Electronic Hamiltonian Research Articles

Fock-Space Schrieffer–Wolff Transformation: Classically-Assisted Rank-Reduced Quantum Phase Estimation Algorithm

We present an extension of many-body downfolding methods to reduce the resources required in the quantum phase estimation (QPE) algorithm. In this paper, we focus on the Schrieffer–Wolff (SW) transformation of the electronic Hamiltonians for molecular systems that provides significant simplifications of quantum circuits for simulations of quantum dynamics. We demonstrate that by employing Fock-space variants of the SW transformation (or rank-reducing similarity transformations (RRST)) one can significantly increase the locality of the qubit-mapped similarity-transformed Hamiltonians. The practical utilization of the SW-RRST formalism is associated with a series of approximations discussed in the manuscript. In particular, amplitudes that define RRST can be evaluated using conventional computers and then encoded on quantum computers. The SW-RRST QPE quantum algorithms can also be viewed as an extension of the standard state-specific coupled-cluster downfolding methods to provide a robust alternative to the traditional QPE algorithms to identify the ground and excited states for systems with various numbers of electrons using the same Fock-space representations of the downfolded Hamiltonian. The RRST formalism serves as a design principle for developing new classes of approximate schemes that reduce the complexity of quantum circuits.

Effect of the Thermal Fluctuations of the Photophysics of GC and CG DNA Steps: A Computational Dynamical Study

Here we refine and assess two computational procedures aimed to include the effect of thermal fluctuations on the electronic spectra and the ultrafast excited state dynamics of multichromophore systems, focusing on DNA duplexes. Our approach is based on a fragment diabatization procedure that, from a given Quantum Mechanical (QM) reference method, can provide the parameters (energy and coupling) of the reference diabatic states on the basis of the isolated fragments, either for a purely electronic excitonic Hamiltonian (FrDEx) or a linear vibronic coupling Hamiltonian (FrD-LVC). After having defined the most cost-effective procedure for DNA duplexes on two smaller fragments, FrDEx is used to simulate the absorption and Electronic Circular Dichroism (ECD) spectra of (GC)5 sequences, including the coupling with the Charge Transfer (CT) states, on a number of structures extracted from classical Molecular Dynamics (MD) simulations. The computed spectra are close to the reference TD-DFT calculations and fully consistent with the experimental ones. We then couple MD simulations and FrD-LVC to simulate the interplay between local excitations and CT transitions, both intrastrand and interstrand, in GC and CG steps when included in a oligoGC or in oligoAT DNA sequence. We predict that for both sequences a substantial part of the photoexcited population on G and C decays, within 50-100 fs, to the corresponding intrastrand CT states. This transfer is more effective for GC steps that, on average, are more closely stacked than CG ones.

Absorption band structure of the photochromic dimethyldihydropyrene/metacyclophanediene couple. Insight from vibronic coupling theory.

A detailed insight behind the structure of absorption bands of the photochromic couple dimethyldihydropyrene (DHP)/metacyclophanediene (CPD) is studied employing vibronic coupling theory. Two separate model molecular Hamiltonians, including a maximum of four electronic states and 18 vibrational modes for DHP and five electronic states and 20 vibrational modes for CPD, are constructed in a diabatic electronic representation. The parameters of the Hamiltonians are estimated from the electronic energies obtained from extensive density functional theory (DFT) and time-dependent DFT calculations. Based on these Hamiltonians' parameters, a detailed analysis of potential energy curves is performed in conjunction with positional and energetic locations of several stationary points in multi-dimensional potential energy surfaces. Based on the results of electronic structure calculations, quantum nuclear dynamics studies on the electronic excited states of DHP and CPD are performed to understand the impact of non-adiabatic effects on the formation of vibronic structures of absorption bands of these photo-isomers.

Relativistic multimode Jahn-Teller and pseudo-Jahn-Teller effects in tetrahedral systems

The relativistic pseudo-Jahn-Teller effect 2T×(t2+e)=(E1/2+G3/2)×(t2+e) and the Jahn-Teller effect 2E×(t2+e)=G3/2×(t2+e) are considered, employing the microscopic (Breit-Pauli) spin–orbit coupling operator. The orbital and spin-orbital parts of electronic Hamiltonian are expanded up to the second order terms in normal modes. Five-dimensional (t2+e) potential energy surfaces are obtained in analytical form as functions of invariants of the Td group.

The parallel-transported (quasi)-diabatic basis.

This article concerns the use of parallel transport to create a diabatic basis. The advantages of the parallel-transported basis include the facility with which Taylor series expansions can be carried out in the neighborhood of a point or a manifold such as a seam (the locus of degeneracies of the electronic Hamiltonian), and the close relationship between the derivative couplings and the curvature in this basis. These are important for analytic treatments of the nuclear Schrödinger equation in the neighborhood of degeneracies. The parallel-transported basis bears a close relationship to the singular-value basis; in this article, both are expanded in power series about a reference point and are shown to agree through second order but not beyond. Taylor series expansions are effected through the projection operator, whose expansion does not involve energy denominators or any type of singularity and in terms of which both the singular-value basis and the parallel-transported basis can be expressed. The parallel-transported basis is a version of Poincaré gauge, well known in electromagnetism, which provides a relationship between the derivative couplings and the curvature and which, along with a formula due to Mead, affords an efficient method for calculating Taylor series of the basis states and the derivative couplings. The case in which fine structure effects are included in the electronic Hamiltonian is covered.

Quantum Simulation of Molecules in Solution.

Quantum chemical calculations on quantum computers have been focused mostly on simulating molecules in the gas phase. Molecules in liquid solution are, however, most relevant for chemistry. Continuum solvation models represent a good compromise between computational affordability and accuracy in describing solvation effects within a quantum chemical description of solute molecules. In this work, we extend the variational quantum eigensolver to simulate solvated systems using the polarizable continuum model. To account for the state dependent solute-solvent interaction we generalize the variational quantum eigensolver algorithm to treat non-linear molecular Hamiltonians. We show that including solvation effects does not impact the algorithmic efficiency. Numerical results of noiseless simulations for molecular systems with up to 12 spin-orbitals (qubits) are presented. Furthermore, calculations performed on a simulated noisy quantum hardware (IBM Q, Mumbai) yield computed solvation free energies in fair agreement with the classical calculations.

Adiabatic theory for high Rydberg molecules

AbstractMolecules in high Rydberg states, or high Rydberg molecules, are special types of excited species. Classically, at least one of the valence electrons of such molecules is excited to a high‐energy near‐threshold orbit with a negative eccentricity. Therefore, the electron spends most of its orbiting time on roaming slowly far away from the parent ionic core. With this classical picture in mind, we develop a general adiabatic theory termed the inverse Born‐Oppenheimer approximation, in contrast to the conventional Born‐Oppenheimer adiabatic theory for molecules in ground or lower excited states, to describe high Rydberg molecules. A thorough theoretical formulation starting from the non‐relativistic molecular Hamiltonian and Schrödinger equation is presented with emphasis on the mathematical rigorousness of this theory. Quantitative calculations of the transition rate constants for many radiative and nonradiative processes involved in high Rydberg molecules are shown explicitly.

A unified framework of transformations based on the Jordan-Wigner transformation.

Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented and many existing transformations-such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations-are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter vector, which can be viewed as a generalization of the above-mentioned mappings. Furthermore, the MSP transformation can adjust flexibly when dealing with different systems. Applying these mappings to the electronic structure Hamiltonians of various molecules, the MSP transformation is found to perform better on a number of Pauli operators and gates needed in the circuit of Hamiltonian simulation. The MSP transformation will reduce the qubit gate requirement for Hamiltonian simulation on noisy intermediate-scale quantum devices, and it will provide a much wider choice of mappings for researchers.

Sum-of-products form of the molecular electronic Hamiltonian and application within the MCTDH method.

We introduce two different approaches to represent the second-quantized electronic Hamiltonian in a sum-of-products form. These procedures aim at mitigating the quartic scaling of the number of terms in the Hamiltonian with respect to the number of spin orbitals and thus enable applications to larger molecular systems. Here, we describe the application of these approaches within the multi-configuration time-dependent Hartree framework. This approach is applied to the calculation of eigenenergies of LiH and electronic ionization spectrum of H2O.

Fidelity Overhead for Nonlocal Measurements in Variational Quantum Algorithms.

Measuring quantum observables by grouping terms that can be rotated to sums of only products of Pauli ẑ operators (Ising form) is proven to be efficient in near term quantum computing algorithms. This approach requires extra unitary transformations to rotate the state of interest so that the measurement of a fragment's Ising form would be equivalent to the measurement of the fragment for the unrotated state. These extra rotations allow one to perform a fewer number of measurements by grouping more terms into the measurable fragments with a lower overall estimator variance. However, previous estimations of the number of measurements did not take into account nonunit fidelity of quantum gates implementing the additional transformations. Through a circuit fidelity reduction, additional transformations introduce extra uncertainty and increase the needed number of measurements. Here we consider a simple model for errors introduced by additional gates needed in schemes involving groupings of commuting Pauli products. For a set of molecular electronic Hamiltonians, we confirm that the numbers of measurements in schemes using nonlocal qubit rotations are still lower than those in their local qubit rotation counterparts, even after accounting for uncertainties introduced by additional gates.

Quantum expectation-value estimation by computational basis sampling

Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically-tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum states in the computational basis on quantum computers. Each of those states is prepared by applying a unitary transformation consisting of at most N CNOT gates, where N is the number of qubits, to a target quantum state whose expectation value is evaluated. Our algorithm is expected to require fewer measurements than conventional methods for a required statistical precision of the expectation value when the target quantum state is concentrated in particular computational basis states. We provide numerical comparisons of our method with existing ones for measuring electronic ground state energies (expectation values of electronic Hamiltonians for the lowest-energy states) of various small molecules. Numerical results show that our method can reduce the numbers of measurements to obtain the ground state energies for a targeted precision by several orders of magnitudes for molecules whose ground states are concentrated. Our results provide another route to measure expectation values of observables, which could accelerate the variational quantum algorithms.

Machine learning in interpretation of electronic core-level spectra

Electronic transitions involving core-level orbitals offer a localized, atomic-site and element specific peek window into statistical systems such as molecular liquids. Although formally understood, the complex relation between structure and spectrum – and the effect of statistical averaging of highly differing spectra of individual structures – render the analysis of an ensemble-averaged core-level spectrum complicated. We explore the applicability of machine learning for molecular structure — core-level spectrum interpretation. We focus on the electronic Hamiltonian using the H2O molecule in the classical-nuclei approximation as our test system. For a systematic view we studied both predicting structures from spectra and, vice versa, spectra from structures, using polynomial approaches and neural networks. We find predicting spectra easier than predicting structures, where a tighter grid (even unphysical) of the spectrum improves prediction, possibly inviting for over-interpretation of the model. The accuracy of the structure prediction worsens when moving outwards from the center of mass of the training set in the structural parameter space, which cannot be overcome by model selection based on generalizability.

Magnetic Orderings from Spin–Orbit Coupled Electrons on Kagome Lattice

We investigate magnetic orderings on kagome lattice numerically from the tight-binding Hamiltonian of electrons, governed by the filling factor and spin-orbit coupling (SOC) of electrons. We find that even a simple kagome lattice model can host both ferromagnetic and noncollinear antiferromagnetic orderings depending on the electron filling, reflecting gap structures in the Dirac and flat bands characteristic to the kagome lattice. Kane--Mele- or Rashba-type SOC tends to stabilize noncollinear orderings, such as magnetic spirals and 120-degree antiferromagnetic orderings, due to the effective Dzyaloshinskii--Moriya interaction from SOC. The obtained phase structure helps qualitative understanding of magnetic orderings in various kagome-layered materials with Weyl or Dirac electrons.

Exact Results for the Tavis-Cummings and Hückel Hamiltonians with Diagonal Disorder.

We present an exact method to calculate the electronic states of one electron Hamiltonians with diagonal disorder. We show that in cases where the disorder has a Cauchy distribution, the disorder averaged one particle Green's function can be calculated directly, using a deterministic, complex (non-Hermitian) Hamiltonian. For this we use the supersymmetric method which has already been used in problems of solid state physics. Using the method we find exact solution for the case of N molecules with site disorder, confined to a microcavity, for any value of N. Our analysis shows that the width of the polaritonic states as a function of N depends on the nature of disorder, and hence it can be used to probe the way molecular energy levels are distributed. We also show how one can find exact results for Hückel type Hamiltonians with on-site Cauchy disorder and demonstrate its use.

High-order contact transformations of molecular Hamiltonians: general approach, fast computational algorithm and convergence of ro-vibrational polyad models

The paper describes methods and fast computational algorithm for building effective Hamiltonians in molecular physics using perturbative approach. Separations of fast and slow variables are considered in the framework of contact transformations (CT). The particular focus is on a systematic derivation of effective models for rovibrational spectroscopy from ab initio-based potential energy surfaces with an exhaustive review of previous studies in this field. We consider applications to several types of polyads coupled by Fermi, Coriolis, Darling-Dennison and other types of resonance interactions with examples for asymmetric top, symmetric top and spherical top molecules. A flexible choice of the modelling operator accounts for strong couplings of various types of nuclear motion in molecules among closely lying levels including vibrational resonance schemes (2:1:2 . ), (2:1:2:1), (4:2:6:3), (3:2:1:2:1:1), etc. that occur for C2v, C3v and Td molecules and their isotopic species. The method is implemented in the MOL_CT programme suite, which offers a complementary tool to variational methods in terms of convergence and computational time. The range of applications is also different. The goal of the CT method is providing mathematical models for analyses of molecular spectra with the high-resolution accuracy using physically meaningful parameters derived from ab initio functions.

Adiabatic state preparation of correlated wave functions with nonlinear scheduling functions and broken-symmetry wave functions

Adiabatic state preparation (ASP) can generate the correlated wave function by simulating the time evolution of wave function under the time-dependent Hamiltonian that interpolates the Fock operator and the full electronic Hamiltonian. However, ASP is inherently unsuitable for studying strongly correlated systems, and furthermore practical computational conditions for ASP are unknown. In quest for the suitable computational conditions for practical applications of ASP, we performed numerical simulations of ASP in the potential energy curves of N2, BeH2, and in the C2v quasi-reaction pathway of the Be atom insertion to the H2 molecule, examining the effect of nonlinear scheduling functions and the ASP with broken-symmetry wave functions with the S2 operator as the penalty term, contributing to practical applications of quantum computing to quantum chemistry. Eventually, computational guidelines to generate the correlated wave functions having the square overlap with the complete-active space self-consistent field wave function close to unity are discussed.

Understanding of the Photodetachment Spectrum of Anionic Mixed Carbon-Boron Cluster C3B5- Following Adiabatic and Nonadiabatic Quantum Chemistry Approaches.

The presence of nonadiabaticity in the photodetachment bands of the anionic mixed carbon-boron cluster C3B5- has been realized through ab initio electronic structure calculations and detailed analyses of quantum dynamics study on top of those electronic structures. In the course of our study, we traverse extensive first principles electronic structure calculations to compute potential energy curves and to trace the energetic locations for the conical intersections in the multidimensional surfaces. All the ab initio calculations are performed on the four low-lying electronic states of the C3B5 cluster, while quantum nuclear dynamics are pursued on those electronic states by applying both time-dependent and time-independent quantum chemistry frameworks. In particular, we rely on the diabatic electronic representation to construct the molecular Hamiltonian. Altogether, the simulated theoretical spectra offer exceptional agreement with the experimental attainments.

Electron Communications and Correlations in Subsystems

Abstract: The quantum entanglement of molecular fragments in reactive systems is approached. The "external" (inter-fragment) and “internal” (intra-fragment) correlation energies are expressed in terms of the DFT average correlation holes resulting from the coupling constant integration of the scaled electron repulsion terms in the electronic Hamiltonian. Information networks in the local and configuration resolutions are examined, and their conditional entropy (covalency) and mutual information (iconicity) descriptors are summarized. The local channels in the single Slater determinant approximation of HF theory are explored in some detail. The multisite events in the bond system for the specified molecular state are tackled, cascade (bridge) propagations are examined, and the Fermi (exchange) correlation of HF theory is discussed. The partial density matrices of interacting fragments are introduced, and their role in shaping the ensemble averages of physical observables and effective communications within reactants is examined.

B800-to-B850 relaxation of excitation energy in bacterial light harvesting: All-state, all-mode path integral simulations.

We report fully quantum mechanical simulations of excitation energy transfer within the peripheral light harvesting complex (LH2) of Rhodopseudomonas molischianum at room temperature. The exciton-vibration Hamiltonian comprises the 16 singly excited bacteriochlorophyll states of the B850 (inner) ring and the 8 states of the B800 (outer) ring with all available electronic couplings. The electronic states of each chromophore couple to 50 intramolecular vibrational modes with spectroscopically determined Huang-Rhys factors and to a weakly dissipative bath that models the biomolecular environment. Simulations of the excitation energy transfer following photoexcitation of various electronic eigenstates are performed using the numerically exact small matrix decomposition of the quasiadiabatic propagator path integral. We find that the energy relaxation process in the 24-state system is highly nontrivial. When the photoexcited state comprises primarily B800 pigments, a rapid intra-band redistribution of the energy sharply transitions to a significantly slower relaxation component that transfers 90% of the excitation energy to the B850 ring. The mixed character B850* state lacks the slow component and equilibrates very rapidly, providing an alternative energy transfer channel. This (and also another partially mixed) state has an anomalously large equilibrium population, suggesting a shift to lower energy by virtue of exciton-vibration coupling. The spread of the vibrationally dressed states is smaller than that of the eigenstates of the bare electronic Hamiltonian. The total population of the B800 band is found to decay exponentially with a 1/e time of 0.5ps, which is in good agreement with experimental results.

Neural Error Mitigation of Near-Term Quantum Simulations

Near-term quantum computers provide a promising platform for finding the ground states of quantum systems, which is an essential task in physics, chemistry and materials science. However, near-term approaches are constrained by the effects of noise, as well as the limited resources of near-term quantum hardware. We introduce neural error mitigation, which uses neural networks to improve estimates of ground states and ground-state observables obtained using near-term quantum simulations. To demonstrate our method’s broad applicability, we employ neural error mitigation to find the ground states of the H2 and LiH molecular Hamiltonians, as well as the lattice Schwinger model, prepared via the variational quantum eigensolver. Our results show that neural error mitigation improves numerical and experimental variational quantum eigensolver computations to yield low energy errors, high fidelities and accurate estimations of more complex observables such as order parameters and entanglement entropy without requiring additional quantum resources. Furthermore, neural error mitigation is agnostic with respect to the quantum state preparation algorithm used, the quantum hardware it is implemented on and the particular noise channel affecting the experiment, contributing to its versatility as a tool for quantum simulation. So-called noisy intermediate-scale quantum devices will be capable of a range of quantum simulation tasks, provided that the effects of noise can be sufficiently reduced. A neural error mitigation approach is developed that uses neural networks to improve the estimates of ground states and ground-state observables of molecules and quantum systems obtained using quantum simulations on near-term devices.