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arXiv:1711.01104 · Physical review. B./Physical review. B

Error estimates for extrapolations with matrix-product states

Claudius Hubig, Jutho Haegeman, Ulrich Schollwöck

Published Jan 16, 2018• 59 citations
arXivHTMLPDF

Abstract

We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of $\ensuremath{\langle}\ensuremath{\psi}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|\ensuremath{\psi}\ensuremath{\rangle}$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the $L=30,S=1/2$ Heisenberg chain, the $L=50$ Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of $10\ifmmode\times\else\texttimes\fi{}4$. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$ at a fraction of the computational effort.

DOI
10.1103/physrevb.97.045125

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We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance \ensuremath⟨\ensuremathψ∣(H\ifmmode^\elseˆ\fi\ensuremath−E)2∣\ensuremathψ\ensuremath⟩\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}\ensuremath⟨\ensuremathψ∣(H\ifmmode^\elseˆ\fi​\ensuremath−E)2∣\ensuremathψ\ensuremath⟩. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of \ensuremath⟨\ensuremathψ∣H\ifmmode^\elseˆ\fi∣\ensuremathψ\ensuremath⟩\ensuremath{\langle}\ensuremath{\psi}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|\ensuremath{\psi}\ensuremath{\rangle}\ensuremath⟨\ensuremathψ∣H\ifmmode^\elseˆ\fi​∣\ensuremathψ\ensuremath⟩ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the L=30,S=1/2L=30,S=1/2L=30,S=1/2 Heisenberg chain, the L=50L=50L=50 Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of 10\ifmmode×\else\texttimes\fi410\ifmmode\times\else\texttimes\fi{}410\ifmmode×\else\texttimes\fi4. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance \ensuremath⟨\ensuremathψ∣(H\ifmmode^\elseˆ\fi\ensuremath−E)2∣\ensuremathψ\ensuremath⟩\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}\ensuremath⟨\ensuremathψ∣(H\ifmmode^\elseˆ\fi​\ensuremath−E)2∣\ensuremathψ\ensuremath⟩ at a fraction of the computational effort.
Explicitly correlated methods, such as the transcorrelated method which shifts a Jastrow or Gutzwiller correlator from the wave function to the Hamiltonian, are designed for high-accuracy calculations of electronic structures, but their application to larger systems has been hampered by the computational cost. We develop improved techniques for the transcorrelated density-matrix renormalization group (DMRG), in which the ground state of the transcorrelated Hamiltonian is represented as a matrix product state (MPS), and demonstrate large-scale calculations of the ground-state energy of the two-dimensional Fermi-Hubbard model. Our developments stem from three technical inventions: (i) constructing matrix product operators (MPOs) of transcorrelated Hamiltonians with low bond dimension and high sparsity, (ii) exploiting the entanglement structure of the ground states to increase the accuracy of the MPS representation, and (iii) optimizing the nonlinear parameter of the Gutzwiller correlator to mitigate the nonvariational nature of the transcorrelated method. We examine systems of size up to $12\ifmmode\times\else\texttimes\fi{}12$ lattice sites, four times larger than previous transcorrelated DMRG studies, and demonstrate that transcorrelated DMRG yields significant improvements over standard nontranscorrelated DMRG for equivalent computational effort. Transcorrelated DMRG reduces the error of the ground-state energy by $2.4\ifmmode\times\else\texttimes\fi{}$--$14\ifmmode\times\else\texttimes\fi{}$, with the smallest improvement seen for a small system at half filling and the largest improvement in a dilute closed-shell system.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. It has been applied to solve many physical problems, including the calculation of ground states and dynamical properties. In this work, we develop a perturbation theory of the DMRG (PT-DMRG) to greatly increase its accuracy in an extremely simple and efficient way. Using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions $\left\{|{\ensuremath{\psi}}_{i}\ensuremath{\rangle}\right\}$ is introduced to describe the perturbations to the ground state obtained by the conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond dimension cutoff are used to define these perturbation terms. The perturbed Hamiltonian is then defined as ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}}_{ij}=\ensuremath{\langle}{\ensuremath{\psi}}_{i}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$; its ground state permits us to calculate physical observables with a considerably improved accuracy compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of a one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of the DMRG is shown to be improved O(10)O(10)O(10) times. Furthermore, for moderate LLL the errors of the DMRG and PT-DMRG both scale linearly with ${L}^{\ensuremath{-}1}$ (with LLL being the length of the chain). The linear relation between the dimension cutoff of the DMRG and that of the PT-DMRG at the same precision shows a considerable improvement in efficiency, especially for large dimension cutoffs. In the thermodynamic limit we show that the errors of the PT-DMRG scale with $\sqrt{{L}^{\ensuremath{-}1}}$. Our work suggests an effective way to define the tangent space of the ground-state MPS, which may shed light on the properties beyond the ground state. This second-order PT-DMRG can be readily generalized to higher orders, as well as applied to models in higher dimensions.
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.
We present a matrix-product state (MPS) based band Lanczos method as solver for quantum cluster methods such as the variational cluster approximation. While a naïve implementation of MPS as cluster solver would barely improve its range of applicability, we show that our approach makes it possible to treat cluster geometries well beyond the reach of exact diagonalization methods. The key modifications we introduce are a continuous energy truncation combined with a convergence criterion that is more robust against approximation errors introduced by the MPS representation and provides a bound to deviations in the resulting Green's function. The potential of the resulting cluster solver is demonstrated by computing the self-energy functional for the single-band Hubbard model at half filling in the strongly correlated regime, on different cluster geometries. Here, we find that only when treating large cluster sizes, observables can be extrapolated to the thermodynamic limit, which we demonstrate at the example of the staggered magnetization. Treating clusters sizes with up to 6 × 6 sites we obtain significant improvement over the extrapolation accessible with exact diagonalization solvers when comparing with quantum Monte Carlo results. Finally, we illustrate the applicability of the MPS cluster solver to more complex models by calculating spectral properties as relevant for the electron-doped cuprate CaCuO 2 .
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 kkk 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)O(dwD^3)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 QRQRQR 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)O(dwkD^2)O(dwkD2) i.e., the most expensive operations are only quadratic in the bond dimension DDD and linear in the number of expansion vectors kkk; (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.