An efficient low-rank approximation to complete active space (CAS) wavefunctions for molecular aggregates is presented. Molecular aggregates usually involve two different characteristic entanglement structures: strong intramolecular entanglement and weak intermolecular entanglement. In the method, low-lying electronic states of molecular aggregates are efficiently expanded by a small number of rank-one basis states that are direct products of monomolecular wavefunctions, each of which is written as a highly entangled state such as the matrix product state (MPS). The complexities raised by strong intramolecular entanglement are therefore encapsulated by the MPS and eliminated from the degree of freedom of the effective Hamiltonian of molecular aggregates. It is demonstrated that the excitation energies of low-lying excited states of a pair of bacteriochlorophyll units with CAS(52e, 50o) are accurately reproduced by only five rank-one basis states. Because the rank-one basis states naturally have diabatic character and reproduce the low-lying spectrum of the CAS space, off-diagonal elements of the Hamiltonian are expected to give accurate diabatic couplings. It is also demonstrated that the energy splitting and the diabatic couplings in anthracene dimer systems are improved by augmenting with additional rank-one basis states.
An efficient low-rank approximation to complete active space (CAS) wavefunctions for molecular aggregates is presented. Molecular aggregates usually involve two different characteristic entanglement structures: strong intramolecular entanglement and weak intermolecular entanglement. In the method, low-lying electronic states of molecular aggregates are efficiently expanded by a small number of rank-one basis states that are direct products of monomolecular wavefunctions, each of which is written as a highly entangled state such as the matrix product state (MPS). The complexities raised by strong intramolecular entanglement are therefore encapsulated by the MPS and eliminated from the degree of freedom of the effective Hamiltonian of molecular aggregates. It is demonstrated that the excitation energies of low-lying excited states of a pair of bacteriochlorophyll units with CAS(52e, 50o) are accurately reproduced by only five rank-one basis states. Because the rank-one basis states naturally have diabatic character and reproduce the low-lying spectrum of the CAS space, off-diagonal elements of the Hamiltonian are expected to give accurate diabatic couplings. It is also demonstrated that the energy splitting and the diabatic couplings in anthracene dimer systems are improved by augmenting with additional rank-one basis states.
Quantum computers promise to solve the electronic structure problem for a large class of molecules. However, the performance of relevant quantum algorithms hinges on preparing initial states with substantial overlap with the target eigenvector. For classically challenging molecules with strong electron correlation, starting from multi-reference states, such as complete active space (CAS) wavefunctions is necessary. Unfortunately, the most advanced state preparation protocols applied to such states result in a gate complexity that scales exponentially with the active space size d. In fact, even encoding a CAS state classically is traditionally believed to be intractable for chemically relevant systems. Here, we draw insights from the recently introduced Quantum Paldus Transform (QPT) to show that there exists an efficient classical representation of CAS states and to design a new state preparation routine outperforming previous ones. The QPT represents a transformation from the Fock basis to a friendlier symmetry-adapted basis. Our main contribution consists in showing that CAS states expanded in this basis can efficiently be represented as a matrix product state (MPS) with a bond dimension scaling as O(d2). One can then efficiently load the MPS on a quantum computer and use the inverse QPT to transform the state to the Fock basis. Moreover, our method can easily be extended to the efficient preparation of CAS states in first quantisation with similar complexity. Crucially, we demonstrate that the complexity of both state preparation protocols only grows polynomially as O(d3) , which constitutes to the best of our knowledge an exponential improvement over the state of the art.
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
This thesis develops a framework for understanding when different excited states of local quantum Hamiltonians can be efficiently represented in a common basis, one containing fewer states than the full Hilbert space. In the analytical part, we introduce metrics that quantify the ''closenes'' of excitations by comparing their truncated density matrices in a fixed single-particle basis, and relate these metrics to their energy differences. We show that large energy separations necessarily imply large differences between density matrices, while small energy differences fall into two categories: one in which the density matrices differ only slightly, and another in which they remain very different, as can occur in the presence of symmetry. We also extend these ideas to bundled matrix product states (MPS), in which multiple excitations are encoded simultaneously, and study how the required bond dimension grows as excitations with different energies are included.
We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.
We report an extension of our previous development that incorporated quantum-chemical density matrix renormalization group (DMRG) into the complete active space second-order perturbation theory (CASPT2) [Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011)]. In the previous study, the combined theory, referred to as DMRG-CASPT2, was built upon the use of pseudo-canonical molecular orbitals (PCMOs) for one-electron basis. Within the PCMO basis, the construction of the four-particle reduced density matrix (4-RDM) using DMRG can be greatly facilitated because of simplicity in the multiplication of 4-RDM and diagonal Fock matrix in the CASPT2 equation. In this work, we develop an approach to use more suited orbital basis in DMRG-CASPT2 calculations, e.g., localized molecular orbitals, in order to extend the domain of applicability. Because the multiplication of 4-RDM and generalized Fock matrix is no longer simple in general orbitals, an approximation is made to it using the cumulant reconstruction neglecting higher-particle cumulants. Also, we present the details of the algorithm to compute 3-RDM of the DMRG wavefunction as an extension of the 2-RDM algorithm of Zgid et al. [J. Chem. Phys. 128, 144115 (2008)] and Chan et al. [J. Chem. Phys. 128, 144117 (2008)]. The performance of the extended DMRG-CASPT2 approach was examined for large-scale multireference systems, such as low-lying excited states of long-chain polyenes and isomerization potential of {[Cu(NH3)3]2O2}(2+).