  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.
