Quantum algorithms for approximate function loading

TL;DR

This paper presents two approximate quantum-state preparation methods inspired by Grover-Rudolph, significantly reducing complexity and gate count for loading classical functions into quantum computers, with a variational approach for broader functions.

Contribution

Introduces two approximate quantum-state preparation methods for NISQ devices, reducing complexity and gate count, and proposes a variational algorithm for non-smooth functions.

Findings

Complexity of Grover-Rudolph based loading is independent of qubit number n.

Significant reduction in two-qubit gate requirements.

High fidelity achieved with fast convergence in studied examples.

Abstract

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 NISQ 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 M\"ott\"onen $\textit{et al}$, results into $\mathcal{O}(2^{k_0(\epsilon)})$, with $n$ the number of qubits and $k_0(\epsilon)$ 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.