Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.
Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.
Matrix product operators (MPOs) are at the heart of the second-generation density matrix renormalization group (DMRG) algorithm formulated in matrix product state language. We first summarize the widely known facts on MPO arithmetic and representations of single-site operators. Second, we introduce three compression methods (rescaled SVD, deparallelization, and delinearization) for MPOs and show that it is possible to construct efficient representations of arbitrary operators using MPO arithmetic and compression. As examples, we construct powers of a short-ranged spin-chain Hamiltonian, a complicated Hamiltonian of a two-dimensional system and, as proof of principle, the long-range four-body Hamiltonian from quantum chemistry.
We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
Matrix product states and matrix product operators (MPOs) are one-dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two-dimensional (2D) tensor network states, known as projected entangled-pair states, rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on the fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.
Theoretical understanding of strongly correlated systems in one spatial dimension has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called matrix product states (MPSs). However, DMRG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the matrix product operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators relevant for the calculation. The MPO approach allows one-dimensional (1D) Hamiltonians to be templated using a small set of finite state automaton rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.
We present a new wavefunction ansatz that combines the technique of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as |Ψ^(N,S,M)_SP−MPS⟩=P_S|Ψ(N,M)_MPS⟩, where P_S is the spin projector for total spin S and |Ψ(N,M)MPS⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of Ŝ^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ(N,M)MPS⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) SP-MPS are closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector P_S naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories. Unlike usual spin projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ(N,M)MPS⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.
Automated construction of U(1)-invariant matrix-product operators from graph representations
We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary U(1) U ( 1 ) -invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a U(1) U ( 1 ) -invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of U(1) U ( 1 ) -invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a S=1 S = 1 Heisenberg chain.