Spiers Memorial Lecture: Quantum chemistry, classical heuristics, and quantum advantage.

We describe the problems of quantum chemistry, the intuition behind classical heuristic methods used to solve them, a conjectured form of the classical complexity of quantum chemistry problems, and the subsequent opportunities for quantum advantage. This article is written for both quantum chemists and quantum information theorists. In particular, we attempt to summarize the domain of quantum chemistry problems as well as the chemical intuition that is applied to solve them within concrete statements (such as a classical heuristic cost conjecture) in the hope that this may stimulate future analysis.

Performant automatic differentiation of local coupled cluster theories: Response properties and ab initio molecular dynamics.

In this work, we introduce a differentiable implementation of the local natural orbital coupled cluster (LNO-CC) method within the automatic differentiation framework of the PySCFAD package. The implementation is comprehensively tuned for enhanced performance, which enables the calculation of first-order static response properties on medium-sized molecular systems using coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)]. We evaluate the accuracy of our method by benchmarking it against the canonical CCSD(T) reference for nuclear gradients, dipole moments, and geometry optimizations. In addition, we demonstrate the possibility of property calculations for chemically interesting systems through the computation of bond orders and Mössbauer spectroscopy parameters for a [NiFe]-hydrogenase active site model, along with the simulation of infrared spectra via ab initio LNO-CC molecular dynamics for a protonated water hexamer.

Benchmarking the exponential ansatz for the Holstein model.

Polarons are quasiparticles formed as a result of lattice distortions induced by charge carriers. The single-electron Holstein model captures the fundamentals of single polaron physics. We examine the power of the exponential ansatz for the polaron ground-state wavefunction in its coupled cluster, canonical transformation, and (canonically transformed) perturbative variants across the parameter space of the Holstein model. Our benchmark serves to guide future developments of polaron wavefunctions beyond the single-electron Holstein model.

Multireference Protonation Energetics of a Dimeric Model of Nitrogenase Iron-Sulfur Clusters.

Characterizing the electronic structure of the iron-sulfur clusters in nitrogenase is necessary to understand their role in the nitrogen fixation process. One challenging task is to determine the protonation state of the intermediates in the nitrogen fixing cycle. Here, we use a dimeric iron-sulfur model to study relative energies of protonation at C, S, or Fe. Using a composite method based on coupled cluster and density matrix renormalization group energetics, we converge the relative energies of four protonated configurations with respect to basis set and correlation level. We find that accurate relative energies require large basis sets as well as a proper treatment of multireference and relativistic effects. We have also tested ten density functional approximations for these systems. Most of them give large errors in their relative energies. The best performing functional in this system is B3LYP, which gives mean absolute and maximum deviations of only 10 and 13 kJ/mol with respect to our correlated wave function estimates, respectively, comparable to the uncertainty in our correlated estimates. Our work provides benchmark results for the calibration of new approximate electronic structure methods and density functionals for these problems.

Assessment of DFT functionals for a minimal nitrogenase [Fe(SH)4H]- model employing state-of-the-art ab initio methods.

We have designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S. We show that the energy difference between these two isomers (∆E) is hard to estimate with quantum-mechanical (QM) methods. For example, different density functional theory (DFT) methods give ∆E estimates that vary by almost 140 kJ/mol, mainly depending on the amount of exact Hartree-Fock included (0%-54%). The model is so small that it can be treated by many high-level QM methods, including coupled-cluster (CC) and multiconfigurational perturbation theory approaches. With extrapolated CC series (up to fully connected coupled-cluster calculations with singles, doubles, and triples) and semistochastic heat-bath configuration interaction methods, we obtain results that seem to be converged to full configuration interaction results within 5 kJ/mol. Our best result for ∆E is 101 kJ/mol. With this reference, we show that M06 and B3LYP-D3 give the best results among 35 DFT methods tested for this system. Brueckner doubles coupled cluster with perturbaitve triples seems to be the most accurate coupled-cluster approach with approximate triples. CCSD(T) with Kohn-Sham orbitals gives results within 4-11 kJ/mol of the extrapolated CC results, depending on the DFT method. Single-reference CC calculations seem to be reasonably accurate (giving an error of ∼5 kJ/mol compared to multireference methods), even if the D1 diagnostic is quite high (0.25) for one of the two isomers.

Block2: A comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond.

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications.

The Chromium Dimer: Closing a Chapter of Quantum Chemistry.

The complex electronic structure and unusual potential energy curve of the chromium dimer have fascinated scientists for decades, with agreement between theory and experiment so far elusive. Here, we present a new ab initio simulation of the potential energy curve and vibrational spectrum that significantly improves on all earlier estimates. Our data support a shift in earlier experimental assignments of a cluster of vibrational frequencies by one quantum number. The new vibrational assignment yields an experimentally derived potential energy curve in quantitative agreement with theory across all bond lengths and across all measured frequencies. By solving this long-standing problem, our results raise the possibility of quantitative quantum chemical modeling of transition metal clusters with spectroscopic accuracy.

A comparison between the one- and two-step spin-orbit coupling approaches based on the ab initio density matrix renormalization group.

The efficient and reliable treatment of both spin-orbit coupling (SOC) and electron correlation is essential for understanding f-element chemistry. We analyze two approaches to the problem: the one-step approach, where both effects are treated simultaneously, and the two-step state interaction approach. We report an implementation of the ab initio density matrix renormalization group with a one-step treatment of the SOC effect, which can be compared to prior two-step treatments on an equal footing. Using a dysprosium octahedral complex and bridged dimer as benchmark systems, we identify characteristics of problems where the one-step approach is beneficial for obtaining the low-energy spectrum.

Differentiable quantum chemistry with PySCF for molecules and materials at the mean-field level and beyond.

We introduce an extension to the PySCF package, which makes it automatically differentiable. The implementation strategy is discussed, and example applications are presented to demonstrate the automatic differentiation framework for quantum chemistry methodology development. These include orbital optimization, properties, excited-state energies, and derivative couplings, at the mean-field level and beyond, in both molecules and solids. We also discuss some current limitations and directions for future work.

Low communication high performance ab initio density matrix renormalization group algorithms.

There has been recent interest in the deployment of ab initio density matrix renormalization group (DMRG) computations on high performance computing platforms. Here, we introduce a reformulation of the conventional distributed memory ab initio DMRG algorithm that connects it to the conceptually simpler and advantageous sum of the sub-Hamiltonian approach. Starting from this framework, we further explore a hierarchy of parallelism strategies that includes (i) parallelism over the sum of sub-Hamiltonians, (ii) parallelism over sites, (iii) parallelism over normal and complementary operators, (iv) parallelism over symmetry sectors, and (v) parallelism within dense matrix multiplications. We describe how to reduce processor load imbalance and the communication cost of the algorithm to achieve higher efficiencies. We illustrate the performance of our new open-source implementation on a recent benchmark ground-state calculation of benzene in an orbital space of 108 orbitals and 30 electrons, with a bond dimension of up to 6000, and a model of the FeMo cofactor with 76 orbitals and 113 electrons. The observed parallel scaling from 448 to 2800 central processing unit cores is nearly ideal.

Constructing tensor network influence functionals for general quantum dynamics.

We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of its bond dimension and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.

Dynamical Phase Transitions in a 2D Classical Nonequilibrium Model via 2D Tensor Networks.

We demonstrate the power of 2D tensor networks for obtaining large deviation functions of dynamical observables in a classical nonequilibrium setting. Using these methods, we analyze the previously unstudied dynamical phase behavior of the fully 2D asymmetric simple exclusion process with biases in both the x and y directions. We identify a dynamical phase transition, from a jammed to a flowing phase, and characterize the phases and the transition, with an estimate of the critical point and exponents.

A coupled cluster framework for electrons and phonons.

We describe a coupled cluster framework for coupled systems of electrons and harmonic phonons. Neutral and charged excitations are accessed via the equation-of-motion version of the theory. Benchmarks on the Hubbard-Holstein model allow us to assess the strengths and weaknesses of different coupled cluster approximations, which generally perform well for weak to moderate coupling. Finally, we report progress toward an implementation for ab initio calculations on solids and present some preliminary results on finite-size models of diamond with a linear electron-phonon coupling. We also report the implementation of electron-phonon coupling matrix elements from crystalline Gaussian type orbitals within the PySCF program package.

Theoretical prediction of magnetic exchange coupling constants from broken-symmetry coupled cluster calculations.

Exchange coupling constants (J) are fundamental to the understanding of spin spectra of magnetic systems. Here, we investigate the broken-symmetry (BS) approaches of Noodleman and Yamaguchi in conjunction with coupled cluster (CC) methods to obtain exchange couplings. J values calculated from CC in this fashion converge smoothly toward the full configuration interaction result with increasing level of CC excitation. We compare this BS-CC scheme to the complementary equation-of-motion CC approach on a selection of bridged molecular cases and give results from a few other methodologies for context.

Recent developments in the PySCF program package.

PySCF is a Python-based general-purpose electronic structure platform that supports first-principles simulations of molecules and solids as well as accelerates the development of new methodology and complex computational workflows. This paper explains the design and philosophy behind PySCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PySCF as a development environment. We then summarize the capabilities of PySCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PySCF across the domains of quantum chemistry, materials science, machine learning, and quantum information science.

Exact parameterization of fermionic wave functions via unitary coupled cluster theory.

A formal analysis is conducted on the exactness of various forms of unitary coupled cluster (UCC) theory based on particle-hole excitation and de-excitation operators. Both the conventional single exponential UCC parameterization and a factorized (referred to here as "disentangled") version are considered. We formulate a differential cluster analysis to determine the UCC amplitudes corresponding to a general quantum state. The exactness of conventional UCC (ability to represent any state) is explored numerically, and it is formally shown to be determined by the structure of the critical points of the UCC exponential mapping. A family of disentangled UCC wave functions is proven to exactly parameterize any state, thus showing how to construct Trotter-error-free parameterizations of UCC for applications in quantum computing. From these results, we construct an exact disentangled UCC parameterization that employs an infinite sequence of particle-hole or general one- and two-body substitution operators.

The electronic complexity of the ground-state of the FeMo cofactor of nitrogenase as relevant to quantum simulations.

We report that a recent active space model of the nitrogenase FeMo cofactor, proposed in the context of simulations on quantum computers, is not representative of the electronic structure of the FeMo cofactor ground-state. A more representative model does not affect much certain resource estimates for a quantum computer such as the cost of a Trotter step, while strongly affecting others such as the cost of adiabatic state preparation. Thus, conclusions should not be drawn from the complexity of quantum or classical simulations of the electronic structure of this system in this active space. We provide a different model active space for the FeMo cofactor that contains the basic open-shell qualitative character, which may be useful as a benchmark system for making resource estimates for classical and quantum computers.

Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics.

We describe a framework to reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system's propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponential complexity of such calculations. We illustrate the method by studying driven diffusion and interacting lattice models in one and two spatial dimensions. Our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.