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

Projective Measurement-Based Quantum Phase Difference Estimation Algorithm for the Direct Computation of Eigenenergy Differences on a Quantum Computer

Quantum computers are capable of calculating the energy difference of two electronic states using the quantum phase difference estimation (QPDE) algorithm. The Bayesian inference-based implementations for the QPDE have been reported so far, but in this approach, the quality of the calculated energy difference depends on the input wave functions being used. Here, we report the inverse quantum Fourier transformation-based QPDE with Na of ancillary qubits, which allows us to compute the difference of eigenenergies based on the single-shot projective measurement. As proof-of-concept demonstrations, we report numerical experiments for the singlet-triplet energy difference of the hydrogen molecule and the vertical excitation energies of halogen-substituted methylenes (CHF, CHCl, CF2, CFCl, and CCl2) and formaldehyde (HCHO).

Computational Analysis of Chemical Reactions Using a Variational Quantum Eigensolver Algorithm without Specifying Spin Multiplicity

The analysis of a chemical reaction along the ground-state potential energy surface in conjunction with an unknown spin state is challenging because electronic states must be separately computed several times using different spin multiplicities to find the lowest energy state. However, in principle, the ground state could be obtained with just a single calculation using a quantum computer without specifying the spin multiplicity in advance. In the present work, ground-state potential energy curves for PtCO were calculated as a proof-of-concept using a variational quantum eigensolver (VQE) algorithm. This system exhibits a singlet-triplet crossover as a consequence of the interaction between Pt and CO. VQE calculations using a statevector simulator were found to converge to a singlet state in the bonding region, while a triplet state was obtained at the dissociation limit. Calculations performed using an actual quantum device provided potential energies within ±2 kcal/mol of the simulated energies after error mitigation techniques were adopted. The spin multiplicities in the bonding and dissociation regions could be clearly distinguished even in the case of a small number of shots. The results of this study suggest that quantum computing can be a powerful tool for the analysis of the chemical reactions of systems for which the spin multiplicity of the ground state and variations in this parameter are not known in advance.

Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer

Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry, we present an equation-of-motion-based method to compute excitation energies following the variational quantum eigensolver algorithm for ground-state calculations on a quantum computer. We perform numerical simulations on H2, H4, H2O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.

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.

Calculation of Core-Excited and Core-Ionized States Using Variational Quantum Deflation Method and Applications to Photocatalyst Modeling

The possibility of performing quantum-chemical calculations using quantum computers has attracted much interest. Variational quantum deflation (VQD) is a quantum-classical hybrid algorithm for the calculation of excited states with noisy intermediate-scale quantum devices. Although the validity of this method has been demonstrated, there have been few practical applications, primarily because of the uncertain effect of calculation conditions on the results. In the present study, calculations of the core-excited and core-ionized states for common molecules based on the VQD method were examined using a classical computer, focusing on the effects of the weighting coefficients applied in the penalty terms of the cost function. Adopting a simplified procedure for estimating the weighting coefficients based on molecular orbital levels allowed these core-level states to be successfully calculated. The O 1s core-ionized state for a water molecule was calculated with various weighting coefficients, and the resulting ansatz states were systematically examined. The application of this technique to functional materials was demonstrated by calculating the core-level states for titanium dioxide (TiO₂) and nitrogen-doped TiO₂ models. The results demonstrate that VQD calculations employing an appropriate cost function can be applied to the analysis of functional materials in conjunction with an experimental approach.

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.