Partial differential equations (PDEs) are relevant for solving real-world problems across many areas. However, their solution may be challenging, especially for large-dimensional or high-resolution problems with high memory demands. This thesis develops new quantum and quantuminspired numerical analysis methods for solving PDEs with potential memory and time savings while maintaining high accuracy. First, we resort to quantum computing, which benefits from exponential encoding advantages and speedups in key operations. Due to the lack of error correction of existing quantum computers, we propose a variational quantum algorithm to solve Hamiltonian PDEs, combining a classical and a quantum computer to exploit the properties of the quantum register. However, the noise sources and limited number of measurements of current quantum devices restrict the scalability of this approach. The high efficiency of the quantum register function encoding motivates its use in developing quantum-inspired algorithms. The second part of the thesis focuses on creating a matrix product state (MPS) finite precision algebra and applying it to quantum-inspired numerical analysis. More concretely, we develop MPS methods to solve static and time-dependent PDEs, motivated by the solution of problems of physical interest: the study of superconducting circuits and the expansion of a particle’s wavefunction in the context of levitodynamics. Using a two-dimensional squeezed harmonic oscillator of up to 230 points as a benchmark, MPS methods for Hamiltonian PDEs show exponential memory advantage compared to vector implementations and asymptotic advantage in time while achieving a low error in the solution approximation. Similarly, the time evolution MPS techniques demonstrate exponential memory compression and comparable accuracy and cost to standard vector methods. We conclude that the MPS framework constitutes a memory-efficient and accurate tool for solving PDEs. These findings present new opportunities for applying quantum-inspired algorithms to a wider range of PDEs and numerical analysis problems, opening exciting avenues for future research and applications

This thesis was funded by MCIN/AEI/10.13039/501100011033 and “FSE invierte en tu futuro” through an FPU Grant FPU19/03590. The author acknowledges support from Spanish Project No. PGC2018-094792-B-I00 (MCIU/AEI/FEDER, UE), CAM/FEDER Project No. S2018/TCS- 4342 (QUITEMAD-CM), Proyecto Sinérgico CAM 2020 Y2020/TCS- 6545 (NanoQuCoCM), Spanish Projects No. PID2021-127968NB-I00 and No. PDC2022-133486-I00, funded by MCIN/AEI/10.13039/5011000 11033 and by the European Union “NextGenerationEU”/PRTR”1, and CSIC Interdisciplinary Thematic Platform (PTI) Quantum Technologies (PTIQTEP+). The author also gratefully acknowledges the Scientific computing Area (AIC), SGAI-CSIC, for their assistance while using the TRUENO and DRAGO Supercomputers for performing the simulations, and Centro de Supercomputación de Galicia (CESGA) for access to the supercomputer FinisTerrae

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica de la Materia Condensada. Fecha de Lectura: 14-03-2025