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Quantum computing presents a promising avenue for solving complex problems, particularly in quantum chemistry, where it could accelerate the computation of molecular properties and excited states. This work focuses on hybrid quantum-classical algorithms for near-term quantum devices, combining the quantum linear response (qLR) method with a polarizable embedding (PE) environment. We employ the self-consistent operator manifold of quantum linear response (q-sc-LR) on top of a unitary coupled cluster (UCC) wave function in combination with a Davidson solver. The latter removes the need to construct the entire electronic Hessian, improving computational efficiency when going towards larger molecules. We introduce a new superposition-state-based technique to compute Hessian-vector products and show that this approach is more resilient towards noise than our earlier gradient-based approach. We demonstrate the performance of the PE-UCCSD model on systems such as butadiene and para-nitroaniline in water and find that PE-UCCSD delivers comparable accuracy to classical PE-CCSD methods on such simple closed-shell systems. We also explore the challenges posed by hardware noise and propose simple error correction techniques to maintain accurate results on noisy quantum computers., Comment: 36 pages, 4 figures

Calculating molecular properties using quantum devices can be done through the quantum linear response (qLR) or, equivalently, the quantum equation of motion (qEOM) formulations. Different parameterizations of qLR and qEOM are available, namely naive, projected, self-consistent, and state-transfer. In the naive and projected parameterizations, the metric is not the identity, and we show that it depends on the redundant orbital rotations. This dependency may lead to divergences in the excitation energies for certain choices of the redundant orbital rotation parameters in an idealized noise-less setting. Further, this leads to significant variance when calculations include statistical noise from finite quantum sampling.

We test the performance of the Polarizable Embedding Variational Quantum Eigensolver Self-Consistent-Field (PE-VQE-SCF) model for computing electric field gradients with comparisons to conventional complete active space self-consistent-field (CASSCF) calculations and experimental results. We compute quadrupole coupling constants for ice VIII and ice IX. We find that the inclusion of the environment is crucial for obtaining results that match the experimental data. The calculations for ice VIII are within the experimental uncertainty for both CASSCF and VQE-SCF for oxygen and lie close to the experimental value for ice IX as well. With the VQE-SCF, which is based on an Adaptive Derivative-Assembled Problem-Tailored (ADAPT) ansatz, we find that the inclusion of the environment and the size of the different basis sets do not directly affect the gate counts. However, by including an explicit environment, the wavefunction and, therefore, the optimization problem becomes more complicated, which usually results in the need to include more operators from the operator pool, thereby increasing the depth of the circuit., Comment: 18 pages, 5 figures

We explore Davidson methods for obtaining excitation energies and other linear response properties within quantum self-consistent linear response (q-sc-LR) theory. Davidson-type methods allow for obtaining only a few selected excitation energies without explicitly constructing the electronic Hessian since they only require the ability to perform Hessian-vector multiplications. We apply the Davidson method to calculate the excitation energies of hydrogen chains (up to H$_{10}$) and analyze aspects of statistical noise for computing excitation energies on quantum simulators. Additionally, we apply Davidson methods for computing linear response properties such as static polarizabilities for H$_2$, LiH, H$_2$O, OH$^-$, and NH$_3$, and show that unitary coupled cluster outperforms classical projected coupled cluster for molecular systems with strong correlation. Finally, we formulate the Davidson method for damped (complex) linear response, with application to the nitrogen K-edge X-ray absorption of ammonia, and the $C_6$ coefficients of H$_2$, LiH, H$_2$O, OH$^-$, and NH$_3$., Comment: 35 pages, 5 figures

The X-ray absorption (XA) spectra of aqueous ammonia and ammonium are computed using a combination of coupled cluster singles and doubles (CCSD) with different quantum mechanical and molecular mechanical embedding schemes. Specifically, we compare frozen Hartree--Fock (HF) density embedding, polarizable embedding (PE), and polarizable density embedding (PDE). Integrating CCSD with frozen HF density embedding is possible within the CC-in-HF framework, which circumvents the conventional system-size limitations of standard coupled cluster methods. We reveal similarities between PDE and frozen HF density descriptions, while PE spectra differ significantly. By including approximate triple excitations, we also investigate the effect of improving the electronic structure theory. The spectra computed using this approach show an improved intensity ratio compared to CCSD-in-HF. Charge transfer analysis of the excitations shows the local character of the pre-edge and main-edge, while the post-edge is formed by excitations delocalized over the first solvation shell and beyond., Comment: 35 pages, 12 figures

Linear response (LR) theory is a powerful tool in classic quantum chemistry crucial to understanding photo-induced processes in chemistry and biology. However, performing simulations for large systems and in the case of strong electron correlation remains challenging. Quantum computers are poised to facilitate the simulation of such systems, and recently, a quantum linear response formulation (qLR) was introduced. To apply qLR to near-term quantum computers beyond a minimal basis set, we here introduce a resource-efficient qLR theory using a truncated active-space version of the multi-configurational self-consistent field LR ansatz. Therein, we investigate eight different near-term qLR formalisms that utilize novel operator transformations that allow the qLR equations to be performed on near-term hardware. Simulating excited state potential energy curves and absorption spectra for various test cases, we identify two promising candidates dubbed ``proj LRSD'' and ``all-proj LRSD''.

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH$_2$ in the series of STO-3G/6-31G/6-31G* basis sets and of H$_4$ and H$_2$O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H$_4$, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems., Comment: 18+14 pages, 4 figures, 1 table; comments welcome

We formulate and implement the Variational Quantum Eigensolver Self Consistent Field (VQE-SCF) algorithm in combination with polarizable embedding (PE), thereby extending PE to the regime of quantum computing. We test the resulting algorithm, PE-VQE-SCF, on quantum simulators and demonstrate that the computational stress on the quantum device is only slightly increased in terms of gate counts compared to regular VQE-SCF. On the other hand, no increase in shot noise was observed. We illustrate how PE-VQE-SCF may lead to the modeling of real chemical systems using a simulation of the reaction barrier of the Diels-Alder reaction between furan and ethene as an example., Comment: 19 pages, 5 figures, submitted to Journal of Chemical Physics

Quantum-mechanical (QM) and classical embedding models approximate a supermolecular quantum chemical calculation. This is particularly useful when the supermolecular calculation has a size that is out of reach for present QM models. Although QM and classical embedding methods share the same goal, they approach this goal from different starting points. In this study, we compare the polarizable embedding (PE) and frozen-density embedding (FDE) models. The former is a classical embedding model, whereas the latter is a density-based QM embedding model. Our comparison focuses on solvent effects on optical spectra of solutes. This is a typical scenario where super-system calculations including the solvent environment become prohibitively large. We formulate a common theoretical framework for PE and FDE models and systematically investigate how PE and FDE approximate solvent effects. Generally, differences are found to be small, except in cases where electron spill-out becomes problematic in the classical frameworks. In these cases, however, atomic pseudopotentials can reduce the electron-spill-out issue., Comment: 39 pages, 7 figures and 3 tables

The polarizable embedding (PE) model is a fragment-based quantum-classical approach aimed at accurate inclusion of environment effects in quantum-mechanical response property calculations. The aim of this tutorial is to give insight into the practical use of the PE model. Starting from a set of molecular structures and until you arrive at the final property, there are many crucial details to consider in order to obtain trustworthy results in an efficient manner. To lower the threshold for new users wanting to explore the use of the PE model, we describe and discuss important aspects related to its practical use. This includes directions on how to generate input files and how to run a calculation.