Quantum State Preparation of Continuous and Discrete Probability Distributions Using an Upsampling Algorithm

Financial, scientific, and machine-learning quantum-computing applications often require repeated loading of classical information into a state vector, due to the quantum-mechanical, no-cloning restriction. Limitations also exist on both dimension of state vector and time to completion of underlying calculation due to quantum decoherence, external noise and stray fields, and scaling issues that inhibit control and precision, particularly with respect to the number of physical qubits needed to implement quantum error correction. These restrictions necessitate construction of an efficient quantum algorithm for preparation of the state vector. In this study we employ the upsampling method of data capture, most often associated with fast Fourier transformation analyses of discrete samples of time series, to construct a quantum algorithm for state-vector preparation. The state-vector amplitudes are square roots of probabilities sampled from a continuous probability distribution having support over the entire real line. In initial construction, the corresponding quantum circuit exhibits exponential gate depth, like other divide-and-conquer algorithms, but can be transformed to an equivalent circuit of polylogarithmic gate depth by adapting the method of quantum forking [I. F. Araujo, et al., Sci. Rep. 11, 6329 (2021).]. Wider application exists for discretizing other continuous variables. We also extend our approach to preparation of state vectors with amplitudes constructed from discrete probability distributions. We present several example applications to demonstrate the utility of our approach.