Comment on "Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs" (Extended Version)

In a recent Letter [Phys. Rev. Lett. 130, 246402 (2023)], Gleis, Li, and von Delft present an algorithm for expanding the bond dimension of a Matrix Product State wave function, giving accuracy similar to 2-site DMRG, but computationally more efficient, closer to the performance of 1-site DMRG. The Controlled Bond Expansion (CBE) algorithm uses the Hamiltonian projected onto two sites, and then further projected onto the two-site tangent space, to extract a set of $k$ vectors that are used to expand the basis between the two sites. CBE achieves this with a complicated sequence of five singular value decompositions (SVDs), in order to project onto the 2-site tangent space and reduce the bond dimension of the tensor network such that the contraction can be done in time $O(dwD^3)$. In this Comment, we show that (1) the projection onto the 2-site tangent space is unnecessary, and is generally not helpful; (2) the sequence of 5 SVDs can be replaced by a single $QR$ decomposition (optionally with one SVD as well), making use of the randomized SVD (RSVD) with high accuracy and significantly improved efficiency, scaling as $O(dwkD^2)$ i.e., the most expensive operations are only quadratic in the bond dimension $D$ and linear in the number of expansion vectors $k$; (3) several statements about the variational properties of the CBE algorithm are incorrect, and the variational properties are essentially identical to existing algorithms including 2-site DMRG and single-site subspace expansion (3S); (4) a similar RSVD approach can be applied to the 3S algorithm, which leads to many advantages over CBE, especially in systems with long range interactions. We also make some comments on the benchmarking MPS algorithms, and the overall computational efficiency with respect to the accuracy of the calculation.