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 inspired by the Grover-Rudolph
algorithm, which partially solve the problem of loading real functions. Indeed,
by allowing for an infidelity ϵ and under certain smoothness
conditions, we prove that the complexity of Grover-Rudolph algorithm can be
reduced from O(2n) to O(2k0​(ϵ)), with n
the number of qubits and k0​(ϵ) 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 quasi-optimized number of hyperparameters. This allows us to
achieve high fidelity in the loaded state with high speed convergence for the
studied examples