Loading classical data into quantum computers represents an essential stage in many relevant quantum algorithms, especially in the field of quantum machine learning. Therefore, the inefficiency of this loading process means a major bottleneck for the application of these algorithms. Here, we introduce two approximate quantum-state preparation methods for the noisy intermediate-scale quantum era inspired by the Grover-Rudolph algorithm, which partially solve the problem of loading real functions. Indeed, by allowing for an infidelity $\epsilon$ and under certain smoothness conditions, we prove that the complexity of the implementation of the Grover-Rudolph algorithm without ancillary qubits, first introduced by Möttönen et al., results into $\mathcal{O}\left(2^{k_0\left(\epsilon \right)}\right)$, with $n$ the number of qubits and $k_0\left(\epsilon \right)$ asymptotically independent of $n$. This leads to a dramatic reduction in the number of required two-qubit gates. Aroused by this result, we also propose a variational algorithm capable of loading functions beyond the aforementioned smoothness conditions. Our variational Ansatz is explicitly tailored to the landscape of the function, leading to a quasioptimized number of hyperparameters. This allows us to achieve high fidelity in the loaded state with high speed convergence for the studied examples.

• Received 21 December 2021
• Revised 18 April 2023
• Accepted 12 July 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033114

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Published by the American Physical Society