Quantum sampling algorithms for quantum state preparation and matrix block-encoding

The problems of quantum state preparation and matrix block-encoding are ubiquitous in quantum computing: they are crucial parts of various quantum algorithms for the purpose for initial state preparation as well as loading problem relevant data. We first present an algorithm based on QRS that prepares a quantum state $|\psi_f\rangle \propto \sum^N_{x=1} f(x)|x\rangle$. When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on $f$ are met. When the preparation of the reference state is not the dominant cost, and the function $f$ and relevant properties are efficiently computable or provided otherwise with cost $o(N)$, the QRS-based method outperforms the generic state preparation algorithm, which has cost $O(N)$. We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix $A = \sum_{ij} A_{ij} |i\rangle \langle j|$. We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements $A_{{\mathbf{ij}}}= 1/\|{\mathbf{i}}-{\mathbf{j}}\|^2$, which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.