An accurate Lanczos method for the matrix product state representation

The Lanczos algorithm naturally finds multiple low-lying eigenstates simultaneously, providing a global and robust treatment of the spectrum without local minima or root flipping. However, when implemented within the matrix product state formalism, its accuracy is severely affected by bond truncation, leading to a pronounced high-error plateau. Here, we show the modified thick-block Lanczos method to enhance the convergence of the Lanczos algorithm within the matrix product state formalism. Numerical benchmarks on the Fermi-Hubbard and Heisenberg models demonstrate that our method achieves the optimal accuracy allowed by the bond dimension. In addition, we benchmark it on a 50-site Fermi-Hubbard model and a 120-site Heisenberg model to demonstrate its scalability. Furthermore, when incorporating the shift-and-invert technique, our method enables the direct targeting of multiple excited states without finding lower eigenstates.