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.

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.

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.

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.

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).

Spin-orbit contributions to the zero-field splitting (ZFS) tensor (DSO tensor) of MIII(acac)3 complexes (M = V, Cr, Mn, Fe and Mo; acac = acetylacetonate anion) are evaluated by means of ab initio (a hybrid CASSCF/MRMP2) and DFT (Pederson-Khanna (PK) and natural orbital-based Pederson-Khanna (NOB-PK)) methods, focusing on the behaviour of DFT-based approaches to the DSO tensors against the valence d-electron configurations of the transition metal ions in octahedral coordination. Both the DFT-based approaches reproduce trends in the D tensors. Significantly, the differences between the theoretical and experimental D (D = DZZ - (DXX + DYY)/2) values are smaller in NOB-PK than in PK, emphasising the usefulness of the natural orbital-based approach to the D tensor calculations of transition metal ion complexes. In the case of d2 and d4 electronic configurations, the DSO(NOB-PK) values are considerably underestimated in the absolute magnitude, compared with the experimental ones. The DSO tensor analysis based on the orbital region partitioning technique (ORPT) revealed that the DSO contributions attributed to excitations from the singly occupied region (SOR) to the unoccupied region (UOR) are significantly underestimated in the DFT-based approaches to all the complexes under study. In the case of d3 and d5 configurations, the (SOR → UOR) excitations contribute in a nearly isotropic manner, which causes fortuitous error cancellations in the DFT-based DSO values. These results indicate that more efforts to develop DFT frameworks should be directed towards the reproduction of quantitative DSO tensors of transition metal complexes with various electronic configurations and local symmetries around metal ions.

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 geff-gtrue 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(Sx2 - Sy2) term is putatively treated to the second order. To show the usefulness of the present approach, we exploit FeIII(Cl)OEP (S = 5/2) (OEP: 2,3,7,8,12,13,17,18-octaethylporphyrin) and CoIIOEP (S = 3/2) well magnetically diluted in the diamagnetic host crystal lattice of NiIIOEP. 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 ReIII,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.

Intact Four-atom Organic Tetracation Stabilized by Charge Localization in the Gas Phase.

Several features distinguish intact multiply charged molecular cations (MMCs) from other species such as monocations and polycations: high potential energy, high electron affinity, a high density of electronic states with various spin multiplicities, and charge-dependent reactions. However, repulsive Coulombic interactions make MMCs quite unstable, and hence small organic MMCs are currently not readily available. Herein, we report that the isolated four-atom molecule diiodoacetylene survives after the removal of four electrons via tunneling. We show that the tetracation remains metastable towards dissociation because of the localization (91-95 %) of the positive charges on the terminal iodine atoms, ensuring minimum Coulomb repulsion between adjacent atoms as well as maximum charge-induced attractive dipole interactions between iodine and carbon. Our approach making use of iodines as the positively charged sites enables small organic MMCs to remain intact.

Quasi-Restricted Orbital Treatment for the Density Functional Theory Calculations of the Spin-Orbit Term of Zero-Field Splitting Tensors.

A quasi-restricted orbital (QRO) approach for the calculation of the spin-orbit term of zero-field splitting tensors (DSO tensors) by means of density functional theory (DFT) importantly features in the fact that it is free from spin contamination problems because it uses spin eigenfunctions for the zeroth order wave functions. In 2011, however, Schmitt and co-workers pointed out that in the originally proposed QRO working equation some possible excitations were not included in their sum-over-states procedure, which causes spurious DSO contributions from closed-shell subsystems located far from the magnetic molecule under study. We have revisited the derivation of the QRO working equation and modified it, making it include all possible types of excitations in the sum-over-states procedure. We have found that the spurious DSO contribution can be eliminated by taking into account contributions from all possible types of singly excited configuration state functions. We have also found that only the SOMO(α) → SOMO(ß) excited configurations have nonzero contributions to the DSO tensors as long as α and ß spin orbitals have the same spatial distributions and orbital energies. For the DSO tensor calculations, by using a ground state wave function free from spin contamination, we propose a natural orbital-based Pederson-Khanna (NOB-PK) method, which utilizes the single determinant wave function consisting of natural orbitals in conjunction with the Pederson-Khanna (PK) type perturbation treatment. Some relevant calculations revealed that the NOB-PK method can afford more accurate DSO tensors than the conventional PK method as well as the QRO approach in MnII complexes and ReIV-based single molecule magnets.

Quantum Chemistry on Quantum Computers: A Polynomial-Time Quantum Algorithm for Constructing the Wave Functions of Open-Shell Molecules.

Quantum computers are capable to efficiently perform full configuration interaction (FCI) calculations of atoms and molecules by using the quantum phase estimation (QPE) algorithm. Because the success probability of the QPE depends on the overlap between approximate and exact wave functions, efficient methods to prepare accurate initial guess wave functions enough to have sufficiently large overlap with the exact ones are highly desired. Here, we propose a quantum algorithm to construct the wave function consisting of one configuration state function, which is suitable for the initial guess wave function in QPE-based FCI calculations of open-shell molecules, based on the addition theorem of angular momentum. The proposed quantum algorithm enables us to prepare the wave function consisting of an exponential number of Slater determinants only by a polynomial number of quantum operations.

Adiabatic quantum computing with spin qubits hosted by molecules.

A molecular spin quantum computer (MSQC) requires electron spin qubits, which pulse-based electron spin/magnetic resonance (ESR/MR) techniques can afford to manipulate for implementing quantum gate operations in open shell molecular entities. Importantly, nuclear spins, which are topologically connected, particularly in organic molecular spin systems, are client qubits, while electron spins play a role of bus qubits. Here, we introduce the implementation for an adiabatic quantum algorithm, suggesting the possible utilization of molecular spins with optimized spin structures for MSQCs. We exemplify the utilization of an adiabatic factorization problem of 21, compared with the corresponding nuclear magnetic resonance (NMR) case. Two molecular spins are selected: one is a molecular spin composed of three exchange-coupled electrons as electron-only qubits and the other an electron-bus qubit with two client nuclear spin qubits. Their electronic spin structures are well characterized in terms of the quantum mechanical behaviour in the spin Hamiltonian. The implementation of adiabatic quantum computing/computation (AQC) has, for the first time, been achieved by establishing ESR/MR pulse sequences for effective spin Hamiltonians in a fully controlled manner of spin manipulation. The conquered pulse sequences have been compared with the NMR experiments and shown much faster CPU times corresponding to the interaction strength between the spins. Significant differences are shown in rotational operations and pulse intervals for ESR/MR operations. As a result, we suggest the advantages and possible utilization of the time-evolution based AQC approach for molecular spin quantum computers and molecular spin quantum simulators underlain by sophisticated ESR/MR pulsed spin technology.

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 the hybrid CASSCF-MRMP2 methods are applied to the calculations of spin-spin and spin-orbit contributions to the zero-field splitting tensors (D tensors) of the halogen-substituted spin-septet 2,4,6-trinitrenopyridines, focusing on the heavy atom effects on the spin-orbit term of the D tensors (D(SO) tensors). The calculations reproduced experimentally determined |D| values within an error of 15%. Halogen substitutions at the 3,5-positions are less influential in the spin-spin dipolar (D(SS)) term of 2,4,6-trinitrenopyridines, although the D(SO) terms are strongly affected by the introduction of heavier halogens. The absolute sign of the D(SO) value (D = D(ZZ) - (D(XX) + D(YY))/2) of 3,5-dibromo derivative 3 is predicted to be negative, which contradicts the Pederson-Khanna (PK) DFT result previously reported. The large negative contributions to the D(SO) value of 3 arise from the excited spin-septet states ascribed mainly to the excitations of in-plane lone pair of bromine atoms → SOMO of π nature. The importance of the excited states involving electron transitions from the lone pair orbital of the halogen atom is also confirmed in the D(SO) tensors of halogen-substituted para-phenylnitrenes. A new scheme based on the orbital region partitioning is proposed for the analysis of the D(SO) tensors as calculated by means of the PK-DFT approach.

Pulsed electron spin nutation spectroscopy of weakly exchange-coupled biradicals: a general theoretical approach and determination of the spin dipolar interaction.

Weakly exchange-coupled biradicals have attracted much attention in terms of their DNP application in NMR spectroscopy for biological systems or the use of synthetic electron-spin qubits. Pulse-ESR based electron spin nutation (ESN) spectroscopy applied to biradicals is generally treated as transition moment spectroscopy from the theoretical side, illustrating that it is a powerful and facile tool to determine relatively short distances between weakly exchange-coupled electron spins. The nutation frequency as a function of the microwave irradiation strength ω(1) (angular frequency) for any cases of weakly exchange-coupled systems can be classified into three categories; D(12) (spin dipolar interaction)-driven, Δg-driven and ω(1)-driven nutation behaviour with the increasing strength of ω(1). For hetero-spin biradicals, Δg effects can be a dominating characteristic in the biradical nutation spectroscopy. Two-dimensional pulse-based electron spin nutation (2D-ESN) spectroscopy operating at the X-band can afford to determine small values of D(12) in weakly exchange-coupled biradicals in rigid glasses. The analytical expressions derived here for ω(1)-dependent nutation frequencies are based on only four electronic spin states relevant to the biradicals, while real biradical systems often have sizable hyperfine interactions. Thus, we have evaluated nuclear hyperfine effects on the nutation frequencies to check the validity of the present theoretical treatment. The experimental spin dipolar coupling of a typical TEMPO-based biradical 1, (2,2,6,6-tetra[((2)H(3))methyl]-[3,3-(2)H(2),4-(2)H(1),5,5-(2)H(2)]piperidin-N-oxyl-4-yl)(2,2,6,6-tetra[((2)H(3))methyl]-[3,3-(2)H(2),4-(2)H(1),5,5-(2)H(2),(15)N]piperidin-(15)N-oxyl-4-yl) terephthalate in a toluene glass, with a distance of 1.69 nm between the two spin sites is D(12) = -32 MHz (the effect of the exchange coupling J(12) is vanishing due to the homo-spin sites of 1, i.e.Δg = 0), while 0 < |J(12)|≦ 1.0 MHz as determined by simulating the random-orientation CW ESR spectra of 1. In addition, we have carried out Q-band pulsed ELDOR (ELectron-electron DOuble Resonance) experiments to confirm whether the obtained values for D(12) and J(12) are accurate. The distance is in a fuzzy region for the distance-measurements capability of the conventional, powerful ELDOR spectroscopy. The strong and weak points of the ESN spectroscopy with a single microwave frequency applicable to weakly exchange-coupled multi-electron systems are discussed in comparison with conventional ELDOR spectroscopy. The theoretical spin dipolar tensor and exchange interaction of the TEMPO biradical, as obtained by sophisticated quantum chemical calculations, agree with the experimental ones.

Time-resolved electron paramagnetic resonance and phosphorescence studies of the lowest excited triplet states of Rh(III) corrole complexes.

The lowest excited triplet (T(1)) ππ* states of gallium (Ga) and various rhodium (Rh) 5,10,15-trispentafluorophenyl corroles (Cors) were studied in the liquid crystal (LC) E-7 and in rigid glasses by time-resolved electron paramagnetic resonance (TR-EPR) spectroscopy. The triplet sublevel energies were experimentally determined by the alignment of the molecules in the LC and by magnetophotoselection in the glass. The sublevel scheme of GaCor was determined by calculating the zero field splitting (ZFS) parameters. Axial ligand effects and quantum chemical calculations were used for the sublevel assignment of RhCors. The anisotropic EPR parameters were used to determine the important higher excited states and the magnitudes of their spin-orbit coupling (SOC) contributions were evaluated. On the basis of these results and analyses, the EPR parameters and triplet lifetime were discussed for each RhCor complex.

A synthetic two-spin quantum bit: g-engineered exchange-coupled biradical designed for controlled-NOT gate operations.

A quantum gate: A system of two coupled electron spins that is useful for simple quantum computing operations has been prepared by synthesis of a biradical 1 and co-crystallization with an isomorphous host molecule. The two weakly exchange-coupled quantum bits (target qubit blue and control qubit red) span four electron spin states. The electron spin transition is denoted by two black arrows.

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.

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.

Ab initio and DFT studies of the spin-orbit and spin-spin contributions to the zero-field splitting tensors of triplet nitrenes with aryl scaffolds.

Spin-orbit and spin-spin contributions to the zero-field splitting (ZFS) tensors (D tensors) of spin-triplet phenyl-, naphthyl-, and anthryl-nitrenes in their ground state are investigated by quantum chemical calculations, focusing on the effects of the ring size and substituted position of nitrene on the D tensor. A hybrid CASSCF/MRMP2 approach to the spin-orbit term of the D tensor (D(SO) tensor), which was recently proposed by us, has shown that the spin-orbit contribution to the entire D value, termed the ZFS parameter or fine-structure constant, is about 10% in all the arylnitrenes under study and less depends on the size and connectivity of the aryl groups. Order of the absolute values for D(SO) can be explained by the perturbation on the energy level and spatial distributions of π-SOMO through the orbital interaction between SOMO of the nitrene moiety and frontier orbitals of the aryl scaffolds. Spin-spin contribution to the D tensor (D(SS) tensor) has been calculated in terms of the McWeeny-Mizuno equation with the DFT/EPR-II spin densities. The D(SS) value calculated with the RO-B3LYP spin density agrees well with the D(Exptl) -D(SO) reference value in phenylnitrene, but agreement with the reference value gradually becomes worse as the D value decreases. Exchange-correlation functional dependence on the D(SS) tensor has been explored with standard 23 exchange-correlation functionals in both RO- and U-DFT methodologies, and the RO-HCTH/407 method gives the best agreement with the D(Exptl) -D(SO) reference value. Significant exchange-correlation functional dependence is observed in spin-delocalized systems such as 9-anthrylnitrene (6). By employing the hybrid CASSCF/MRMP2 approach and the McWeeny-Mizuno equation combined with the RO-HCTH/407/EPR-II//U-HCTH/407/6-31G* spin densities for D(SO) and D(SS), respectively, a quantitative agreement with the experiment is achieved with errors less than 10% in all the arylnitrenes under study. Guidelines to the putative approaches to D(SS) tensor calculations are given.

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.

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.