Efficient algorithm for fidelity estimation of two quantum states

The fidelity estimation between two quantum states is crucial for quantum computation and information science. However, an efficacious method for this, especially for mixed states and higher-dimensional density matrices, remains elusive. While there are many existing algorithms on computing the fidelity between two pure states, there is not much work on how to obtain the fidelity between two mixed states. Here, an efficient quantum algorithm for the fidelity estimation is proposed, based primarily on the density matrix exponentiation and interferometeric scheme for mixed states, with a time complexity of $O(\kappa^2N^2/\epsilon^7)$, where $N$ is the system size, $\kappa$ is the condition number of the density matrices and $\epsilon$ is a precision error. This algorithm may serve as a resource-efficient technique to deduce fidelity of any two (pure or mixed) unknown or known quantum states, when the density matrices of the quantum states commute with each other.