The Quantum Zeno Monte Carlo method for the Hamiltonian eigenstate properties

In this work, we develop the Quantum Zeno Monte Carlo method to find Hamiltonian eigenstate properties such as the ground state energy and the Green's function. Our method is classical-quantum hybrid algorithm and is based on the quantum zeno effect, which is the phenomenon that repeated measurements slow down the speed of transition. Our method finds Hamiltonian eigenstates by implementing the quantum zeno-like procedure using the Monte Carlo method. Unlike popular variational algorithms, our method does not rely on specific ansatz, so it is free of barren plateau problem. Moreover, we show that our method can find Hamiltonian eigenstate properties within a polynomial quantum cost, if the Hamiltonian satisfies moderate conditions. We demonstrate our method in two ways. First, by applying our method for a small system using the quantum processing unit and the noisy simulator, we show that the method is applicable for current NISQ hardware. Second, we verify the polynomial solvability of our method by applying it to the half-filled Hubbard model with various sizes through exact simulator.