Theory Seminar: Quantum and quantum-inspired methods for numerical analysis
Paula Garcia Molina, IFF-CSIC
Herbert-Walther-lecture hall (G 0.25) at MPQ
Wednesday, November 16th 2022, 11:30am (MEZ)
Even though the great success of classical numerical methods to approximate a great variety of problems of continuous mathematics with multiple applications, they are limited by the memory of current classical computers. As an alternative arises the field of quantum numerical analysis, which uses the advantages of quantum computing to overcome these limitations. In this area, it is key to find an efficient representation of functions and operators in a quantum register. We propose an encoding based on Fourier analysis. Using this representation, we create a variational quantum algorithm to solve Hamiltonian partial differential equations (PDEs), resorting to space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We obtain low infidelities in the resolution of PDEs both under ideal and noisy circumstances, demonstrating the high compression of information in a quantum computer. However, practical fidelities are limited bythe noise and the errors in the evaluation of the cost function in real computers. This motivates the use of quantum-inspired techniques, such as tensor networks, to solve these problems, as they can benefit from the same encoding and have shown heuristic advantages for this kind of problems. Thus, we extend our encoding of functions and operators to matrix product states and operators and use it to develop imaginary time evolution and global optimization quantum-inspired algorithms for the resolution of Hamiltonian PDEs. We benchmark these methods and find that the global optimization methods are a suitable alternative to the classical methods, as they allow us to reach larger discretizations while still obtaining a high precision.