Optimal fidelity estimation for density matrix

Fidelity estimation is a necessary tool for evaluating noise in quantum measurement and quantum computation. The traditional fidelity estimation is to calculate the distance between two density matrices by employing direct fidelity estimation, which consumes too much copies of state. To reduce the number of copies of the state, we develop optimal fidelity estimation by proposing an optimal model. It calculates the minimum number of copies of state given a fixed value for the fidelity deviation. The result shows it saves a large number of copies of state compared with traditional approach (Direct Fidelity estimation) that is developed several years ago.The number of copies of the state employed increases slower than linear increase with increase of the dimension of density matrix when pauli measurement basis is applied. In addition, it consumes roughly a constant number of copies of the state with the increase of dimension of density matrix when the measurement bases are freely chosen.