Full quantum eigensolvers based on variance

The advancement of quantum computation paves a novel way for addressing the issue of eigenstates. In this paper, two full quantum eigenvalue solvers based on quantum gradient descent are put forward. Compared to the existing classical-quantum hybrid approaches such as the variance-variational quantum eigenvalue solver, our method enables faster convergent computations on quantum computers without the participation of classical algorithms. As any eigenstate of a Hamiltonian has zero variance, this paper takes the variance as the objective function and utilizes the quantum gradient descent method to optimize it, demonstrating the optimization of the objective function on the quantum simulator. With the swift progress of quantum computing hardware, the two variance full quantum eigensolvers proposed in this paper are anticipated to be implemented on quantum computers, thereby offering an efficient and potent calculation approach for solving eigenstate problems. Employing this algorithm, we showcase 2 qubits of deuterium and hydrogen molecule. Furthermore, we numerically investigate the energy and variance of the Ising model in larger systems, including 3, 4, 5, 6, and 10 qubits.