Tensor product methods for strongly correlated molecular systems

TBA

The Full Configuration Interaction QMC method is a stochastic method to diagonalise extremely large matrices, of the type normally encountered in quantum many body problems, including Fermionic systems, in which the underlying Hilbert space is antisymmetric. In principle it allows the wave functions to be sampled in extremely large Hilbert spaces (in some cases favourable cases exceeding $10^{108}$ Slater determinants). The manifestation of the Fermion sign problem in the method is rather subtle, and arises through the necessity to annihilate walkers of the opposite sign whenever they encounter each other. In practice, in order to enhance the rate of walker annihilation, an (initiator) approximation is introduced to be keep the walker distribution compact, and this leads to what is called the "initiator bias", which limits the the accuracy of the method when applied to large molecules. We have recently discovered a simple and effective means to unbias for this initiator bias, enabling essentially exact results to be obtained on hirthto unattainable problems. We illustrate this for the benzene molecule, (30 electrons correlated in 108 orbitals, in a Hilbert space of 10^35), where a result exceeding the best quantum chemical method is obtained in a few hours on a medium sized cluster. Perspectives on the development will be discussed.

A wide class of coupled-cluster methods is introduced, based on Arponen's extended coupled-cluster theory. This class of methods is formulated in terms of a coordinate transformation of the cluster operators. The mathematical framework for the error analysis of coupled-cluster methods based on Arponen's bivariational principle is presented, in which the concept of local strong monotonicity of the flipped gradient of the energy is central. A general mathematical result is presented, describing sufficient conditions for coordinate transformations to preserve the local strong monotonicity. The result is applied to the presented class of methods, which include the standard and quadratic coupled-cluster methods, and also Arponen's canonical version of extended coupled-cluster theory. Some numerical experiments are presented, and the use of canonical coordinates for diagnostics is discussed.

Tensor network methods have become workhorses in the study of strongly correlated systems in condensed matter physics and increasingly also in the context of molecular systems. While their variational character provides results of remarkable accuracy, the capture quantum states featuring low entanglement well, a set of states often referred to as the 'physical corner'. While this physical describes ground states of local Hamiltonian problems from the condensed matter context well, the same cannot be said for problems involving molecules in quantum chemistry or systems undergoing time evolution. In this talk I will provide an overview over tensor network methods augmented by fermionic mode transformations in the context of molecular problems [1], condensed-matter simulations [2], and time evolution [3]. If time allows, I might briefly mention two new applications of tensor networks in being a design principle for building quantum devices [4] and in machine learning [5,6]. [1] Fermionic orbital optimisation in tensor network states C. Krumnow, L. Veis, Ö. Legeza, J. Eisert Phys. Rev. Lett. 117, 210402 (2016) [2] Towards overcoming the entanglement barrier when simulating long-time evolution C. Krumnow, J. Eisert, Ö. Legeza arXiv:1904.11999 [3] Dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems C. Krumnow, L. Veis, J. Eisert, Ö. Legeza arXiv:1906.00205 [4] Simulating topological tensor networks with Majorana qubits C. Wille, R. Egger, J. Eisert, A. Altland Phys. Rev. B 99, 115117 (2019) [5] Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning I. Glasser, R. Sweke, N. Pancotti, J. Eisert, J. I. Cirac arXiv:1907.03741, NeurIPS (2019) [6] Tensor network approaches for learning non-linear dynamical laws A. Goessmann, M. Goette, I. Roth, G. Kutyniok, J. Eisert, R. Sweke Submitted to ICML2020 (2020)

We have been developing gausslet basis sets for use in electronic structure, which improve the computational scaling of DMRG. I will review the basic ideas and report on recent progress.

As pointed out by W. Kohn [1] Schrödinger's wavefunction $\psi$ for a system of $N$ interacting electrons loses its meaning when $N \gtrsim 10^3$. This is due to the exponential increase of the dimensions of Hilbert space with $N$. This situation is unsatisfactory and the exponential wall problem (EWS) needs a resolution. It is shown that the EWP is avoided by using the action function $ln \psi$ in defining a wavefunction in operator space rather than $\psi$ in Hilbert space [2]. This can be done for weakly as well as strongly correlated electrons. A hint how to deal with the logarithm is obtained from imperfect classical gases. Periodic solids have been treated this way with accurate results, e.g., for binding energies. [1] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999) [2] P. Fulde, J. Chem. Phys. 150, 030901 (2019)

One of the greatest challenges in electronic-structure theory is the multireference or static/strong correlation problem. For the siglereference situation (or dynamically correlated situation), the gold standard of quantum chemistry is the coupled-cluster method with singles and doubles and perturbative triples, CCSD(T). The reason is that the standard coupled-cluster (CC) hierarchy of methods is polynomially scaling and size-extensive. However, no unique and truly successful multireference coupled-cluster method exists. Indeed, one can say that the hunt for such a method is also a great challenge of quantum chemistry. I will argue that the proper mathematical framework for coupled-cluster type methods is the bivariational principle as introduced by Arponen [J. Arponen, Ann. Phys. 151 (1983), 311--382] and Löwdin [P.-O. Löwdin, J. Math. Phys. 24 (1983), 70--87], a formal generalization of the usual Rayleigh--Ritz variational principle to operators that are not necessarily self-adjoint (i.e., non-Hermitian in the finite-dimensional case). Its mathematical analysis uses tools for nonlinear functional analysis and monotone operator theory, most importantly Zarantonello's theorem [A. Laestadius and S. Kvaal, SIAM J. Numer. Anal. 56 (1028), 660--683]. While unconventional, the use of the bivariational principle has the important benefit that convergence and quadratic energy errors can be proven for general singlereference methods. On the other hand, virtually all published multireference coupled-cluster methods are based on the Bloch equation for the wave operator, the multireference equivalent of the projected similarity transformed Schrödinger equation, the latter being the traditional background for the standard CC hierarchy of methods. In this talk, I will propose to instead explore the bivariational principle for statically correlated systems. I will outline an hierarchy of experimental multireference CC methods, both in multistate and singlestate versions, and present recent numerical results on insertion of Be into H$_2$. We connect with the world of the density-matrix renormalization group method and tensor-network states (TNS) by casting the formulation of such methods in a bivariational language. This yields convenient formulations of TNS based methods, applicable to non-selfadjoint problems, orbital optimization (also for non-selfadjoint problems), and, most importantly, hybridization with coupled-cluster type methods. The opportunity for a rigorous mathematical approach to energy errors and convergence will also be discussed. This work has received funding from the ERC-STG-2014 under grant agreement No 639508.

Reliable theoretical treatment of chemical reactions, molecular excited states, and many other chemical phenomena requires the determination of multireference (or strongly correlated) electronic state of molecular system. Our group has been working on method developments and applications of high efficient DMRG method with complete active space modeling of the multireference states. I will show recent extensions of the quantum chemical DMRG methods. In addition, I will show another type of multireference approach based on Boltzmann machine ansatz that has been recently introduced.

In QC-DMRG, the choice of the ordering of the orbital basis is crucial for efficient calculations with reduced bond dimensions. The Fiedler order which is based on an entanglement analysis of the ground state is a universal ordering method which is widely used in practice. In this talk, a new ordering scheme is proposed based on the discovery of an inversion symmetry of higher-order singular values for ground states of noninteracting Hamiltonians. The decay of these singular values govern the approximability of the wave function in the tensor train (or matrix product state) format. This symmetry gives a natural criterion to optimize for a QC-DMRG calculation. Numerical experiments on model systems will be presented comparing our approach with the canonical and the Fiedler orders. This is a joint work with Gero Friesecke (TU Munich).

Karol Kowalski, Nicholas Bauman Pacific Northwest National Laboratory It has recently been demonstrated [1] that the sub-system embedding sub-algebras approach to the CC theory (SES-CC) can be employed to redefine how CC ground-state energies are calculated. In contrast to the standard energy expression, one can obtain CC energy by diagonalizing effective Hamiltonian in the active space corresponding to the so-called internal excitations defined by the corresponding sub-system embedding sub-algebra. From this perspective, solving CC equations is equivalent to the Hamiltonian renormalization procedure that provides a rigorous separation of external fermionic degrees of freedom from those corresponding to the internal excitations (inside the active space). We illustrate the SES-CC formalism properties based on the single-reference CC formulation and double unitary CC Ansatz (DUCC Ansatz) [2,3]. We also discuss the possible utilization of downfolded Hamiltonians in the context of DMRG formalisms and quantum computing. [1] K. Kowalski, “Properties of coupled cluster equations originating in excitation sub-algebras,” J. Chem. Phys. 148, 094104 (2018). [2] N.P. Bauman, E.J. Bylaska, S. Krishnamoorthy, G.H. Low, N. Wiebe, C.E. Granade, M. Roetteler, M. Troyer, K. Kowalski, “Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalism,” J. Chem. Phys. 151, 014107 (2019). [3] N.P. Bauman, G.H. Low, K. Kowalski, “Quantum simulations of excited states with active-space downfolded Hamiltonians,” J. Chem. Phys. 151, 234114 (2019).

In this talk I will present ways in with strong electron correlation in realistic materials can be treated with a cost that only scales as a low order polynomial of the size of the system. The algorithms will consist of two steps, the first in which all valence orbitals of systems will be included in an active space and nearly exact ground state wavefunctions will be calculated. In the second step, the out of active space correlation will be calculated using multireference perturbation theory and multireference configuration interaction. The traditional limitations of these methods will be overcome by new algorithms that allow one to proceed without needing to calculate the reduced density matrices of the active space wavefunction.

The tailored coupled-cluster (TCC) approach is a simple and very appealing way to target multi-configurational states by means of a single-reference type coupled-cluster method. The most recent method developed under the TCC umbrella is the coupled-cluster method tailored by matrix product state (DMRG-TCC). The choice of the active space is a crucial step in any TCC method; we investigate the sensitivity of the DMRG-TCC method with respect to the active space choice and elaborate on potentially automated choices based on entropy profiles.

Describing strongly interacting electrons is one of the crucial challenges of modern quantum physics. A comprehensive solution to this electron correlation problem would simultaneously exploit both the pairwise interaction and its spatial decay. By taking a quantum information perspective, we explain how this structure of realistic Hamiltonians gives rise to two conceptually different notions of correlation and entanglement. The first one describes correlations between orbitals while the second one refers more to the particle picture. We illustrate those two concepts of orbital and particle correlation and present measures thereof. Intriguingly, our results for different molecular systems reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. We also speculate on a promising relation between orbital and particle correlation and argue that this may replace the obscure but widely used concept of static and dynamic correlation.

A density matrix functional embedding theory [1] based on a (one-electron) unitary Householder transformation [2] will be presented. Unlike in density matrix embedding theory (DMET)[3], the bath becomes, in the present formalism, an explicit functional of the one-electron reduced density matrix (RDM), thus leading to a substantial simplification of the embedding problem. Connections with DMET and density/RDM functional theories (DFT/RDMFT) will be discussed. [1] S. Sekaran, M. Tsuchiizu, M. Saubanère, and E. Fromager, to be submitted (2021). [2] F. Rotella and I. Zambettakis, Applied mathematics letters, vol. 12, no. 4, pp. 29-34, 1999. [3] G. Knizia and G. K.-L. Chan, Phys. Rev. Lett. 109, 186404 (2012).

Despite the recent developments in simulating correlated quantum systems a rigorous solution of interacting quantum systems is only rarely possible. A common approximation scheme consists in treating a small part of the system, called the cluster, in a rigorous manner, while performing approximations to the rest, namely the bath. In this presentation I provide an overview over various cluster embedding schemes and provide a detailed description of the self consistent cluster embedding (SCCE) developed at HQS. To this end I start with an introduction to the cluster perturbation theory (CPT), the dynamical mean field theory(DMFT) including its cluster extension (CDMFT). I then provide a short introduction to inverse mean field theories (iMF) and finally combine the idea of iMF and cluster embedding to our SCCE method.

I will describe progress towards real materials modeling with tensor networks in real-space. I will also describe some new and useful codes that we have developed in this area.

Selected configuration interaction (SCI) methods are currently enjoying a resurgence due to several recent developments which improve either the overall computational efficiency or the compactness of the resulting SCI vector. These recent advances have made it possible to get full CI (FCI) quality results for much larger orbital active spaces, compared to conventional approaches. However, due to the starting assumption that the FCI vector has only a small number of significant Slater determinants, SCI quickly becomes intractable for systems with strong correlation. In this talk, I will describe a method for developing SCI algorithms in a way which exploits local molecular structure to significantly reduce the number of SCI variables. The proposed method is defined by first grouping the orbitals into clusters over which we can define many particle cluster states. We then directly perform the SCI algorithm in a basis of tensor products of cluster states instead of Slater determinants. While the approach is general for arbitrarily defined cluster states, we find significantly improved performance by defining cluster states through a self-consistent Tucker decomposition of the global (and sparse) SCI vector. Numerical results show that TPSCI can be used to reduce the number of variables in the variational space quite significantly as compared to other selected CI approaches.

We present the relativistic version of the DMRG-tailored coupled cluster method (DMRG-TCC), aimed at calculations of strongly correlated systems containing heavy atoms. The method allows to include dynamical correlation at lower computational costs with respect to DMRG in a very large active space, as we demonstrate on the system of TlH. We also present a (non-relativistic) implementation of DMRG-TCCSD and TCCSD(T) based on the domain-based local pair natural orbital approach (DLPNO) in Orca, which is able to treat large molecules, for example oxo-Mn(Salen) or Iron(II)-porphyrin complex.

The density matrix renormalization group (DMRG), with matrix product states (MPSs) being the underlying variational ansatz, is one of the most powerful numerical methods for studying quantum many-body systems. The Gutzwiller projected wave functions provide another important family of variational ansatz in condensed matter and quantum chemistry. I will discuss our recent progress on devising methods to convert Gutzwiller projected wave functions into MPSs and using them to initialize DMRG simulations. I will show that the performance of DMRG can be drastically improved by initializing with a properly chosen Gutzwiller ansatz. This also allows to quantify the closeness of the initial Gutzwiller ansatz and the final converged state after DMRG sweeps, thereby sheds light on whether the Gutzwiller ansatz captures the essential entanglement features of the actual ground state for a given Hamiltonian.

The success of DMRG as a method in recent years can be considered to be built on two pillars of understanding. These come from the ideas of renormalization and entanglement, and from the form of the matrix product state as a wave function ansatz. We take these two concepts, and motivated by their use in tensor networks, take them in a different direction to develop new methods with potential in electronic structure. In the first part, we will show how the idea of Schmidt decomposition approaches can be extended to the idea of generating 1-body or 2-body 'bath' states, rather than the n-body entanglement used in tensor networks. We will use these to develop the idea of a local embedding in the quantum chemistry of real materials, as well as renormalized spectral approaches via a combination with diagrammatic techniques. In the second part, we will examine the form of the MPS ansatz, and ask whether modifications to its form can extend the model naturally to higher dimensions in a tractable way. From this, we will introduce the 'Quantum Gaussian Process State' - a new wave function ansatz, originally developed from machine learning principles, but equally-well motivated from a tensor network perspective. Refs: Fully Algebraic and Self-consistent Effective Dynamics in a Static Quantum Embedding, PV Sriluckshmy, M Nusspickel, E Fertitta, GH Booth arXiv:2012.05837 (2021) A wave function perspective and efficient truncation of renormalised second-order perturbation theory, OJ Backhouse, M Nusspickel, GH Booth, J. Chem. Theory Comput. 16, 1090 (2020) Efficient excitations and spectra within a perturbative renormalization approach, OJ Backhouse, GH Booth J. Chem. Theory Comput. 16, 10, 6294–6304 (2020) Gaussian Process States: A data-driven representation of quantum many-body physics, A Glielmo, Y Rath, G Csanyi, A De Vita, GH Booth Phys. Rev. X 10, 041026 (2020)

Thanks to the advent of ultrafast spectroscopic techniques, the dynamics of a molecule can be resolved experimentally on the natural time scales of both its electronic and nuclear motions. In principle, the exact quantum dynamics of a molecular system [1,2] in a given basis can be simulated with Full Configuration Interaction (full-CI) methods. The exponential increase of the computational cost of full-CI hinders, however, straightforward applications to large molecular systems. In this contribution, we show how to limit this unfavorable scaling by expressing the wave function as a matrix product state, a parameterization employed in the Density Matrix Renormalization Group algorithm (DMRG) [3]. The different strategies that have been designed to integrate the resulting equation of motion are broadly defined as time-dependent DMRG (TD-DMRG). We generalize a recently developed tangent space-based TD-DMRG algorithm [4] to electronic- [5] and vibrational [6] quantum chemical Hamiltonians. We apply the resulting theory to simulate the nuclear and electronic dynamics, possibly coupled together, of molecules with several dozens of degrees of freedom. We assess the accuracy of the simulations by comparison with state-of-the-art CI results. We also generalize TD-DMRG to imaginary-time and show that the resulting method can optimize the ground-state wave function of non-Hermitian Hamiltonians. We apply this imaginary-time TD-DMRG variant to the similarity transformed transcorrelated Hamiltonian [7] and demonstrate that the resulting method, transcorrelated DMRG (DMRG) [8], can enhance the DMRG convergence for molecular calculations. References: [1] Meyer H. D., WIRES Comp. Mol Sci. 2011, 2, 351. [2] Sato T., Ishikawa K. L., Phys. Rev. A. 2013, 88, 023402. [3] Baiardi A., Reiher M., J. Chem. Phys., 2020, 152, 040903. [4] Haegeman J., Lubich C., Oseledets I., Vandereycken F., Phys. Rev. B. 2016, 94, 1. [5] Baiardi A., Reiher M., J. Chem. Theory Comput. 2019, 15, 3481. [6] Baiardi A., Arxiv 2020, 2010.02049. [7] Boys S. F., Handy N. C., Proc. Roy. Soc. A. 1969, 310, 1500. [8] Baiardi A., Reiher M., J. Chem. Phys. 2020, 153, 164115.

We present an efficient numerical method to calculate spectral densities of complex impurities based on the Density Matrix Renormalization Group (DMRG) and use it as the impurity solver of the Dynamical Mean Field Theory (DMFT). By using a self-consistent bath configuration with very low entanglement, we take full advantage of the DMRG to calculate dynamical response functions paving the way to treat large effective impurities such as those corresponding to multi-orbital interacting models and multi-site or multi-momenta clusters. It also solves for complex impurities using ab-initio input which opens its realm of applications to real materials. This method leads to reliable calculations of non-local self energies on the real frequency axis directly, at zero temperature, arbitrary dopings and interactions and at all energy scales. We will show results for multi-site and multi-orbital models in which interesting features arise such as emergent low-energy bound states.

Computational modelling has undoubtedly become an integral part of chemical research. For instance, providing an understanding of the nature of a chemical bond and how it relates to the chemical reactivity of a molecule is of central interest for chemistry. To rationalize such properties at a molecular level requires a tailor-made quantum chemical framework that provides the means to accurately describe the central piece of information relevant to chemistry, viz., the electronic structure. Its determination relies on the solution of the quantum mechanical equations for the electronic many-body problem. Whereas solving the electronic Schrödinger equation (often) yields sufficiently accurate results for light elements, i.e., those of the upper part of the periodic table, this is generally not the case for heavy elements. For the latter, an explicit consideration of the theory of special relativity is required. In this talk, the development and application of matrix-product-state based multiconfigurational quantum chemical methods [1-7] is presented that are designed to accommodate the needs to account for the correlated motion of the electrons as well as for relativistic effects in a rigorous manner. [1] S. Knecht, Ö. Legeza, M. Reiher, J. Chem. Phys.,140, 041101 (2014). [2] S. Knecht, S. Keller, J. Autschbach, M. Reiher, J. Chem. Theory Comput., 12, 5881 (2016). [3] S. Battaglia, S. Keller, S. Knecht, J. Chem. Theory Comput., 14, 2353 (2018). [4] S. Knecht, H. J. Aa. Jensen, T. Saue, Nat. Chem.,11, 40 (2019). [5] S. Knecht, Nachr. Chem., 67, 57 (2019). [6] S. Knecht, in preparation, 2020. [7] J. Paquier, S. Knecht, J. Toulouse, work in progress, 2020.

While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of non-interacting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right, but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this scaling can be improved by adapting methods based on matrix product states, which have been enormously successful for low-dimensional interacting quantum systems, to the case of free fermions. Compared to the case of interacting systems, our methods achieve an exponential speedup in the entanglement entropy of the state. We demonstrate their use to solve systems of up to one million sites with an effective MPS bond dimension of 10^15.