Efficient Ab Initio Auxiliary-Field Quantum Monte Carlo Calculations in Gaussian Bases via Low-Rank Tensor Decomposition

Self-Refinement of Auxiliary-Field Quantum Monte Carlo via Non-Orthogonal Configuration Interaction

For optimal accuracy, auxiliary-field quantum Monte Carlo (AFQMC) requires trial states consisting of multiple Slater determinants. We develop an efficient algorithm to select the determinants from an AFQMC random walk eliminating the need for other methods. When determinants contribute significantly to the nonorthogonal configuration interaction energy, we include them in the trial state. These refined trial wave functions significantly reduce the phaseless bias and sampling variance of the local energy estimator. With 100 to 200 determinants, we lower the error of AFQMC by up to a factor of 10 for second-row elements that are not accurately described with a Hartree-Fock trial wave function. For the HEAT set, we improve the average error to within chemical accuracy. For benzene, the largest studied system, we reduce AFQMC error by 80% with 214 Slater determinants and find a 10-fold increase of the time to solution. We show that phaseless errors prevail in systems with static correlation or strong spin contamination. For such systems, improved trial states enable stable free-projection AFQMC calculations, achieving chemical accuracy even in the strongly correlated regime.

Beyond CCSD(T) Accuracy at Lower Scaling with Auxiliary Field Quantum Monte Carlo

We introduce a black-box auxiliary field quantum Monte Carlo (AFQMC) approach to perform highly accurate electronic structure calculations using configuration interaction singles and doubles (CISD) trial states. This method consistently provides more accurate energy estimates than coupled cluster singles and doubles with perturbative triples (CCSD(T)), often regarded as the gold standard in quantum chemistry. This level of precision is achieved at a lower asymptotic computational cost, scaling as O(N6) compared to the O(N7) scaling of CCSD(T). We provide numerical evidence supporting these findings through results for challenging main group and transition metal-containing molecules.

Quantum Algorithms for the Variational Optimization of Correlated Electronic States with Stochastic Reconfiguration and the Linear Method

Solving the electronic Schrodinger equation for strongly correlated ground states is a long-standing challenge. We present quantum algorithms for the variational optimization of wave functions correlated by products of unitary operators, such as Local Unitary Cluster Jastrow (LUCJ) ansatzes, using stochastic reconfiguration (SR) and the linear method (LM). While an implementation on classical computing hardware would require exponentially growing compute cost, the cost (number of circuits and shots) of our quantum algorithms is polynomial in system size. We find that classical simulations of optimization with the linear method consistently find lower energy solutions than with the L-BFGS-B optimizer across the dissociation curves of the notoriously difficult N2 and C2 dimers; LUCJ predictions of the ground-state energies deviate from exact diagonalization by 1 kcal/mol or less at all points on the potential energy curve. While we do characterize the effect of shot noise on the LM optimization, these noiseless results highlight the critical but often overlooked role that optimization techniques must play in attacking the electronic structure problem (on both classical and quantum hardware), for which even mean-field optimization is formally NP hard. We also discuss the challenge of obtaining smooth curves in these strongly correlated regimes, and propose a number of quantum-friendly solutions ranging from symmetry-projected ansatz forms to a symmetry-constrained optimization algorithm.

Tensor hypercontraction for fully self-consistent imaginary-time GF2 and GWSOX methods: Theory, implementation, and role of the Green's function second-order exchange for intermolecular interactions

We present an efficient MPI-parallel algorithm and its implementation for evaluating the self-consistent correlated second-order exchange term (SOX), which is employed as a correction to the fully self-consistent GW scheme called scGWSOX (GW plus the SOX term iterated to achieve full Green's function self-consistency). Due to the application of the tensor hypercontraction (THC) in our computational procedure, the scaling of the evaluation of scGWSOX is reduced from O(nτnAO5) to O(nτN2nAO2). This fully MPI-parallel and THC-adapted approach enabled us to conduct the largest fully self-consistent scGWSOX calculations with over 1100 atomic orbitals with only negligible errors attributed to THC fitting. Utilizing our THC implementation for scGW, scGF2, and scGWSOX, we evaluated energies of intermolecular interactions. This approach allowed us to circumvent issues related to reference dependence and ambiguity in energy evaluation, which are common challenges in non-self-consistent calculations. We demonstrate that scGW exhibits a slight overbinding tendency for large systems, contrary to the underbinding observed with non-self-consistent RPA. Conversely, scGWSOX exhibits a slight underbinding tendency for such systems. This behavior is both physical and systematic and is caused by exclusion-principle violating diagrams or corresponding corrections. Our analysis elucidates the role played by these different diagrams, which is crucial for the construction of rigorous, accurate, and systematic methods. Finally, we explicitly show that all perturbative fully self-consistent Green's function methods are size-extensive and size-consistent.

Toward Large-Scale AFQMC Calculations: Large Time Step Auxiliary-Field Quantum Monte Carlo

We report modifications of the ph-AFQMC algorithm that allow the use of large time steps and reliable time step extrapolation. Our modified algorithm eliminates size-consistency errors present in the standard algorithm when large time steps are employed. We investigate various methods to approximate the exponential of the one-body operator within the AFQMC framework, distinctly demonstrating the superiority of Krylov methods over the conventional Taylor expansion. We assess various propagators within AFQMC and demonstrate that the Split-2 propagator is the optimal method, exhibiting the smallest time-step errors. For the HEAT set molecules, the time-step extrapolated energies deviate on average by only 0.19 kcal/mol from the accurate small time-step energies. For small water clusters, we obtain accurate complete basis-set binding energies using time-step extrapolation with a mean absolute error of 0.07 kcal/mol compared to CCSD(T). Using large time-step ph-AFQMC for the N2 dimer, we show that accurate bond lengths can be obtained while reducing CPU time by an order of magnitude.

Scalable Quantum Monte Carlo with Direct-Product Trial Wave Functions

The computational demand posed by applying multi-Slater determinant trials in phaseless auxiliary-field quantum Monte Carlo methods (MSD-AFQMC) is particularly significant for molecules exhibiting strong correlations. Here, we propose using direct-product wave functions as trials for MSD-AFQMC, aiming to reduce computational overhead by leveraging the compactness of multi-Slater determinant trials in direct-product form (DP-MSD). This efficiency arises when the active space can be divided into noncoupling subspaces, a condition we term "decomposable active space". By employing localized-active space self-consistent field wave functions as an example of such trials, we demonstrate our proposed approach across a range of molecular systems, each exhibiting varying degrees of complexity in their electronic structures. Our findings indicate that the compact DP-MSD trials can reduce computational costs substantially, by up to 36 times for the C2H6N4 molecule where the two double bonds between nitrogen N=N are clearly separated by a C-C single bond, while maintaining accuracy when active spaces are decomposable. In the case of larger systems such as the benzene dimer, characterized by weak coupling between the two monomers, we observed a decrease in computational cost compared to using a complete active space trial, yet we retained the same level of accuracy. However, for systems where these active subspaces strongly couple, a scenario we refer to as "strong subspace coupling", the method's accuracy decreases compared to that achieved with a complete active space approach. We anticipate that our method will be beneficial for systems with noncoupling to weakly coupling subspaces that require local multireference treatments.

Toward Linear Scaling Auxiliary-Field Quantum Monte Carlo with Local Natural Orbitals

We develop a local correlation variant of auxiliary-field quantum Monte Carlo (AFQMC) based on local natural orbitals (LNO-AFQMC). In LNO-AFQMC, independent AFQMC calculations are performed for each localized occupied orbital using a truncated set of tailored orbitals. Because the size of this space does not grow with the system size for a target accuracy, the method has linear scaling. Applying LNO-AFQMC to molecular problems containing a few hundred to a thousand orbitals, we demonstrate convergence of total energies with significantly reduced costs. The savings are more significant for larger systems and larger basis sets. However, even for our smallest system studied, we find that LNO-AFQMC is cheaper than canonical AFQMC, in contrast with many other reduced-scaling methods. Perhaps most significantly, we show that energy differences converge much more quickly than total energies, making the method ideal for applications in chemistry and material science. Our work paves the way for linear scaling AFQMC calculations of strongly correlated systems, which would have a transformative effect on ab initio quantum chemistry.

Block-Invariant Symmetry Shift: Preprocessing Technique for Second-Quantized Hamiltonians to Improve Their Decompositions to Linear Combination of Unitaries

Computational cost of energy estimation for molecular electronic Hamiltonians via quantum phase estimation (QPE) grows with the difference between the largest and smallest eigenvalues of the Hamiltonian. In this work, we propose a preprocessing procedure that reduces the norm of the Hamiltonian without changing its eigenspectrum for the target states of a particular symmetry. The new procedure, block-invariant symmetry shift (BLISS), builds an operator T̂ such that the cost of implementing H ^ - T ^ is reduced compared to that of Ĥ, yet H ^ - T ^ acts on the subspaces of interest the same way as Ĥ does. BLISS performance is demonstrated for a linear combination of unitaries (LCU)-based QPE approaches on a set of small molecules. Using the number of electrons as the symmetry specifying the target set of states, BLISS provided a factor of 2 reduction of 1-norm for several LCU decompositions compared to their unshifted versions.

Bridging physical intuition and hardware efficiency for correlated electronic states: the local unitary cluster Jastrow ansatz for electronic structure

A prominent goal in quantum chemistry is to solve the molecular electronic structure problem for ground state energy with high accuracy. While classical quantum chemistry is a relatively mature field, the accurate and scalable prediction of strongly correlated states found, e.g., in bond breaking and polynuclear transition metal compounds remains an open problem. Within the context of a variational quantum eigensolver, we propose a new family of ansatzes which provides a more physically appropriate description of strongly correlated electrons than a unitary coupled cluster with single and double excitations (qUCCSD), with vastly reduced quantum resource requirements. Specifically, we present a set of local approximations to the unitary cluster Jastrow wavefunction motivated by Hubbard physics. As in the case of qUCCSD, exactly computing the energy scales factorially with system size on classical computers but polynomially on quantum devices. The local unitary cluster Jastrow ansatz removes the need for SWAP gates, can be tailored to arbitrary qubit topologies (e.g., square, hex, and heavy-hex), and is well-suited to take advantage of continuous sets of quantum gates recently realized on superconducting devices with tunable couplers. The proposed family of ansatzes demonstrates that hardware efficiency and physical transparency are not mutually exclusive; indeed, chemical and physical intuition regarding electron correlation can illuminate a useful path towards hardware-friendly quantum circuits.

Multicomponent Cholesky Decomposition: Application to Nuclear-Electronic Orbital Theory

The Cholesky decomposition technique is commonly used to reduce the memory requirement for storing two-particle repulsion integrals in quantum chemistry calculations that use atomic orbital bases. However, when quantum methods use multicomponent bases, such as nuclear-electronic orbitals, additional challenges are introduced due to asymmetric two-particle integrals. This work proposes several multicomponent Cholesky decomposition methods for calculations using nuclear-electronic orbital density functional theory. To analyze the errors in different Cholesky decomposition components, benchmark calculations using water clusters are carried out. The largest benchmark calculation is a water cluster (H2O)27 where all 54 protons are treated quantum mechanically. This study provides energetic and complexity analyses to demonstrate the accuracy and performance of the proposed multicomponent Cholesky decomposition method.

Relativistic resolution-of-the-identity with Cholesky integral decomposition

In this study, we present an efficient integral decomposition approach called the restricted-kinetic-balance resolution-of-the-identity (RKB-RI) algorithm, which utilizes a tunable RI method based on the Cholesky integral decomposition for in-core relativistic quantum chemistry calculations. The RKB-RI algorithm incorporates the restricted-kinetic-balance condition and offers a versatile framework for accurate computations. Notably, the Cholesky integral decomposition is employed not only to approximate symmetric large-component electron repulsion integrals but also those involving small-component basis functions. In addition to comprehensive error analysis, we investigate crucial conditions, such as the kinetic balance condition and variational stability, which underlie the applicability of Dirac relativistic electronic structure theory. We compare the computational cost of the RKB-RI approach with the full in-core method to assess its efficiency. To evaluate the accuracy and reliability of the RKB-RI method proposed in this work, we employ actinyl oxides as benchmark systems, leveraging their properties for validation purposes. This investigation provides valuable insights into the capabilities and performance of the RKB-RI algorithm and establishes its potential as a powerful tool in the field of relativistic quantum chemistry.

Benchmark Phaseless Auxiliary-Field Quantum Monte Carlo Method for Small Molecules

We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO2, and O2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

Non-Hertz-Millis scaling of the antiferromagnetic quantum critical metal via scalable Hybrid Monte Carlo

A key component of the phase diagram of many iron-based superconductors and electron-doped cuprates is believed to be a quantum critical point (QCP), delineating the onset of antiferromagnetic spin-density wave order in a quasi-two-dimensional metal. The universality class of this QCP is believed to play a fundamental role in the description of the proximate non-Fermi liquid behavior and superconducting phase. A minimal model for this transition is the O(3) spin-fermion model. Despite many efforts, a definitive characterization of its universal properties is still lacking. Here, we numerically study the O(3) spin-fermion model and extract the scaling exponents and functional form of the static and zero-momentum dynamical spin susceptibility. We do this using a Hybrid Monte Carlo (HMC) algorithm with a novel auto-tuning procedure, which allows us to study unprecedentedly large systems of 80 × 80 sites. We find a strong violation of the Hertz-Millis form, contrary to all previous numerical results. Furthermore, the form that we do observe provides good evidence that the universal scaling is actually governed by the analytically tractable fixed point discovered near perfect "hot-spot'" nesting, even for a larger nesting window. Our predictions can be directly tested with neutron scattering. Additionally, the HMC method we introduce is generic and can be used to study other fermionic models of quantum criticality, where there is a strong need to simulate large systems.

Hybrid Auxiliary Field Quantum Monte Carlo for Molecular Systems

We propose a quantum Monte Carlo approach to solve the many-body Schrödinger equation for the electronic ground state. The method combines optimization from variational Monte Carlo and propagation from auxiliary field quantum Monte Carlo in a way that significantly alleviates the sign problem. In application to molecular systems, we obtain highly accurate results for configurations dominated by either dynamic or static electronic correlation.

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within "chemical accuracy" (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob's ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 - N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Efficient quantum analytic nuclear gradients with double factorization

Efficient representations of the Hamiltonian, such as double factorization, drastically reduce the circuit depth or the number of repetitions in error corrected and noisy intermediate-scale quantum (NISQ) algorithms for chemistry. We report a Lagrangian-based approach for evaluating relaxed one- and two-particle reduced density matrices from double factorized Hamiltonians, unlocking efficiency improvements in computing the nuclear gradient and related derivative properties. We demonstrate the accuracy and feasibility of our Lagrangian-based approach to recover all off-diagonal density matrix elements in classically simulated examples with up to 327 quantum and 18 470 total atoms in QM/MM simulations with modest-sized quantum active spaces. We show this in the context of the variational quantum eigensolver in case studies, such as transition state optimization, ab initio molecular dynamics simulation, and energy minimization of large molecular systems.

Quantum chemistry simulation of ground- and excited-state properties of the sulfonium cation on a superconducting quantum processor

The computational description of correlated electronic structure, and particularly of excited states of many-electron systems, is an anticipated application for quantum devices. An important ramification is to determine the dominant molecular fragmentation pathways in photo-dissociation experiments of light-sensitive compounds, like sulfonium-based photo-acid generators used in photolithography. Here we simulate the static and dynamical electronic structure of the H3S+ molecule, taken as a minimal model of a triply-bonded sulfur cation, on a superconducting quantum processor of the IBM Falcon architecture. To this end, we generalize a qubit reduction technique termed entanglement forging or EF [A. Eddins et al., Phys. Rev. X Quantum, 2022, 3, 010309], currently restricted to the evaluation of ground-state energies, to the treatment of molecular properties. While in a conventional quantum simulation a qubit represents a spin-orbital, within EF a qubit represents a spatial orbital, reducing the number of required qubits by half. We combine the generalized EF with quantum subspace expansion [W. Colless et al., Phys. Rev. X, 2018, 8, 011021], a technique used to project the time-independent Schrodinger equation for ground- and excited-states in a subspace. To enable experimental demonstration of this algorithmic workflow, we deploy a sequence of error-mitigation techniques. We compute dipole structure factors and partial atomic charges along ground- and excited-state potential energy curves, revealing the occurrence of homo- and heterolytic fragmentation. This study is an important step towards the computational description of photo-dissociation on near-term quantum devices, as it can be generalized to other photodissociation processes and naturally extended in different ways to achieve more realistic simulations.

ipie: A Python-Based Auxiliary-Field Quantum Monte Carlo Program with Flexibility and Efficiency on CPUs and GPUs

We report the development of a python-based auxiliary-field quantum Monte Carlo (AFQMC) program, ipie, with preliminary timing benchmarks and new AFQMC results on the isomerization of [Cu₂O₂]²⁺. We demonstrate how implementations for both central and graphical processing units (CPUs and GPUs) are achieved in ipie. We show an interface of ipie with PySCF as well as a straightforward template for adding new estimators to ipie. Our timing benchmarks against other C++ codes, QMCPACK and Dice, suggest that ipie is faster or similarly performing for all chemical systems considered on both CPUs and GPUs. Our results on [Cu₂O₂]²⁺ using selected configuration interaction trials show that it is possible to converge the ph-AFQMC isomerization energy between bis(μ-oxo) and μ-η²:η² peroxo configurations to the exact known results for small basis sets with 10⁵-10⁶ determinants. We also report the isomerization energy with a quadruple-zeta basis set with an estimated error less than a kcal/mol, which involved 52 electrons and 290 orbitals with 10⁶ determinants in the trial wave function. These results highlight the utility of ph-AFQMC and ipie for systems with modest strong correlation and large-scale dynamic correlation.

Twenty Years of Auxiliary-Field Quantum Monte Carlo in Quantum Chemistry: An Overview and Assessment on Main Group Chemistry and Bond-Breaking

In this work, we present an overview of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) approach from a computational quantum chemistry perspective and present a numerical assessment of its performance on main group chemistry and bond-breaking problems with a total of 1004 relative energies. While our benchmark study is somewhat limited, we make recommendations for the use of ph-AFQMC for general main-group chemistry applications. For systems where single determinant wave functions are qualitatively accurate, we expect the accuracy of ph-AFQMC in conjunction with a single-determinant trial wave function to be between that of coupled-cluster with singles and doubles (CCSD) and CCSD with perturbative triples (CCSD(T)). For these applications, ph-AFQMC should be a method of choice when canonical CCSD(T) is too expensive to run. For systems where multireference (MR) wave functions are needed for qualitative accuracy, ph-AFQMC is far more accurate than MR perturbation theory methods and competitive with MR configuration interaction (MRCI) methods. Due to the computational efficiency of ph-AFQMC compared to MRCI, we recommended ph-AFQMC as a method of choice for handling dynamic correlation in MR problems. We conclude with a discussion of important directions for future development of the ph-AFQMC approach.

Adiabatic models for the quantum dynamics of surface scattering with lattice effects

In this contribution, we review models for the lattice effects in quantum dynamics calculations on surface scattering, which is important to modeling heterogeneous catalysis for achieving an interpretation of experimental measurements. Unlike dynamics models for reactions in the gas phase, those for heterogeneous reactions have to include the effects of the surface. For manageable computational costs in calculations, the effects of static surface (SS) are firstly modeled as this is simply and easily implemented. Then, the SS model has to be improved to include the effects of the flexible surface, that is the lattice effects. To do this, various surface models have been designed where the coordinates of the surface atoms are introduced in the Hamiltonian operator, especially those of the top surface atom. Based on this model Hamiltonian operator, extensive multi-dimension quantum dynamics calculations can be performed to recover the lattice effects. Here, we first review an overview of the techniques in constructing the Hamiltonian operator, which is a sum of the kinetic energy operator (KEO) and potential energy surface (PES). Since the PES containing the coordinates of the surface atoms in a cell is still expensive, the SS model is often accepted. We consider a mathematical model, called the coupled harmonic oscillator (CHO) model, to introduce the concepts of adiabatic and diabatic representations for separating the molecule and surface. Under the adiabatic model, we further introduce the expansion model where the potential function is Taylor expanded around the optimized geometry of the surface. By an expansion model truncated at the first and second order, various coupling surface models between the molecule and surface are derived. Moreover, by further and deeply understanding the adiabatic representation, an effective Hamiltonian operator is obtained by optimizing the total wave function in factorized form. By this factorized form of wave function and effective Hamiltonian operator, the geometry phase of the surface wave function is theoretically found. This theoretical prediction may be measured by carefully designing experiments. Finally, discussions on the adiabatic representation, the PES construction, and possibility of the classical-dynamics solutions are given. Based on these discussions, a simple outlook on the dynamics of photocatalytics is finally given.

A Localized-Orbital Energy Evaluation for Auxiliary-Field Quantum Monte Carlo

Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has recently emerged as a promising method for the production of benchmark-level simulations of medium- to large-sized molecules because of its accuracy and favorable polynomial scaling with system size. Unfortunately, the memory footprints of standard energy evaluation algorithms are nontrivial, which can significantly impact timings on graphical processing units (GPUs) where memory is limited. Previous attempts to reduce scaling by taking advantage of the low-rank structure of the Coulombic integrals have been successful but exhibit high prefactors, making their utility limited to very large systems. Here we present a complementary cubic-scaling route to reduce memory and computational scaling based on the low rank of the Coulombic interactions between localized orbitals, focusing on the application to ph-AFQMC. We show that the error due to this approximation, which we term localized-orbital AFQMC (LO-AFQMC), is systematic and controllable via a single variable and that the method is computationally favorable even for small systems. We present results demonstrating robust retention of accuracy versus both experiment and full ph-AFQMC for a variety of test cases chosen for their potential difficulty for localized-orbital-based methods, including the singlet-triplet gaps of the polyacenes benzene through pentacene, the heats of formation for a set of Platonic hydrocarbon cages, and the total energy of ferrocene, Fe(Cp)₂. Finally, we reproduce our previous result for the gas-phase ionization energy of Ni(Cp)₂, agreeing with full ph-AFQMC to within statistical error while using less than 1/15th of the computer time.

Selected configuration interaction wave functions in phaseless auxiliary field quantum Monte Carlo

We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared towards optimizing computational performance for longer CI expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in AFQMC calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.

Taming the Sign Problem in Auxiliary-Field Quantum Monte Carlo Using Accurate Wave Functions

We explore different ways of incorporating accurate trial wave functions into free projection auxiliary-field quantum Monte Carlo (fp-AFQMC). States employed include coupled-cluster singles and doubles, multi-Slater, and symmetry-projected mean-field wave functions. We adapt a recently proposed fast multi-Slater local energy evaluation algorithm for fp-AFQMC, making the use of long expansions from selected configuration interaction methods feasible. We demonstrate how these wave functions serve to mitigate the sign problem and accelerate convergence in quantum chemical problems, allowing the application of fp-AFQMC to systems of substantial sizes. Our calculations on the widely studied model Cu₂O₂²⁺ system show that many previously reported isomerization energies differ substantially from the near-exact fp-AFQMC value.

A diagrammatic approach for automatically deriving analytical gradients of tensor hyper-contracted electronic structure methods

We introduce a diagrammatic approach to facilitate the automatic derivation of analytical nuclear gradients for tensor hyper-contraction (THC) based electronic structure methods. The automatically derived gradients are guaranteed to have the same scaling in terms of both operation count and memory footprint as the underlying energy calculations, and the computation of a gradient is roughly three times as costly as the underlying energy. The new diagrammatic approach enables the first cubic scaling implementation of nuclear derivatives for THC tensors fitted in molecular orbital basis (MO-THC). Furthermore, application of this new approach to THC-MP2 analytical gradients leads to an implementation, which is at least four times faster than the previously reported, manually derived implementation. Finally, we apply the new approach to the 14 tensor contraction patterns appearing in the supporting subspace formulation of multireference perturbation theory, laying the foundation for developments of analytical nuclear gradients and nonadiabatic coupling vectors for multi-state CASPT2.

Toward the Minimal Floating Operation Count Cholesky Decomposition of Electron Repulsion Integrals

As quantum chemistry calculations deal with molecular systems of increasing size, the memory requirement to store electron-repulsion integrals (ERIs) greatly outpaces the physical memory available in computing hardware. The Cholesky decomposition of ERIs provides a convenient yet accurate technique to reduce the storage requirement of integrals. Recent developments of a two-step algorithm have drastically reduced the memory operation (MOP) count, leaving the floating operation (FLOP) count as the last frontier of cost reduction in the Cholesky ERI algorithm. In this report, we introduce a dynamic integral tracking, reusing, and compression/elimination protocol embedded in the two-step Cholesky ERI method. Benchmark studies suggest that this technique becomes particularly advantageous when the basis set consists of many computationally expensive high-angular-momentum basis functions. With this dynamic-ERI improvement, the Cholesky ERI approach proves to be a highly efficient algorithm with minimal FLOP and MOP count.

Robust Approximation of Tensor Networks: Application to Grid-Free Tensor Factorization of the Coulomb Interaction

Approximation of a tensor network by approximating (e.g., factorizing) one or more of its constituent tensors can be improved by canceling the leading-order error due to the constituents' approximation. The utility of such robust approximation is demonstrated for robust canonical polyadic (CP) approximation of a (density-fitting) factorized two-particle Coulomb interaction tensor. The resulting algebraic (grid-free) approximation for the Coulomb tensor, closely related to the factorization appearing in pseudospectral and tensor hypercontraction approaches, is efficient and accurate, with significantly reduced rank compared to the naive (nonrobust) approximation. Application of the robust approximation to the particle-particle ladder term in the coupled-cluster singles and doubles reduces the size complexity from O (N⁶) to O (N⁵) with robustness ensuring negligible errors in chemically relevant energy differences using CP ranks approximately equal to the size of the density-fitting basis.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

Quantum simulations employing connected moments expansions

Further advancement of quantum computing (QC) is contingent on enabling many-body models that avoid deep circuits and excessive use of CNOT gates. To this end, we develop a QC approach employing finite-order connected moment expansions (CMX) and affordable procedures for initial state preparation. We demonstrate the performance of our approach employing several quantum variants of CMX through the classical emulations on the H₂ molecule potential energy surface and the Anderson model with a broad range of correlation strength. The results show that our approach is robust and flexible. Good agreement with exact solutions can be maintained even at the dissociation and strong correlation limits.

Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer

Molecular quantum computing simulations are currently limited by the use of minimal Gaussian bases, a problem we overcome using a canonical transcorrelated Hamiltonian to accelerate basis convergence, with unitary coupled cluster as an example.