Global optimization of MPS in quantum-inspired numerical analysis

Paula Garc\'ia-MolinaLuca TagliacozzoJuan Jos\'e Garc\'ia-Ripoll

Mar 2023

摘要

This work discusses the solution of partial differential equations (PDEs)using matrix product states (MPS). The study focuses on the search for thelowest eigenstates of a Hamiltonian equation, for which five algorithms areintroduced: imaginary-time evolution, steepest gradient descent, an improvedgradient descent, an implicitly restarted Arnoldi method, and density matrixrenormalization group (DMRG) optimization. The first four methods areengineered using a framework of limited-precision linear algebra, whereoperations between MPS and matrix product operators (MPOs) are implemented withfinite resources. All methods are benchmarked using the PDE for a quantumharmonic oscillator in up to two dimensions, over a regular grid with up to$2^{28}$ points. Our study reveals that all MPS-based techniques outperformexact diagonalization techniques based on vectors, with respect to memoryusage. Imaginary-time algorithms are shown to underperform any type of gradientdescent, both in terms of calibration needs and costs. Finally, Arnoldi likemethods and DMRG asymptotically outperform all other methods, including exactdiagonalization, as problem size increases, with an exponential advantage inmemory and time usage.

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