A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPSs). Instead of the site tensors themselves, we propose to optimize sequentially the ``message'' tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPSs closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep, while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins and on free electrons on the lattice.
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(dwD3). 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(dwkD2) 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.
In the approaches based on matrix-product states (MPSs), such as the density-matrix renormalization group (DMRG) method, the ordering of the sites crucially affects the computational accuracy. We investigate the performance of an algorithm that searches for the optimal site order by iterative local site rearrangement. We improve the algorithm by expanding the range of site rearrangement and apply it to a one-dimensional quantum Heisenberg model with random site permutation. The results indicate that increasing the range of the site rearrangement significantly improves the computational accuracy of the DMRG method. In particular, increasing the rearrangement range from two to three sites reduces the average relative error in the ground-state energy by 65% to 94% in the cases we tested. We also discuss the computational cost of the algorithm and its application as a preprocessing for MPS-based calculations.
DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state Ψ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in HΨ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
Abstract. The density matrix renormalization group (DMRG) algorithm is a popular alternating minimization scheme for solving high-dimensional optimization problems in the tensor train format. Classical DMRG, however, is based on sequential minimization, which raises challenges in its implementation on parallel computing architectures. To overcome this, we propose a novel additive two-level DMRG algorithm that combines independent, local minimization steps with a global update step using a subsequent coarse-space minimization. Our proposed algorithm, which is directly inspired by additive Schwarz methods from the domain decomposition literature, is particularly amenable to implementation on parallel, distributed architectures since both the local minimization steps and the construction of the coarse-space can be performed in parallel. Numerical experiments on strongly correlated molecular systems demonstrate that the method achieves competitive convergence rates while achieving significant parallel speedups.