Application of an upsampling algorithm to quantum state preparation of continuous and discrete probability distributions

Upsampling of time series data is often associated with classical fast Fourier transform analysis, but it also can be employed in the preparation of a quantum state vector. The quantum circuit of preparation derived from functional data sampled in this way exhibits exponential gate depth, like other divide-and-conquer algorithms, but is transformable to an equivalent circuit of polylogarithmic gate depth. In this study, we derive a polylogarthmic upsampling algorithm for construction of a state vector with amplitudes that are square roots of probabilities sampled from a continuous probability distribution having support over the entire real line. As examples, we prepare state vectors associated with Gaussian and Laplace distributions; state vectors for other continuous probability distributions such as Cauchy and Student's t follow in a similar manner. We also extend the algorithm to include preparation of a state vector whose amplitudes are square roots of an arbitrary distribution of discrete probabilities. Our analyses focus on univariate distributions, but can be extended to multivariate forms. The polylogarithmic upsampling algorithm has financial and scientific application.