Ground State Energy Research Articles

A computational study and analysis of Variational Quantum Eigensolver over multiple parameters for molecules and ions

AbstractMaterial discovery is a phenomenon practiced since the evolution of the world. The discovery of materials has led to significant development in varied fields such as Science, Engineering and Technology. Computationally simulating molecules has been an area of interest in the industry as well as academia. However, simulating large molecules can be computationally expensive in terms of computing power and complexity. Quantum computing is a recent development that can improve the efficiency in predicting properties of atoms and molecules which will be useful for material design. The Variational Quantum Eigensolver (VQE) is one such quantum algorithm used to calculate the ground state energy of molecules or ions. In this study, we have done a comparative analysis of the parameters that constitute the VQE algorithm. This includes components such as basis, qubit mapping, ansatz, and optimizers used. We have also developed a database consisting of 79 single atoms and their variations of oxidation states and 33 molecules with the data of their Hamiltonian and ground state energy and dipole moment.

A circuit-generated quantum subspace algorithm for the variational quantum eigensolver.

Recent research has shown that wavefunction evolution in real and imaginary time can generate quantum subspaces with significant utility for obtaining accurate ground state energies. Inspired by these methods, we propose combining quantum subspace techniques with the variational quantum eigensolver (VQE). In our approach, the parameterized quantum circuit is divided into a series of smaller subcircuits. The sequential application of these subcircuits to an initial state generates a set of wavefunctions that we use as a quantum subspace to obtain high-accuracy groundstate energies. We call this technique the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By benchmarking CSVQE on a range of quantum chemistry problems, we show that it can achieve significant error reduction in the best case compared to conventional VQE, particularly for poorly optimized circuits, greatly improving convergence rates. Furthermore, we demonstrate that when applied to circuits trapped at local minima, CSVQE can produce energies close to the global minimum of the energy landscape, making it a potentially powerful tool for diagnosing local minima.

A qubit-efficient variational selected configuration-interaction method

Abstract Finding the ground-state energy of molecules is an important and challenging computational problem for which quantum computing can potentially find efficient solutions. The variational quantum eigensolver (VQE) is a quantum algorithm that tackles the molecular groundstate problem and is regarded as one of the flagships of quantum computing. Yet, to date, only very small molecules were computed via VQE, due to high noise levels in current quantum devices. Here we present an alternative variational quantum scheme that requires significantly less qubits than VQE. The reduction in the qubit number allows for shallower circuits to be sufficient, rendering the method more resistant to noise. The proposed algorithm, termed variational quantum selected-configuration-interaction (VQ-SCI), is based on: (a) representing the target groundstate as a superposition of Slater determinant configurations, encoded directly upon the quantum computational basis states; and (b) selecting a-priory only the most dominant configurations. This is demonstrated through a set of groundstate calculations of the H2, LiH, BeH2, H2O, NH3 and C2H4 molecules in the sto-3g basis set, performed on IBM quantum devices. We show that the VQ-SCI reaches the full configuration interaction energy within chemical accuracy using the lowest number of qubits reported to date. Moreover, when the SCI matrix is generated ‘on the fly’, the VQ-SCI requires exponentially less memory than classical SCI methods. This offers a potential remedy to a severe memory bottleneck problem in classical SCI calculations. Finally, the proposed scheme is general and can be straightforwardly applied for finding the groundstate of any Hermitian matrix, outside the chemical context.

Improved modularity and new features in ipie: Toward even larger AFQMC calculations on CPUs and GPUs at zero and finite temperatures.

ipie is a Python-based auxiliary-field quantum Monte Carlo (AFQMC) package that has undergone substantial improvements since its initial release [Malone et al., J. Chem. Theory Comput. 19(1), 109-121 (2023)]. This paper outlines the improved modularity and new capabilities implemented in ipie. We highlight the ease of incorporating different trial and walker types and the seamless integration of ipie with external libraries. We enable distributed Hamiltonian simulations of large systems that otherwise would not fit on a single central processing unit node or graphics processing unit (GPU) card. This development enabled us to compute the interaction energy of a benzene dimer with 84 electrons and 1512 orbitals with multi-GPUs. Using CUDA and cupy for NVIDIA GPUs, ipie supports GPU-accelerated multi-slater determinant trial wavefunctions [Huang et al. arXiv:2406.08314 (2024)] to enable efficient and highly accurate simulations of large-scale systems. This allows for near-exact ground state energies of multi-reference clusters, [Cu2O2]2+ and [Fe2S2(SCH3)4]2-. We also describe implementations of free projection AFQMC, finite temperature AFQMC, AFQMC for electron-phonon systems, and automatic differentiation in AFQMC for calculating physical properties. These advancements position ipie as a leading platform for AFQMC research in quantum chemistry, facilitating more complex and ambitious computational method development and their applications.

Qudit-based variational quantum eigensolver using photonic orbital angular momentum states.

Solving the electronic structure problem is a notorious challenge in quantum chemistry and material science. Variational quantum eigensolver (VQE) is a promising hybrid classical-quantum algorithm for finding the lowest-energy configuration of a molecular system. However, it typically requires many qubits and quantum gates with substantial quantum circuit depth to accurately represent the electronic wave function of complex structures. Here, we propose an alternative approach to solve the electronic structure problem using VQE with a single qudit. Our approach exploits a high-dimensional orbital angular momentum state of a heralded single photon and notably reduces the required quantum resources compared to conventional multi-qubit-based VQE. We experimentally demonstrate that our single-qudit-based VQE can efficiently estimate the ground state energy of hydrogen (H2) and lithium hydride (LiH) molecular systems corresponding to two- and four-qubit systems, respectively. We believe that our scheme opens a pathway to perform a large-scale quantum simulation for solving more complex problems in quantum chemistry and material science.

Unveiling Mechanism of Temperature‐Dependent Self‐Trapped Exciton Emission in 1D Hybrid Organic–Inorganic Tin Halide for Advanced Thermography

AbstractLead‐free hybrid metal halide phosphors/crystals showing self‐trapped exciton (STE) emission have been recently explored for thermography due to the strong temperature dependence of their photoluminescence (PL) lifetime (τ). However, realizing high‐spatial‐resolution thermography using polycrystalline powders or crystals presents a challenge. Moreover, the underlying mechanism of temperature‐dependent STE remains elusive. Herein, a homogeneous 1D ODASn2I6 (ODA, 1,8‐octanediamine) nm‐scale thin film exhibiting efficient STE emission is investigated. The PL decay shows a strong temperature dependence from 275 K (τ ≈ 1.31 µs) to 350 K (τ ≈ 0.65 µs) yielding a thermal sensitivity of 0.014 K−1. By employing temperature‐dependent transient absorption spectroscopy, detailed information is obtained about the relaxation processes prior to the STE formation. Simultaneous analyses of steady‐state and time‐resolved spectroscopies lead to a self‐consistent model where the thermally activated phonon‐assisted nonradiative pathway explains the temperature dependence of the PL lifetime via a conical intersection between the ground state and STE potential energy surfaces. Finally, a discernible 50 ns variation in PL lifetimes across different heated regimes over a distance of 1.15 mm is successfully demonstrated with fluorescence lifetime imaging microscopy, underscoring the substantial potential of ODASn2I6 thin film for high‐spatial‐resolution thermography.

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {a}$$\\end{document} in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\\pi \\rho \\mathfrak {a}$$\\end{document} and shows that corrections are smaller than Cρa(ρa3)1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\rho \\mathfrak {a} (\\rho {{\\mathfrak {a}}}^3)^{1/2}$$\\end{document}, for a sufficiently large constant C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C > 0$$\\end{document}. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order ρa(ρa3)1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\mathfrak {a}(\\rho {{\\mathfrak {a}}}^3)^{1/2}$$\\end{document}, in agreement with the Lee–Huang–Yang prediction.

Mapping quantum circuits to shallow-depth measurement patterns based on graph states

Abstract The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions—a so-called standard form—which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus, enabling a significant decrease in circuit depth for certain applications. We show that groups of mutually-commuting operators can be implemented using fully-parallel, i.e. non-adaptive, measurements within our protocol. In addition, we discuss how groups of mutually commuting observables can be simulatenously measured by adjusting the resource state, rather than performing a costly basis transformation prior to the measurement as it is done in the circuit model. Finally, we demonstrate the utility of our technique on two examples of high practical relevance—the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver (VQE) for the ground-state energy estimation of the water molecule. For the VQE, we find a reduction of the depth by a factor of 4 to 5 using measurement patterns vs. the standard circuit model. At the same time, since we incorporate the simultaneous measurements, our patterns allow us to save shots by a factor of at least 3.5 compared to measuring Pauli strings individually in the circuit model.

Stochastic Resolution of Identity to CC2 for Large Systems: Ground State and Triplet Excitation Energy Calculations.

An implementation of stochastic resolution of identity to the CC2 (sRI-CC2) ground state energy followed by triplet excitation energy calculations is presented. A set of stochastic orbitals is introduced to further decouple the expensive 4-index electron repulsion integrals on the basis of RI approximation. A Laplace transformation of the orbital energy difference denominators into numerical summations is adopted to obtain a third-order overall scaling. We select a series of hydrogen dimer chains with nearly thousands of electrons, as well as some other molecules, for sRI-CC2 energies and test the accuracy and time consumption in comparison with those of RI-CC2. Our sRI-CC2 results reproduce a modest agreement with the RI-CC2 in Q-Chem program package and it allows a steep scaling reduction from O(N5) to O(N3). Besides, the unrestricted sRI-CC2 calculations also fit well with the restricted results. Thus, our sRI-CC2 implementation of ground state energy and triplet excitation energy provides a cost-efficient alternative approach, especially for some large-sized systems.

Model and Energy Bounds for a Two-Dimensional System of Electrons Localized in Concentric Rings.

We study a two-dimensional system of interacting electrons confined in equidistant planar circular rings. The electrons are considered spinless and each of them is localized in one ring. While confined to such ring orbits, each electron interacts with the remaining ones by means of a standard Coulomb interaction potential. The classical version of this two-dimensional quantum model can be viewed as representing a system of electrons orbiting planar equidistant concentric rings where the kinetic energy may be discarded when one is searching for the lowest possible energy. Within this framework, the lowest possible energy of the system is the one that minimizes the total Coulomb interaction energy. This is the equilibrium energy that is numerically determined with high accuracy by using the simulated annealing method. This process allows us to obtain both the equilibrium energy and position configuration for different system sizes. The adopted semi-classical approach allows us to provide reliable approximations for the quantum ground state energy of the corresponding quantum system. The model considered in this work represents an interesting problem for studies of low-dimensional systems, with echoes that resonate with developments in nanoscience and nanomaterials.

A hybrid method integrating Green's function Monte Carlo and projected entangled pair states

Abstract This paper introduces a hybrid approach combining Green’s function Monte Carlo (GFMC) method with projected entangled pair state (PEPS) ansatz. This hybrid method regards PEPS as a trial state and a guiding wave function in GFMC. By leveraging PEPS’s proficiency in capturing quantum state entanglement and GFMC’s efficient parallel architecture, the hybrid method is well-suited for the accurate and efficient treatment of frustrated quantum spin systems. As a benchmark, we applied this approach to study the frustrated J 1-J 2 Heisenberg model on a square lattice with periodic boundary conditions (PBC). Compared with other numerical methods, our approach integrating PEPS and GFMC shows competitive accuracy in the performance of ground-state energy. This paper provides systematic and comprehensive discussion of the approach of our previous work [1].

Variational quantum imaginary time evolution for matrix product state Ansatz with tests on transcorrelated Hamiltonians.

The matrix product state (MPS) Ansatz offers a promising approach for finding the ground state of molecular Hamiltonians and solving quantum chemistry problems. Building on this concept, the proposed technique of quantum circuit MPS (QCMPS) enables the simulation of chemical systems using a relatively small number of qubits. In this study, we enhance the optimization performance of the QCMPS Ansatz by employing the variational quantum imaginary time evolution (VarQITE) approach. Guided by McLachlan's variational principle, the VarQITE method provides analytical metrics and gradients, resulting in improved convergence efficiency and robustness of the QCMPS. We validate these improvements numerically through simulations of H2, H4, and LiH molecules. In addition, given that VarQITE is applicable to non-Hermitian Hamiltonians, we evaluate its effectiveness in preparing the ground state of transcorrelated Hamiltonians. This approach yields energy estimates comparable to the complete basis set (CBS) limit while using even fewer qubits. In particular, we perform simulations of the beryllium atom and LiH molecule using only three qubits, maintaining high fidelity with the CBS ground state energy of these systems. This qubit reduction is achieved through the combined advantages of both the QCMPS Ansatz and transcorrelation. Our findings demonstrate the potential practicality of this quantum chemistry algorithm on near-term quantum devices.

More quantum chemistry with fewer qubits

Quantum computation is one of the most promising new paradigms for the simulation of physical systems composed of electrons and atomic nuclei, with applications in chemistry, solid-state physics, materials science, and molecular biology. This requires a truncated representation of the electronic structure Hamiltonian using a finite number of orbitals. While it is, in principle, obvious how to improve on the representation by including more orbitals, this is usually unfeasible in practice (e.g., because of the limited number of qubits available) and severely compromises the accuracy of the obtained results. Here, we propose a quantum algorithm that improves on the representation of the physical problem by virtue of second-order perturbation theory. In particular, our quantum algorithm evaluates the second-order energy correction through a series of time-evolution steps under the unperturbed Hamiltonian. An important application is to go beyond the active-space approximation, allowing to include corrections of virtual orbitals, known as multireference perturbation theory. Here, we exploit the fact that the unperturbed Hamiltonian is diagonal for virtual orbitals and show that the number of qubits is independent of the number of virtual orbitals. This gives rise to more accurate energy estimates without increasing the number of qubits. Moreover, we demonstrate numerically for realistic chemical systems that the total runtime has highly favorable scaling in the number of virtual orbitals compared to previous studies. Numerical calculations confirm the necessity of the multireference perturbation theory energy corrections to reach accurate ground-state energy estimates. Our perturbation theory quantum algorithm can also be applied to symmetry-adapted perturbation theory. As such, we demonstrate that perturbation theory can help to reduce the quantum hardware requirements for quantum chemistry. Published by the American Physical Society 2024

Effect of magnetic and electric field on the ground state properties of weak coupling piezoelectric polaron in a quantum well with asymmetric Gaussian confinement

In this study, we employed the linear combination operation and a modified Lee-Low-Pines unitary transformation method to explore the ground state properties of an interacting electron coupled with piezo-acoustic phonons within a quantum well featuring asymmetric Gaussian confinement potential well under the influence of magnetic and electric fields. We calculated various properties, including the ground state energy, ground state binding energy, magnetic moment, magnetic susceptibility, and polarizability of the weakly coupled piezoelectric polaron. The effects of the magnetic and electric fields, electron-phonon weak coupling strength, Debye cut-off wavenumber, and confinement depth and range on these properties were thoroughly analyzed. Our findings reveal that the ground state energy decreases with increasing the Debye wavenumber, confinement range, and electron-phonon weak coupling strength. In contrast, the ground state binding energy shows the opposite trend, increasing with these parameters. Furthermore, the magnetic and electric fields, confinement depth and range, electron-phonon coupling strength, and Debye wavenumber are critical factors that significantly impact the magnetic and optical properties of the piezoelectric polaron.

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

Investigation of spin polarized semiconductor Cs2XMoBr6 (X = Li, Na) compounds for spintronic and optoelectronic applications

This study has applied the density functional theory (DFT) to calculate the structural, magnetoelectronic, optical, and thermoelectric characteristics of Cs2XMoBr6 (X = Li, Na). The magnetic stability of studied compounds is examined by minimizing their total ground state energies. Also, both compounds exhibit cubic stability by tolerance factor as well as negative enthalpy of formation. The analysis of spin-dependent electronic properties reveals the semiconductor nature of mentioned materials in both spin-up/dn versions. The computed Eg of 0.90/1.99 eV in spin-up/dn sections for Cs2NaMoBr6 while Cs2LiMoBr6 reveals Eg of 1.99/1.21 eV in spin-up/dn version. The optical properties show that incident light is optimally absorbed in visible to ultraviolet areas, indicating their potential use in UV-based photo sensors and other photovoltaic gadgets. The Mo-atom primarily participate to the total magnetic moment (μB(Tot)) for Cs2XMoBr6. Further, the material's high figure of merit (ZT) >0.7 within the 300 - 800 K temperature range for Cs2XMoBr6 indicate its high efficiency in thermoelectric (TE) applications. Semiconducting electronic bands with highly asymmetric spin polarization as well as greater TE efficiency, Cs2XMoBr6 materials are appropriate for uses in spintronics and energy harvesting.

Exciting DeePMD: Learning excited-state energies, forces, and non-adiabatic couplings.

We extend the DeePMD neural network architecture to predict electronic structure properties necessary to perform non-adiabatic dynamics simulations. While learning the excited state energies and forces follows a straightforward extension of the DeePMD approach for ground-state energies and forces, how to learn the map between the non-adiabatic coupling vectors (NACV) and the local chemical environment descriptors of DeePMD is less trivial. Most implementations of machine-learning-based non-adiabatic dynamics inherently approximate the NACVs, with an underlying assumption that the energy-difference-scaled NACVs are conservative fields. We overcome this approximation, implementing the method recently introduced by Richardson [J. Chem. Phys. 158, 011102 (2023)], which learns the symmetric dyad of the energy-difference-scaled NACV. The efficiency and accuracy of our neural network architecture are demonstrated through the example of the methaniminium cation CH2NH2+.

Stability of spin-spiral magnetic structures in Mn2PtSn

The stability of a long-periodic homogeneous spin-spiral configuration in an inverse tetragonal Heusler compound, Mn2PtSn, is studied with the help of density functional theory calculations. The energetically most stable collinear magnetic state in this system is the ferrimagnetic one. However, the existence of negative phonon frequency makes this configuration dynamically unstable. The energy dispersion plots reveal that an energy minimum exists at q=0.1 along [100] and [110] propagating directions, which correspond to a stable non-collinear configuration compared to the collinear spin states. The inclusion of spin–orbit coupling further reduces the ground-state energy without changing the q-vector of the energy minima. The cycloidal spiral configuration, where the spins rotate at an angle of 36° along the propagating direction, is found to be more stable than the screw spiral configuration. The calculated density of state plots further supports the stability of the non-collinear cycloidal spin order. This stable, non-collinear spin-spiral configuration of Mn2PtSn makes this compound a prospective material for spintronics device applications.

Rotaxane Based Molecular Junctions Have Emerged as Promising Components For Electronic and Optoelectronic Applications

Background: Rotaxane molecule consists of three main parts: wheel, axle, and stoppers. This study was conducted to increase the number of wheels to get the electronic, electrical and thermal properties of Nano molecular junctions based on rotaxane. Materials and Methods: Calculations were performed to evaluate the transport properties and the geometrically optimization of these molecular junctions using a combination of DFT, SIESTA code, Gollum code and a non-equilibrium Green’s function formalism. Each molecule was attached to opposing 35-atom (111) directed pyramidal gold electrodes. DFT and GGA were used to compute the ground state energy of various molecular junctions. Results: The results of the T(E) showed high values, and this is evidence of that constructive interference that is controlled and reinforced by changing the number of wheels in the rotaxane molecules, which affected the HOMO-LUMO gap and thus contributed to raising or lowering the T(E) values. The relationship between G/Go and thermopower showed that the highest G/Go yields to low thermopower. The decay constant is a fundamental property that influences the transmission of electrons, conductance properties, and threshold voltage in the rotaxane molecules. Conclusion: Molecular junctions based on Rotaxane showed high values of the T(E), which increase the electrical conductivity, making them promising for electronic applications. The smaller value of threshold voltage is a useful property for different applications. The results of power factor bring to us an important conclusion, which is that the impact of G/G0 is more dominant than that of thermopower in the case of rotaxane molecular junctions.

Ferrimagnetic half metals TaMnZ (Z = As, Sb, Bi) for thermoelectric and spintronic applications – Material computations

Half metals possess coupling behavior of metals and semiconductors with wide heat transport and spin transport properties. This research article covers the structural, mechanical, electronic, magnetic and thermoelectric properties of the half Heusler alloys TaMnZ (Z = As, Sb, Bi). The equilibrium lattice constants and corresponding ground state energies show that the studied alloys are stable in the ferrimagnetic cubic phase. The band dispersion plots suggest that the compounds TaMnAs, TaMnSb and TaMnBi are half metals with indirect band gap having the band gap energies of 1.13 eV, 1.24 eV and 1.12 eV respectively in the spin down channel. As the materials have 100 % spin polarization with an integer magnetic moment of −1 μB, they are suitable for making spintronic devices such as spin transistors and spin-flip flops. The temperature-dependent thermoelectric properties of the alloys have been studied by classical Boltzmann theory. The materials have low lattice thermal conductivity which was confirmed by Slack’s equation. The obtained thermoelectric figure of merit for p-type TaMnAs, TaMnSb and TaMnBi is 0.7, 0.6 and 0.8 respectively at 1200 K. Similarly, the figure of merit obtained for n-type TaMnAs, TaMnSb and TaMnBi is 0.6, 0.64 and 0.8 respectively at the same temperature. The favourable heat and spin transport properties of these alloys show that they are the potential characters to figure out better thermoelectric and spintronic performance.