In this work ground and excited electronic states of Heisenberg cluster models, in the form of con- figuration interaction many-body wave functions, are characterized within the spin-adapted Graph- ical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin Slocal ≥ 21 are presented, including J1–J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and bio-mimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reorder- ing, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block- diagonal structure of the Hamiltonian matrix it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route towards wave functions of reduced multi-reference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest.
In this work ground and excited electronic states of Heisenberg cluster models, in the form of con- figuration interaction many-body wave functions, are characterized within the spin-adapted Graph- ical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin Slocal ≥ 21 are presented, including J1–J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and bio-mimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reorder- ing, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block- diagonal structure of the Hamiltonian matrix it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route towards wave functions of reduced multi-reference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest.
The electronic, magnetic, and structural properties of small clusters are studied in the framework of the single-band Hubbard Hamiltonian. Results for various ground-state and excited-state many-body properties are presented, which were calculated exactly by means of Lanczos's numerical diagonalization method. A full geometry optimization is performed for N\ensuremath{\le}8 atoms by considering all possible nonequivalent cluster structures with fixed nearest-neighbor bond lengths. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb interaction strength U/t and number of electrons \ensuremath{\nu}. The resulting interplay between electron correlations, magnetism, and cluster structure is analyzed and the main trends as a function of N, U/t, and \ensuremath{\nu} are derived. The stability of cluster ferromagnetism is studied from two complementary points of view. First, for N\ensuremath{\le}8 and \ensuremath{\nu}=N+1, we determine exactly the stability of the ferromagnetic ground state with respect to electronic excitations and structural changes. It is shown that in small clusters the structural changes can be as important to the temperature dependence of the magnetization as the purely electronic excitations. Second, we determine the stability of the saturated ferromagnetic state with respect to single spin flips as a function of the band filling \ensuremath{\nu}/N. In this case a few selected larger clusters (7\ensuremath{\le}N\ensuremath{\le}43) in the strongly correlated limit (U/t\ensuremath{\rightarrow}+\ensuremath{\infty}) are considered. It is shown that the \ensuremath{\nu}/N dependence of the spin-flip energy \ensuremath{\Delta}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{sf}}$ shows interesting electronic-shell-like oscillations, which reflect the characteristics of the single-particle energy-level structure and its dependence on the symmetry and size of the cluster. Finally, we conclude by discussing some of the limitations of the model together with relevant extensions. \textcopyright{} 1996 The American Physical Society.
We present a numerical, non-perturbative framework for constructing effective Hamiltonians that describe the dynamics of low-energy degrees of freedom within a restricted Hilbert space in quantum many-body systems. The approach is based on block diagonalization guided by a minimal-deformation principle imposed within a selected target sector. The formulation is designed to remain compatible with the numerical linked-cluster expansion. For gapped systems, the relation between minimal deformation and cluster additivity requires careful treatment when excited eigenstates contain finite admixtures of the ground state. After establishing a cluster-additive basis that reproduces the Hörmann-Schmidt construction, the minimal-deformation criterion uniquely determines the effective Hamiltonian within each excitation sector. The same criterion also provides a practical numerical procedure for selecting relevant low-energy eigenstates, including regimes characterized by strong level mixing and avoided crossings. The framework is illustrated using two spin models: the one-dimensional transverse-field Ising model as a benchmark and the two-dimensional Shastry-Sutherland model with Dzyaloshinskii-Moriya interactions, relevant to SrCu _2 2 (BO _3 3 ) _2 2 . In both cases, the resulting effective Hamiltonians accurately capture the excitation dynamics and the associated band structures.
The general formalism of the block-correlated coupled cluster (BCCC) method, an alternative multireference coupled cluster method for calculating the ground-state electronic structures of molecular systems, has been presented. The BCCC theory is constructed in terms of a complete set of many-electron states of individual blocks, assumed that the whole system could be partitioned into a set of blocks. The reference state in the BCCC is selected as a tensor product of the most important many-electron state of each system block. By truncating the cluster operator to a certain n-block correlation level, an approximate but size-extensive BCCC method, denoted as BCCCn, is defined. For reducing the computational effort but without much loss of accuracy, the reduced density matrix is introduced to generate an optimal subset of many-electron states for each block. I have implemented the BCCCn (n=2,3) methods within the S=1/2 Heisenberg Hamiltonian, and applied them to calculate the ground-state energies of one-dimensional spin chains and quasi-one-dimensional two-leg spin ladders. The calculated results show that with the appropriate partition of the studied systems the BCCC3 method can yield quite satisfactory ground-state energies for these spin systems.
Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.
In recent years, numerous novel materials have emerged in fields such as high-temperature superconductivity and quantum communication, including unconventional superconductors, topological insulators, and spin liquids. These materials exhibit significant quantum phenomena such as quantum interference, tunneling, fluctuations, and entanglement, often arising from strong correlation effects among electrons within molecules. In the community of quantum chemistry, strong correlation effects are typically classified into dynamic electron correlation and static electron correlation. Dynamic electron correlation, caused by the instantaneous movement of electrons leading to charge density fluctuations, presents computational challenges due to the necessity of introducing a multi-particle form of the wave function. This departure from the mean-field single-particle perspective results in a substantial increase in the computational cost for calculating electron correlation levels. On the other hand, static electron correlation arises from the near-degeneracy of numerous frontier molecular orbitals, resulting in a quantum superposition of electron configurations. It requires the use of multi-configuration methods and faces the well-known “exponential wall” challenge, where computational complexity exponentially grows with the system size. Therefore, due to the involvement of both dynamic and static electron correlations in strongly correlated systems, achieving precise electronic structure calculations is exceptionally difficult, representing a significant challenge in theoretical chemistry and computational physics. Traditionally, to describe the static correlation, quantum chemists often define crucial correlated orbitals and their electrons within the complete active space (CAS). Full configuration interaction (FCI) calculations are performed within the CAS, while the region outside the CAS is described by mean-field methods. While these multi-configuration methods allow for arbitrary electron arrangement within the CAS, effectively capturing the static correlation of the system, they struggle to handle the dynamic correlations beyond the CAS. Therefore, combining them with multi-reference configuration interaction (MRCI), multi-reference perturbation theory (MRPT), multi-reference coupled cluster theory (MRCC), and other multi-reference methods becomes essential to accurately handle strong correlation systems. Constructing multi-configuration wave functions to obtain static correlation by multi-configuration methods, and then generating excitation configurations based on these multi-configuration wave functions to obtain dynamic correlation has become the mainstream of the development of current wave function methods. Challenges in this direction include the exponential growth in computational complexity with an increase in CAS size and the precision of multi-reference methods affecting the accuracy of capturing dynamic correlations beyond the active space. Over the last decade, various multi-configuration methods, such as selected configuration interaction (sCI), Density matrix renormalization group (DMRG), full configuration interaction quantum Monte Carlo (FCIQMC), and neural-network quantum states (NQS), have successfully expanded the computable active space by explicitly or implicitly extracting and compressing complete configuration space information. Furthermore, combining these multi-configuration methods with traditional multi-reference approaches, such as MRCI, MRPT and MRCC, enables capturing dynamic correlations caused by electronic instantaneous interactions beyond the active space. This achieves higher precision and quantitative descriptions for mid-sized strongly correlated molecular systems (around 50 active orbitals). Additionally, novel methods like Block-Correlated Coupled Cluster Theory (BCCC) have made progress in attempting to discuss static and dynamic correlations within a unified framework. This article primarily introduces the advancements in wave function theory electronic structure methods for strongly correlated molecular systems in recent years. The first section details multi-configuration methods that have expanded the CAS size, including sCI, DMRG, FCIQMC, NQS, and outlines their developmental trajectories. The second section delves into the combination of these multi-configuration methods with traditional multi-reference approaches, leveraging larger active spaces to address dynamic correlations and achieve higher precision results. Finally, the article discusses the prospects of these methods in three directions: Handling static and dynamic correlations within a unified framework, integrating with relativistic effects, and combining with machine learning.
We present a symmetry-adapted extension of sample-based quantum diagonalization (SQD) that rigorously embeds space-group symmetry into the many-body subspace sampled by quantum hardware. The method is benchmarked on the two-leg ladder Hubbard model using both molecular orbital and momentum bases. Energy convergence is shown to be improved in the momentum basis compared to the molecular orbital basis for both the spin-quintet ground state and the spin-singlet excited state. We clarify the relationship between the compactness of the many-body wave function and the sparsity of the representation matrices of symmetry operations. Furthermore, the enhancement of the superconducting correlation function due to the Coulomb interaction is demonstrated. Our method highlights the importance of symmetry structure in random-sampling quantum simulation of correlated systems