arXiv:2005.11349 · The Journal of Chemical Physics
Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation
This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: (i) selected configuration interaction (sCI) realized both via an a posteriori (ap-sCI) and variational selection of the most important determinants, (ii) an a posteriori singular value decomposition (SVD) of the FCI tensor (SVD-FCI), and (iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that the DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods (except DMRG) perform best with a delocalized basis, and the efficiency of sCI and SVD-FCI is closer to that of DMRG. For larger analogs of the models, the DMRG consistently requires the fewest parameters but still scales exponentially in 2D and 3D systems, and the performance of SVD-FCI is essentially equivalent to that of ap-sCI.
This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: (i) selected configuration interaction (sCI) realized both via an a posteriori (ap-sCI) and variational selection of the most important determinants, (ii) an a posteriori singular value decomposition (SVD) of the FCI tensor (SVD-FCI), and (iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that the DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods (except DMRG) perform best with a delocalized basis, and the efficiency of sCI and SVD-FCI is closer to that of DMRG. For larger analogs of the models, the DMRG consistently requires the fewest parameters but still scales exponentially in 2D and 3D systems, and the performance of SVD-FCI is essentially equivalent to that of ap-sCI.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
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
Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (10 1–2 ) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (10 2–3 ) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein–Nesbet perturbation theory (ENPT), without recourse to the expensive high order n -electron reduced density matrices ( n -RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system–environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.
High Resolution Image Download MS PowerPoint Slide Strongly correlated molecules remain out of reach for most electronic-structure solvers because the exact wave function spans a determinant space that grows exponentially with the number of orbitals. State-of-the-art selected-CI (e.g., CIPSI and HCI) and ML-CI methods mitigate search by exploiting Hamiltonian sparsity and learned prescreening. Nonetheless, in strongly multireference regimes, the excitation-based candidate lists can grow rapidly, which may increase memory and screening costs. We present the Hamiltonian-guided autoregressive selected-configuration interaction (HAAR-SCI), a learn–sample–compress workflow that runs on a single GPU. A gated Transformer samples determinants autoregressively; Gumbel Top-K noise encourages exploration, and GPU min-heap kernels keep only configurations with the largest Hamiltonian couplings. The network is retrained after each expansion, and iterations stop when successive energies differ by ≤1 mHa. Across an 18-molecule benchmark set reaching 116 spin–orbitals, HAAR-SCI attains a mean absolute error of 0.51 mHa while using much fewer determinants than heat-bath CI. It traces the entire N 2 dissociation curve within 0.67 mHa and achieves HCI accuracy for the 40-spin-orbital [Fe 2 S 2 (SCH 3 ) 4 ] 2– cluster with a determinant reduction of 72%, demonstrating its power on systems considered intractable for conventional selected-CI solvers. A final probability-mass pruning compresses typical wave functions by a further 10–50×, retaining <0.01% of the Hilbert space yet still capturing >99.9% of the correlation energy. HAAR-SCI thus offers a compact and truly scalable route to chemical-accuracy quantum chemistry on commodity hardware.
Abstract Following the Governed Fault-Tolerant Quantum Computation of molecular ground states in the companion work [1]—H₂, LiH, H₂O, and BeH₂ at equilibrium and its strong-multireference transition state, all committed inside chemical accuracy (the BeH₂ equilibrium commit, at ΔE = +0.595 µHa, sitting 7.7× inside wavenumber accuracy—1 cm⁻¹ ≈ 0.00456 mHa, the precision regime of high-resolution molecular spectroscopy—the only QPU chemistry commit on record inside that band)—we extend Governed Fault-Tolerant Quantum Computing (GFTQC), executed under the governance of Seed IQ (the Adaptive Multiagent Autonomous Control platform of AIX Global), from chemical accuracy on those molecules to exact reproduction of a full-configuration-interaction benchmark: the linear hydrogen chain, the community's canonical test of strong electron correlation, established by Motta et al. [2] as a single full-configuration-interaction (FCI) equation of state of H₁₀ onto which fifteen-plus independent classical many-body methods were brought into agreement, and the accepted target any exact solver must hit.On Motta's own basis (STO-6G) the committed energy of H₁₀ reproduces his published FCI reference exactly, to twelve decimal places where his published precision is eight, at every point of the dissociation curve—at R = 1.8 bohr −5.424385376333 Ha against the published −5.42438538—each value internally certified exact. We then extend the benchmark in the two directions that turn a model system into chemistry: in length, carrying the exact energy from H₁₀ out to a chain of H₁₀₀; and in basis, computing it across four Gaussian bases from minimal to triple-zeta—STO-6G, 6-31G, cc-pVDZ, cc-pVTZ, the correlation-consistent sets of production electronic-structure theory—recovering progressively more correlation as the basis is enriched, each energy committed exact. Throughout, single-reference coupled cluster is exact near equilibrium but ceases to represent the state as the chain stretches and the dominant-determinant weight |c_HF|² collapses, whereas the governed commit stays exact at every length and every basis.These extensions lie beyond exact classical reach—the configuration space exceeds 10⁵⁸ determinants for H₁₀₀, unstorable on and unsolvable by any high-performance computer—and beyond any reported quantum method; to our knowledge the exact hydrogen-chain energies beyond Motta's classical H₁₀, in production bases and out to H₁₀₀, have not been computed before by any method. No error mitigation, post-selection, or classical fitting is used; every energy is read from the executed fault-tolerant state over qubit-wise-commuting measurement lanes on IBM Boston (Heron r3), accessed by AIX Global, and internally certified exact by a proprietary mechanism not disclosed here. This is the quantum value of fault tolerance made concrete on a strongly-correlated system: the exact answer, delivered where classical computing cannot store the problem and single-reference methods cannot represent the state at all. These results establish a new state of the art for exact electronic-structure computation on hardware: the first exact hydrogen-chain energies beyond H₁₀, by any method, classical or quantum.
This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.