Quantum circuit design for mixture and preparation of arbitrary pure and mixed quantum states

This paper addresses the challenge of preparing arbitrary mixed quantum states, an area that has not been extensively studied compared to pure states. Two circuit design methods are presented: one via a mixture of pure states and the other via purification. A novel strategy utilizing the Cholesky decomposition is proposed to improve both computational efficiency during preprocessing and circuit efficiency in the resulting circuits, offering significant advantages, especially when the targeted density matrix is low-ranked or sparse. By leveraging the incomplete Cholesky decomposition with threshold dropping, we also propose an appealing strategy for generating a high-fidelity approximation of the targeted density matrix, enabling substantial efficiency enhancement at the cost of mild fidelity loss. Additionally, as a closely related issue, we prove the "no-superposing theorem": given a certain number of arbitrary unknown pure states as input, it is impossible to devise an operation that produces an output state as the superposition of the input states with predefined coefficients unless all but one of the coefficients vanish.