Approximate Quantum Algorithms for Efficient Classical-to-Quantum Data Loading and Encoding

Classical information loading is a crucial task for many quantum algorithms, playing a fundamental role in the field of quantum machine learning. Consequently, the inefficiency of this loading process becomes a significant bottleneck for the application of these algorithms. In this context, we present and compare algorithms for the amplitude and dynamic encodings of classical data into a quantum computer. 

In the case of amplitude encoding, we introduce two approximate quantum-state preparation methods for the NISQ era, drawing inspiration from the Grover-Rudolph algorithm. The first method reduces the number of gates required when no ancillary qubits are used, while the second proposes a variational algorithm capable of loading real functions beyond the Grover-Rudolph algorithm. We also examine the encoding of polynomial functions, either through their matrix product state representation or a scheme that involves the block encoding of the linear function using the Walsh-Hadamard transform and a polynomial transformation of the amplitudes, achieved through the quantum singular value transformation (QSVT). 

On the other hand, in the context of dynamic encoding, we enhance the matrix exponentiation techniques when a limited number of copies of a quantum state is available. This is achieved by introducing imperfect quantum copies, which significantly improve the performance of previous proposals.

The Authors acknowledge support from EU FET Open project EPIQUS (899368) and HORIZON-CL4- 2022-QUANTUM01-SGA project 101113946 OpenSuperQPlus100 of the EU Flagship on Quantum Technologies, the Spanish Ramón y Cajal Grant RYC-2020-030503-I, project Grant No. PID2021-125823NA-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ERDF Invest in your Future”, and from the IKUR Strategy under the collaboration agreement between Ikerbasque Foundation and BCAM on behalf of the Department of Education of the Basque Government. This project has also received support from the Spanish Ministry of Economic Affairs and Digital Transformation through the QUANTUM ENIA project call - Quantum Spain, and by the EU through the Recovery, Transformation and Resilience Plan - NextGenerationEU.