Tree Tensor Network State with Variable Tensor Order: An Efficient Multireference Method for Strongly Correlated Systems

We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chemistry. TTNS is a variational method to efficiently approximate complete active space (CAS) configuration interaction (CI) wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrices in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrices belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the number of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topology based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement.

[1]  J. I. Cirac,et al.  Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states , 2007 .

[2]  Markus Reiher,et al.  New electron correlation theories for transition metal chemistry. , 2011, Physical chemistry chemical physics : PCCP.

[3]  Markus Reiher,et al.  Relativistic DMRG calculations on the curve crossing of cesium hydride. , 2005, The Journal of chemical physics.

[4]  G. Evenbly,et al.  Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law , 2009, 0903.5017.

[5]  M. Reiher,et al.  Unravelling the quantum-entanglement effect of noble gas coordination on the spin ground state of CUO. , 2013, Physical chemistry chemical physics : PCCP.

[6]  Jürgen Gauss,et al.  State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve. , 2004, The Journal of chemical physics.

[7]  Ors Legeza,et al.  Simulating strongly correlated quantum systems with tree tensor networks , 2010, 1006.3095.

[8]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[9]  W. Hackbusch Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[10]  Garnet Kin-Lic Chan,et al.  Approximating strongly correlated wave functions with correlator product states , 2009, 0907.4646.

[11]  M. Reiher,et al.  Entanglement Measures for Single- and Multireference Correlation Effects. , 2012, The journal of physical chemistry letters.

[12]  F. Verstraete,et al.  Post-matrix product state methods: To tangent space and beyond , 2013, 1305.1894.

[13]  S. White,et al.  Measuring orbital interaction using quantum information theory , 2005, cond-mat/0508524.

[14]  O. Legeza,et al.  Quantum data compression, quantum information generation, and the density-matrix renormalization group method , 2004, cond-mat/0401136.

[15]  F. Verstraete,et al.  Complete-graph tensor network states: a new fermionic wave function ansatz for molecules , 2010, 1004.5303.

[16]  G. Vidal,et al.  Classical simulation of quantum many-body systems with a tree tensor network , 2005, quant-ph/0511070.

[17]  Stephen R. Langhoff,et al.  Full configuration‐interaction study of the ionic–neutral curve crossing in LiF , 1988 .

[18]  Paul W. Ayers,et al.  The density matrix renormalization group for ab initio quantum chemistry , 2013, The European Physical Journal D.

[19]  Marcel Nooijen,et al.  On the spin and symmetry adaptation of the density matrix renormalization group method. , 2008, The Journal of chemical physics.

[20]  T. Yanai,et al.  High-performance ab initio density matrix renormalization group method: applicability to large-scale multireference problems for metal compounds. , 2009, The Journal of chemical physics.

[21]  Jeppe Olsen,et al.  Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces , 1988 .

[22]  M. Fiedler A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[23]  Bjrn Engquist Encyclopedia of Applied and Computational Mathematics , 2016 .

[24]  I. McCulloch From density-matrix renormalization group to matrix product states , 2007, cond-mat/0701428.

[25]  G. Zaránd,et al.  Density matrix numerical renormalization group for non-Abelian symmetries , 2008, 0802.4332.

[26]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[27]  Richard L. Martin,et al.  Ab initio quantum chemistry using the density matrix renormalization group , 1998 .

[28]  J. Zolésio,et al.  Springer series in Computational Mathematics , 1992 .

[29]  Markus Reiher,et al.  Orbital Entanglement in Bond-Formation Processes. , 2013, Journal of chemical theory and computation.

[30]  Philippe Corboz,et al.  Fermionic multiscale entanglement renormalization ansatz , 2009, 0907.3184.

[31]  Guifre Vidal,et al.  Simulation of interacting fermions with entanglement renormalization , 2009, Physical Review A.

[32]  Uzi Vishkin,et al.  Recursive Star-Tree Parallel Data Structure , 1993, SIAM J. Comput..

[33]  B. A. Hess,et al.  Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach , 2002, cond-mat/0204602.

[34]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[35]  U. Schollwoeck The density-matrix renormalization group in the age of matrix product states , 2010, 1008.3477.

[36]  F. Verstraete,et al.  Exploring frustrated spin systems using projected entangled pair states , 2009, 0901.2019.

[37]  Garnet Kin-Lic Chan,et al.  Spin-adapted density matrix renormalization group algorithms for quantum chemistry. , 2012, The Journal of chemical physics.

[38]  M. Reiher,et al.  Quantum-information analysis of electronic states of different molecular structures , 2010, 1008.4607.

[39]  Entropic analysis of quantum phase transitions from uniform to spatially inhomogeneous phases. , 2006, Physical review letters.

[40]  Garnet Kin-Lic Chan,et al.  Efficient tree tensor network states (TTNS) for quantum chemistry: generalizations of the density matrix renormalization group algorithm. , 2013, The Journal of chemical physics.

[41]  G. Chan,et al.  Entangled quantum electronic wavefunctions of the Mn₄CaO₅ cluster in photosystem II. , 2013, Nature chemistry.

[42]  Ors Legeza,et al.  Entanglement production by independent quantum channels , 2005, cond-mat/0512270.

[43]  Roman Orus,et al.  A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States , 2013, 1306.2164.

[44]  Sandeep Sharma,et al.  The density matrix renormalization group in quantum chemistry. , 2011, Annual review of physical chemistry.

[45]  G. Vidal Class of quantum many-body states that can be efficiently simulated. , 2006, Physical review letters.

[46]  Östlund,et al.  Thermodynamic limit of density matrix renormalization. , 1995, Physical review letters.

[47]  J. Sólyom,et al.  Optimizing the density-matrix renormalization group method using quantum information entropy , 2003 .

[48]  U. Schollwoeck The density-matrix renormalization group , 2004, cond-mat/0409292.

[49]  Markus Reiher,et al.  The Density Matrix Renormalization Group Algorithm in Quantum Chemistry , 2010 .

[50]  B. A. Hess,et al.  QC-DMRG study of the ionic-neutral curve crossing of LiF , 2002 .

[51]  Xiang Density-matrix renormalization-group method in momentum space. , 1996, Physical review. B, Condensed matter.

[52]  Markus Reiher,et al.  Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings. , 2005, The Journal of chemical physics.

[53]  M. Fiedler Algebraic connectivity of graphs , 1973 .

[54]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[55]  Reinhold Schneider,et al.  Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors , 2013, SIAM J. Matrix Anal. Appl..