Quantum Computing Beyond Ground-State Electronic Structure: A Review of Progress Toward Quantum Chemistry Out of the Ground State.

To understand the properties and interactions of materials, and determining the ground state energies is one of the important challenges in quantum chemistry, materials science, and quantum mechanics, where quantum computing can play an important role for studying the properties of materials. In this study, we have explored the quantum processor application to compute the He molecule ground state energy which utilizes the Variational Quantum Eigensolver (VQE) algorithm. In here, we have implemented VQE on a state-of-the-art quantum processor, optimizing a parameterized quantum circuit to minimize the energy expectation value of the He molecule's Hamiltonian on the four qubits processor. The obtained results of this work show a significant improvement in accuracy compared to classical computational methods, such as Hartree-Fock and density functional theory, which demonstrate the compute potential of quantum algorithms in quantum many-body problems. Thus, these results demonstrate the advantages of quantum computing in achieving high accuracy in simulations of molecular and material properties, and pave the way for future applications in more complex systems. This work highlights the potential of quantum processors in the fields of quantum chemistry, computational physics, and data science.

Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today’s quantum devices. Here we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground state energy. Our algorithm, termed “local active space unitary coupled cluster” (LAS-UCC), scales linearly with system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene and resources estimates are provided for linear chains of up to 20 H2 molecules.

Small hydrocarbon radicals play a key role in combustion chemistry, in hydrocarbon crackers and in certain regions of the atmosphere as well as in the interstellar space. The emergence of such radicals requires high energy environments where the energy can be transfused by heat or by light. Thermally induced processes thereby occur almost exclusively in the electronic ground state of a molecule whereas photoexcited molecules can react in different electronic states resulting in much more complex reaction mechanisms. The subject of this thesis is the investigation of the photophysical processes, the reaction mechanisms and the kinetics of the reactions that occur upon excitation of small hydrocarbon radicals to their first electronically excited state. Furthermore, the characterization and the localization of several electronically excited states of small hydrocarbon radicals was conducted. All the work described in this thesis is based on computational simulations that were carried out in parallel to experimental investigations in our group. In order to calculate the molecular dynamics of photoexcited molecules ab initio, a computer program was written which calculates the movements of the atoms during a photoreaction. Thereby it also determines in which electronic state the molecule is at any time. This program interacts with available quantum chemical software packages in order to obtain the energies, energy gradients and nonadiabatic couplings between the electronic states of the molecule under investigation. Before we inspected the nonadiabatic molecular dynamics of the ethyl radical, C2H5, we simulated the dissociation dynamics of a microcanonical ensemble of ethyl radicals in the electronic ground state with UHF and DFT potentials. The chosen ex-

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

The calculation of non-covalent interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, the use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms (Chem. Sci., 2022, 13, 3094), this provides a recipe for complete SAPT(VQE) interaction energies up to second order, which is a well established truncation. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from a quantum computer, simulated through ideal statevectors. The errors of the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions. In addition, we present heme-nitrosyl model complexes as a system class for near term quantum computing simulations. They are strongly correlated, biologically relevant and difficult to simulate with classical quantum chemical methods. This is illustrated with density functional theory (DFT) as the predicted interaction energies exhibit a strong sensitivity with respect to the choice of functional. Thus, this work paves the way to obtain accurate interaction energies on a NISQ-era quantum computer with few quantum resources. It is the first step in alleviating one of the major challenges in quantum chemistry, where in-depth knowledge of both the method and system is required a priori to reliably generate accurate interaction energies.

BOOK REVIEW article Front. Chem., 03 September 2013Sec. Theoretical and Computational Chemistry https://doi.org/10.3389/fchem.2013.00014

Quantum chemistry calculations using energy derivatives on quantum computers

It has been claimed that quantum computers can mimic quantum systems efficiently in the polynomial scale. Traditionally, those simulations are carried out numerically on classical computers, which are inevitably confronted with the exponential growth of required resources, with the increasing size of quantum systems. Quantum computers avoid this problem, and thus provide a possible solution for large quantum systems. In this paper, we first discuss the ideas of quantum simulation, the background of quantum simulators, their categories, and the development in both theories and experiments. We then present a brief introduction to quantum chemistry evaluated via classical computers followed by typical procedures of quantum simulation towards quantum chemistry. Reviewed are not only theoretical proposals but also proof-of-principle experimental implementations, via a small quantum computer, which include the evaluation of the static molecular eigenenergy and the simulation of chemical reaction dynamics. Although the experimental development is still behind the theory, we give prospects and suggestions for future experiments. We anticipate that in the near future quantum simulation will become a powerful tool for quantum chemistry over classical computations.

Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today's quantum devices. Here, we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground-state energy. Our algorithm, termed "local active space unitary coupled cluster" (LAS-UCC), scales linearly with the system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene, and resource estimates are provided for linear chains of up to 20 H2 molecules.

Although quantum computation (QC) is regarded as a promising numerical method for computational quantum chemistry, current applications of quantum-chemistry calculations on quantum computers are limited to small molecules. This limitation can be ascribed to technical problems in building and manipulating more qubits and the associated complicated operations of quantum gates in a quantum circuit when the size of the molecular system becomes large. As a result, reducing the number of required qubits is necessary to make QC practical. Currently, the minimal STO-3G basis set is commonly used in benchmark studies because it requires the minimum number of spin orbitals. Nonetheless, the accuracy of using STO-3G is generally low and thus cannot provide useful predictions. We propose to adopt Daubechies wavelet functions as an accurate and efficient method for QCs of molecular electronic properties. We demonstrate that a minimal basis set constructed from Daubechies wavelet basis can yield accurate results through a better description of the molecular Hamiltonian, while keeping the number of spin orbitals minimal. With the improved Hamiltonian through Daubechies wavelets, we calculate vibrational frequencies for H$_2$ and LiH using quantum-computing algorithm to show that the results are in excellent agreement with experimental data. As a result, we achieve quantum calculations in which accuracy is comparable with that of the full configuration interaction calculation using the cc-pVDZ basis set, whereas the computational cost is the same as that of a STO-3G calculation. Thus, our work provides a more efficient and accurate representation of the molecular Hamiltonian for efficient QCs of molecular systems, and for the first time demonstrates that predictions in agreement with experimental measurements are possible to be achieved with quantum resources available in near-term quantum computers.

Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.

The variational quantum eigensolver (VQE) is a promising algorithm to demonstrate quantum advantage on near-term noisy-intermediate-scale quantum (NISQ) computers. One central problem of VQE is the effect of noise, especially physical noise, on realistic quantum computers. We systematically study the effect of noise for the VQE algorithm by performing numerical simulations with various local noise models, including amplitude damping, dephasing, and depolarizing noise. We show that the ground state energy will deviate from the exact value as the noise probability increase, and typically, the noise will accumulate as the circuit depth increase. The results suggest that the noisy quantum system can remain entanglement at the noise level of NISQ devices by comparing the VQE solution with the mean-field solution for the many-body ground state problem. We build a noise model to capture the noise in a real quantum computer, and the corresponding numerical simulation is consistent with experimental results on IBM Quantum computers through cloud. Our work sheds new light on the practical research of noisy VQE, and the deep understanding of the noise effect of VQE will also help develop error mitigation techniques on near-term quantum computers.

Variational quantum eigensolvers (VQEs) are among the most promising quantum algorithms for solving electronic structure problems in quantum chemistry, particularly during the noisy intermediate-scale quantum (NISQ) era. In this study, we investigate the capabilities and limitations of VQE algorithms implemented on current quantum hardware for determining molecular ground-state energies, focusing on the adaptive derivative-assembled pseudo-Trotter ansatz VQE (ADAPT-VQE). To address the significant computational challenges posed by molecular Hamiltonians, we explore various well known strategies to simplify the Hamiltonian, optimize the ansatz, and improve classical parameter optimization through modifications of the COBYLA optimizer. These enhancements are integrated into a tailored quantum computing implementation designed to minimize the circuit depth and computational cost. Using benzene as a benchmark system, we demonstrate the application of these optimizations on an IBM quantum computer. Despite these improvements, our results highlight the limitations imposed by current quantum hardware, particularly the impact of quantum noise on state preparation and energy measurement. The noise levels in today's devices prevent meaningful evaluations of molecular Hamiltonians with sufficient accuracy to produce reliable quantum chemical insights. Finally, we extrapolate the requirements for future quantum hardware to enable practical and scalable quantum chemistry calculations using VQE algorithms. This work provides an assessment of current quantum algorithms for molecular modeling on real quantum hardware, highlighting the impact of noise and hardware limitations on the achievable accuracy.

Quantum computing, as an emerging computing paradigm, is expected to tackle problems such as quantum chemistry, optimization, quantum chemistry, information security, and artificial intelligence, which are intractable with using classical computing. Quantum computing hardware and software continue to develop rapidly, but they are not expected to realize universal quantum computation in the next few years. Therefore, the use of quantum hardware to solve practical problems in the near term has become a hot topic in the field of quantum computing. Exploration of the applications of near-term quantum hardware is of great significance in understanding the capability of quantum hardware and promoting the practical process of quantum computing. Hybrid quantum-classical algorithm (also known as variational quantum algorithm) is an appropriate model for near-term quantum hardware. In the hybrid quantum-classical algorithm, classical computers are used to maximize the power of quantum devices. By combining quantum computing with machine learning, the hybrid quantum-classical algorithm is expected to achieve the first practical application of quantum computation and play an important role in the studying of quantum computing. In this review, we introduce the framework of hybrid quantum-classical algorithm and its applications in quantum chemistry, quantum information, combinatorial optimization, quantum machine learning, and other fields. We further discuss the challenges and future research directions of the hybrid quantum-classical algorithm.

As quantum hardware continues to improve, more and more application scientists have entered the field of quantum computing. However, even with the rapid improvements in the last few years, quantum devices, especially for quantum chemistry applications, still struggle to perform calculations that classical computers could not calculate. In lieu of being able to perform specific calculations, it is important have a systematic way of estimating the resources necessary to tackle specific problems. Standard arguments about computational complexity provide hope that quantum computers will be useful for problems in quantum chemistry but obscure the true impact of many algorithmic overheads. These overheads will ultimately determine the precise point when quantum computers will perform better than classical computers. We have developed QREChem to provide logical resource estimates for ground state energy estimation in quantum chemistry through a Trotter-based quantum phase estimation approach. QREChem provides resource estimates which include the specific overheads inherent to problems in quantum chemistry by including heuristic estimates of the number of Trotter steps and number of necessary ancilla, allowing for more accurate estimates of the total number of gates. We utilize QREChem to provide logical resource estimates for a variety of small molecules in various basis sets, obtaining estimates in the range of 107–1015 for total number of T gates. We also determine estimates for the FeMoco molecule and compare all estimates to other resource estimation tools. Finally, we compare the total resources, including hardware and error correction overheads, demonstrating the need for fast error correction cycle times.