamiltonian model forfurther analysis of our algorithm. By numerical simulations, we show that thetotal number $n_c$ of iterations is proportional to $\simeq {\calO}(D^{-1}\epsilon^{-0.19})$, where $D$ is the difference between the two lowesteigenvalues, and $\epsilon$ is an error defined as the probability that thefinally obtained system state is in an unexpected (i.e. not the lowest)eigenstate.

A quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian assisted with an ancillary qubit system