Quantum Algorithm for Numerical Energy Gradient Calculations at the Full Configuration Interaction Level of Theory

A Bayesian phase difference estimation (BPDE) algorithm allows us to compute the energy gap of two electronic states of a given Hamiltonian directly by utilizing the quantum superposition of their wave functions. Here we report an extension of the BPDE algorithm to the direct calculation of the energy difference of two molecular geometries. We apply the BPDE algorithm for the calculation of numerical energy gradients based on the two-point finite-difference method, enabling us to execute geometry optimization of one-dimensional molecules at the full-CI level on a quantum computer. Results of numerical quantum circuit simulations of the geometry optimization of the H₂ molecule with the STO-3G and 6-31G basis sets, the LiH and BeH₂ molecules at the full-CI/STO-3G level, and the N₂ molecule at the CASCI(6e,6o)/6-311G* level are given.

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 C_2v 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.

Quantum Algorithm for Full Configuration Interaction Calculations without Controlled Time Evolutions

A quantum phase estimation algorithm allows us to perform full configuration interaction (full-CI) calculations on quantum computers with polynomial costs against the system size under study, but it requires quantum simulation of the time evolution of the wave function conditional on an ancillary qubit, which makes the algorithm implementation on real quantum devices difficult. Here, we discuss an application of the Bayesian phase difference estimation algorithm that is free from controlled time evolution operations to the full-CI calculations.

Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps

Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.

Smallest Organic Tetracation in the Gas Phase: Stability of Multiply Charged Diiodoacetylene Produced in Intense Femtosecond Laser Fields

Coulomb explosion imaging, which is the reconstruction of a molecular structure by measuring the three-dimensional momenta of atomic ions formed by a Coulomb explosion of multiply charged molecular cations (MMCs), has been utilized widely. In contrast, intact MMCs, whose properties and reactions are interesting from both fundamental and applied scientific perspectives, themselves have been little explored to date. This study demonstrates that the four-atom molecule diiodoacetylene (DIA) can survive as a long-lived species in the gas phase after the removal of four electrons in intense femtosecond laser fields. The electron configurations of the equilibrium structures of the electronic ground states calculated by the complete active space self-consistent field (CASSCF) method reveal the stability of multiply charged DIA. The dissociation energies are estimated to be 3.01, 3.59, 2.57, 1.82, and 1.61 eV for neutral, cation radical, dication, trication radical, and tetracation, respectively. A fairly deep potential well suggests that a DIA tetracation is metastable toward dissociation, whereas the repulsive potential of a pentacation radical confirms its absence in the mass spectrum. With their sufficiently long lifetimes, minimum number of atoms, and simple dissociation paths, DIA MMCs are promising candidates for further experimental and theoretical investigations of multiply charged ion chemistry.

Pulse Energy and Pulse Duration Effects in the Ionization and Fragmentation of Iodomethane by Ultraintense Hard X Rays

The interaction of intense femtosecond x-ray pulses with molecules sensitively depends on the interplay between multiple photoabsorptions, Auger decay, charge rearrangement, and nuclear motion. Here, we report on a combined experimental and theoretical study of the ionization and fragmentation of iodomethane (CH_{3}I) by ultraintense (∼10^{19}  W/cm^{2}) x-ray pulses at 8.3 keV, demonstrating how these dynamics depend on the x-ray pulse energy and duration. We show that the timing of multiple ionization steps leading to a particular reaction product and, thus, the product's final kinetic energy, is determined by the pulse duration rather than the pulse energy or intensity. While the overall degree of ionization is mainly defined by the pulse energy, our measurement reveals that the yield of the fragments with the highest charge states is enhanced for short pulse durations, in contrast to earlier observations for atoms and small molecules in the soft x-ray domain. We attribute this effect to a decreased charge transfer efficiency at larger internuclear separations, which are reached during longer pulses.

Quantum Algorithm for the Direct Calculations of Vertical Ionization Energies

Recently, a quantum algorithm that is capable of directly calculating the energy gap between two electronic states having different spin quantum numbers without inspecting the total energy of the individual electronic states was proposed. This quantum algorithm guarantees an exponential speedup, like quantum phase estimation (QPE)-based full-CI, with much lower costs. In this work, we propose a modified quantum circuit for the direct calculations of spin state energy gaps to reduce the number of qubits and quantum gates, extending the quantum algorithm to the direct calculation of vertical ionization energies. Numerical quantum circuit simulations for the ionization of light atoms (He, Li, Be, B, C, and N) and small molecules (HF, BF, CF, CO, O₂, NO, CN, F₂, H₂O, and NH₃) revealed that the proposed quantum algorithm affords the vertical ionization energies within 0.1 eV of precision.

A quantum algorithm for spin chemistry: a Bayesian exchange coupling parameter calculator with broken-symmetry wave functions

The Heisenberg exchange coupling parameter J (H = −2JS_i · S_j) characterises the isotropic magnetic interaction between unpaired electrons, and it is one of the most important spin Hamiltonian parameters of multi-spin open shell systems. The J value is related to the energy difference between high-spin and low-spin states, and thus computing the energies of individual spin states are necessary to obtain the J values from quantum chemical calculations. Here, we propose a quantum algorithm, B̲ayesian ex̲change coupling parameter calculator with b̲roken-symmetry wave functions (BxB), which is capable of computing the J value directly, without calculating the energies of individual spin states. The BxB algorithm is composed of the quantum simulations of the time evolution of a broken-symmetry wave function under the Hamiltonian with an additional term jS², the wave function overlap estimation with the SWAP test, and Bayesian optimisation of the parameter j. Numerical quantum circuit simulations for H₂ under a covalent bond dissociation, C, O, Si, NH, OH⁺, CH₂, NF, O₂, and triple bond dissociated N₂ molecule revealed that the BxB can compute the J value within 1 kcal mol⁻¹ of errors with less computational costs than conventional quantum phase estimation-based approaches.

A quantum algorithm “Bayesian exchange coupling parameter calculator with broken-symmetry wave function (BxB)” enables us to calculate Heisenberg exchange coupling parameter J without inspecting total energies of individual spin states, within 1 kcal mol⁻¹ of energy tolerance.

A probabilistic spin annihilation method for quantum chemical calculations on quantum computers

A probabilistic spin annihilation method based on the quantum phase estimation algorithm is presented for quantum chemical calculations on quantum computers. This approach can eliminate more than one spin component from the spin contaminated wave functions by single operation. Comparison with the spin annihilation operation on classical computers is given.

Trityl-Aryl-Nitroxide-Based Genuinely g-Engineered Biradicals, As Studied by Dynamic Nuclear Polarization, Multifrequency ESR/ENDOR, Arbitrary Wave Generator Pulse Microwave Waveform Spectroscopy, and Quantum Chemical Calculations

Trityl and nitroxide radicals are connected by π-topologically controlled aryl linkers, generating genuinely g-engineered biradicals. They serve as a typical model for biradicals in which the exchange (J) and hyperfine interactions compete with the g-difference electronic Zeeman interactions. The magnetic properties underlying the biradical spin Hamiltonian for solution, including J's, have been determined by multifrequency CW-ESR and ¹H ENDOR spectroscopy and compared with those obtained by quantum chemical calculations. The experimental J values were in good agreement with the quantum chemical calculations. The g-engineered biradicals have been tested as a prototype for AWG (Arbitrary Wave Generator)-based spin manipulation techniques, which enable GRAPE (GRAdient Pulse Engineering) microwave control of spins in molecular magnetic resonance spectroscopy for use in molecular spin quantum computers, demonstrating efficient signal enhancement of specific weakened hyperfine signals. Dynamic nuclear polarization (DNP) effects of the biradicals for 400 MHz nuclear magnetic resonance signal enhancement have been examined, giving efficiency factors of 30 for ¹H and 27.8 for ¹³C nuclei. The marked DNP results show the feasibility of these biradicals for hyperpolarization.

Quantum Chemistry on Quantum Computers: A Method for Preparation of Multiconfigurational Wave Functions on Quantum Computers without Performing Post-Hartree-Fock Calculations

The full configuration interaction (full-CI) method is capable of providing the numerically best wave functions and energies of atoms and molecules within basis sets being used, although it is intractable for classical computers. Quantum computers can perform full-CI calculations in polynomial time against the system size by adopting a quantum phase estimation algorithm (QPEA). In the QPEA, the preparation of initial guess wave functions having sufficiently large overlap with the exact wave function is recommended. The Hartree-Fock (HF) wave function is a good initial guess only for closed shell singlet molecules and high-spin molecules carrying no spin-β unpaired electrons, around their equilibrium geometry, and thus, the construction of multiconfigurational wave functions without performing post-HF calculations on classical computers is highly desired for applying the method to a wide variety of chemistries and physics. In this work, we propose a method to construct multiconfigurational initial guess wave functions suitable for QPEA-based full-CI calculations on quantum computers, by utilizing diradical characters computed from spin-projected UHF wave functions. The proposed approach drastically improves the wave function overlap, particularly in molecules with intermediate diradical characters.

Quantum chemistry on quantum computers: quantum simulations of the time evolution of wave functions under the S2 operator and determination of the spin quantum number S

Quantum computers have an enormous impact on quantum chemical calculations. Approaches to calculate the energies of atoms and molecules on quantum computers by utilizing quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) have been well documented, and dozens of methodological improvements to decrease computational costs and to mitigate errors have been reported until recently. However, the possible methodological implementation of observables on quantum computers such as calculating the spin quantum numbers of arbitrary wave functions, which is a crucial issue in quantum chemistry, has been discussed less. Here, we propose a quantum circuit to simulate the time evolution of wave functions under an S2 operator, exp(-iS2t)|Ψ, and integrate it into the QPE circuit enabling us to determine the spin quantum number of the arbitrary wave functions. We demonstrate that the spin quantum numbers of up to three spins can be determined by only one qubit measurement in QPE.

Analyses of sizable ZFS and magnetic tensors of high spin metallocomplexes

The fictitious spin-1/2 Hamiltonian approach is the putative method to analyze the fine-structure/hyperfine ESR spectra of high spin metallocomplexes having sizable zerofield splitting (ZFS), thus giving salient principal g-values far from around g = 2 without explicitly providing their ZFS parameters in most cases. Indeed, the significant departure of the g-values from g = 2 is indicative of the occurrence of their high spin states, but naturally they never agree with true g-values acquired by quantum chemical calculations such as sophisticated DFT or ab initio MO calculations. In this work, we propose facile approaches to determine the magnetic tensors of high spin metallocomplexes having sizable ZFS, instead of performing advanced high-field/high-frequency ESR spectroscopy. We have revisited analytical expressions for the relationship between effective g-values and true principal g-values for high spins. The useful analytical formulas for the g^eff-g^true relationships are given for S's up to 7/2. The genuine Zeeman perturbation formalism gives the exact solutions for S = 3/2, and for higher S's it is much more accurate than the pseudo-Zeeman perturbation approach documented so far (A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, 1970; J. R. Pilbrow, J. Magn. Reson., 1978, 31, 479; F. Trandafir et al., Appl. Magn. Reson., 2007, 31, 553; M. Fittipaldi et al., J. Phys. Chem. B, 2008, 112, 3859), in which the E(S_x² - S_y²) term is putatively treated to the second order. To show the usefulness of the present approach, we exploit Fe^III(Cl)OEP (S = 5/2) (OEP: 2,3,7,8,12,13,17,18-octaethylporphyrin) and Co^IIOEP (S = 3/2) well magnetically diluted in the diamagnetic host crystal lattice of Ni^IIOEP. The advantage of single-crystal ESR spectroscopy lies in the fact that the molecular information on the principal axes of the magnetic tensors is crucial in comparing with reliable theoretical results. In high spin states of metallocomplexes with sizable ZFS in pseudo-octahedral symmetry, their fine-structure ESR transitions for the principal z-axis orientation appear in the lower field far from g = 2 at the X-band, disagreeing with the putative intuitive picture obtained using relevant ESR spectroscopy. A Re^III,IV dinuclear complex in a mixed valence state exemplifies the cases, whose fine-structure/hyperfine ESR spectra of the neat crystals have been analyzed in their principal-axis system. The DFT-based/ab initio MO calculations of the magnetic tensors for all the high spin entities in this work were carried out.

ESR analyses of picket fence MnII and 6th ligand coordinated FeIII porphyrins (S = 5/2) and a CoII(hfac) complex (S = 3/2) with sizable ZFS parameters revisited: a full spin Hamiltonian approach and quantum chemical calculations

The conventional X-band ESR spectra are fully reanalyzed by a full spin Hamiltonian approach.

Behaviour of DFT-based approaches to the spin–orbit term of zero-field splitting tensors: a case study of metallocomplexes, MIII(acac)3 (M = V, Cr, Mn, Fe and Mo)

Zero-field splitting tensors of M^III(acac)₃ complexes are calculated using ab initio and DFT methods.

Structural Determination of a DNA Oligomer for a Molecular Spin Qubit Lloyd Model of Quantum Computers

The global molecular and local spin-site structures of a DNA duplex 22-oligomer with site-directed four spin-labeling were simulated by molecular mechanics (MM) calculations combined with Q-band pulsed electron-electron double resonance (PELDOR) spectroscopy. This molecular-spin bearing DNA oligomer is designed to give a complex testing ground for the structural determination of molecular spins incorporated in the DNA duplex, which serves as a platform for 1D periodic arrays of two or three non-equivalent electron spin qubit systems, (AB)n or (ABC)n, respectively, enabling to execute quantum computing or quantum information processing (Lloyd model of electron spin versions): A, B and C designate non-equivalent addressable spin qubits for quantum operations. The non-equivalence originates in difference in the electronic g-tensor. It is not feasible to determine the optimal structures for such DNA oligomers having molecular flexibility only by the MM calculations because there are many local minima in energy for their possible molecular structures. The spin-distance information derived from the PELDOR spectroscopy helps determine the optimal structures out of the possible ones acquired by the MM calculations. Based on the MM searched structures, we suggest the optimal structures for semi-macromolecules having site-directed multi-spin qubits. We emphasize that for our four molecular spins embedded in the DNA oligomer the Fajer’s error analysis in PELDOR-based distance measurements was of essential importance.

Adiabatic quantum computing with spin qubits hosted by molecules

Molecular spin QCs for adiabatic quantum computing: a phthalocyanine derivative with three electron qubits and a glutaconic acid radical with one electron bus qubit and two nuclear client qubits.

An ab initio MO study of heavy atom effects on the zero-field splitting tensors of high-spin nitrenes: how the spin–orbit contributions are affected

The CASSCF and hybrid CASSCF–MRMP2 methods reproduce the ZFS tensors of spin-septet 2,4,6-trinitrenopyridines, focusing on the heavy atom effects on the spin–orbit terms of the tensors.

A new series of bis(ene-1,2-dithiolato)tungsten(IV), -(V), -(VI) complexes as reaction centre models of tungsten enzymes: preparation, crystal structures and spectroscopic properties

The carbomethoxy substituted dithiolene ligand (L(COOMe)) enabled us to develop a series of new bis(ene-1,2-dithiolato)tungsten complexes including W(IV)O, W(IV)(OSiBuPh(2)), W(VI)O(2), W(VI)O(OSiBuPh(2)) and W(VI)O(S) core structures. By using these tungsten complexes, a systematic study of the terminal monodentate ligand effects has been performed on the structural, spectroscopic properties and reactivity. The structure and spectroscopic properties of the tungsten complexes have also been compared to those of the molybdenum complexes coordinated by the same ligand to investigate the effects of the metal ion (W vs. Mo). X-ray crystallographic analyses of the tungsten(IV) complexes have revealed that the tungsten centres adopt a distorted square pyramidal geometry with a dithiolene ligand having an ene-1,2-dithiolate form. On the other hand, the dioxotungsten(VI) complex exhibits an octahedral structure consisting of the bidentate L(COOMe) and two oxo groups, in which π-delocalization was observed between the W(VI)O(2) and ene-1,2-dithiolate units. The tungsten(IV) and dioxotungsten(VI) complexes are isostructural with the molybdenum counter parts. DFT calculation study of the W(VI)O(S) complex has indicated that the W=S bond of 2.2 Å is close to the bond length between the tungsten centre and ambiguously assigned terminal monodentate atom in aldehyde oxidoreductase of the tungsten enzyme. Resonance Raman (rR) spectrum of the W(VI)O(S) complex has shown the two inequivalent L(COOMe) ligands with respect to their bonding interactions with the tungsten centre, reproducing the appearance of two ν(C=C) stretches in the rR spectrum of aldehyde oxidoreductase. Sulfur atom transfer reaction from the W(VI)O(S) complex to triphenylphosphines has also been studied kinetically to demonstrate that the tungsten complex has a lower reactivity by about one-order of magnitude, when compared with its molybdenum counterpart.

Treatment with lactose (galactose)-restricted and medium-chain triglyceride-supplemented formula for neonatal intrahepatic cholestasis caused by citrin deficiency

Citrin plays a role in the transfer of NADH-reducing equivalent from cytosol to mitochondria as part of the malate-aspartate shuttle in liver. Citrin deficiency may cause an impairment of glycolysis due to an increase in the cytosolic NADH/NAD ratio leading to an energy shortage in the liver. Mutations of the SLC25A13 gene are responsible for neonatal intrahepatic cholestasis (NICCD) and adult-onset type II citrullinemia (CTLN2). Most patients with NICCD show a resolution of symptoms within the first year of life, but some patients present with severe symptoms and require liver transplantation. We treated four patients including three siblings with NICCD by lactose (galactose)-restricted and medium-chain triglyceride (MCT)-supplemented formula. This formula rapidly improved the clinical condition and laboratory findings. Early treatment was more effective and did not require long-term administration. Lactose (galactose)-restriction can avoid further increase in the cytosolic NADH/NAD ratio in the liver and MCT supplementation can provide energy to hepatic cells by producing an excess of acetyl-CoA in mitochondria. Early treatment with lactose (galactose)-restricted and MCT-supplemented formula is recommended for patients with NICCD and possibly for patients with CTLN2.

Random-orientation high-spin electron spin resonance spectroscopy and comprehensive spectral analyses of the quintet dicarbene and dinitrene with meta-topological linkers: origins of peculiar line-broadening in fine-structure ESR spectra in organic rigid glasses

In high-spin chemistry, random-orientation fine-structure (FS) electron spin resonance (ESR) spectroscopy entertains advantages as the most facile and convenient method to identify high-spin systems, as frequently reported in the literature. Random-orientation ESR spectroscopy applicable to organic high-spin entities can date back to the Wasserman and co-workers' attempt on the first spin-quintet dicarbene, m-phenylenebis(phenylmethlene) (m-PBPM), in the 2-MTHF glass in 1963 and 1967, following their successful work on randomly oriented triplet-state ESR spectroscopy. The FS ESR spectrum of m-PBPM in the 2-MTHF glass, however, has never fully been analyzed due to a peculiar line-broadening appearing at many canonical peaks. Organic high-spin spectra from most quintet dinitrenes also suffer from similar phenomena. Seemingly intrinsic line-diffusing or -broadening phenomena adversely affect the reliable determination of FS parameters for organic high-spin entities in rigid glasses. In high-spin chemistry, the line-broadening has been an obstacle that masks key FS transitions in many cases. Thus, both the origin of the broadening and the comprehensive spectral analysis have been a long-standing issue. In this report, we examine the origin of the line-broadening appearing in the FS ESR spectra, illustrated by a comprehensive spectral analysis for m-PBPM in the quintet ground state and the first-documented quintet-state dinitrene, m-phenylenebis(nitrene) (m-PBN) in the 2-MTHF glass. A complete analysis of the random-orientation FS spectra from m-PBPM diluted in the benzophenone crystal has shown that the g-anisotropy of m-PBPM is not prominent. Also the higher-order FS terms such as S(i)(2)S(j)(2) group-theoretically allowed for S = 2 are not necessary in spite of the argument for a hydrocarbon-based tetraradical (S = 2) in the ground state. Our new approach to the line-broadening analysis invokes both exact analytical solutions for the resonance fields of canonical peaks and the magnetic-parameters gradient method. The D- and E-values of m-PBPM acquired by the spectral simulation in this study give different molecular structures of the quintet dicarbene in the benzophenone crystal lattice (D = +0.0703(0) cm(-1), E = +0.0212(0) cm(-1)) and in the 2-MTHF glass (D = +0.0780(0) cm(-1), E = +0.0221(0) cm(-1)). Microscopic origins of the line-broadening observed for high-spin oligocarbenes or oligonitrenes generated by photolysis in organic glasses have been proposed.