Relativistic VQE calculations of molecular electric dipole moments on trapped ion quantum hardware
Palak Chawla, Shweta, Kasturi Ranjan Swain, Tushti Patel, Renu Bala, Shetty, Disha, Kenji Sugisaki, Sudhindu Bikash Mandal, Riu, Jordi, Jan Nogué, V. S. Prasannaa, B. P. Das
The quantum-classical hybrid variational quantum eigensolver (VQE) algorithm is among the most actively studied topics in atomic and molecular calculations on quantum computers, yet few studies address properties other than energies or account for relativistic effects. This work presents high-precision 18-qubit relativistic VQE simulations for calculating the permanent electric dipole moments (PDMs) of BeH to RaH molecules on traditional computers, and 6- and 12-qubit PDM computations for SrH on IonQ quantum devices. To achieve high precision on current noisy intermediate scale era quantum hardware, we apply various resource reduction methods, including Reinforcement Learning and causal flow preserving ZX-Calculus routines, along with error mitigation and post-selection techniques. Our approach reduces the two-qubit gate count in our 12-qubit circuit by 99.71%, with only a 2.35% trade-off in precision for PDM when evaluated classically within a suitably chosen active space. On the current generation IonQ Forte-I hardware, the error in PDM is -1.17% relative to classical calculations and only 1.21% compared to the unoptimized circuit.
The quantum-classical hybrid variational quantum eigensolver (VQE) algorithm is among the most actively studied topics in atomic and molecular calculations on quantum computers, yet few studies address properties other than energies or account for relativistic effects. This work presents high-precision 18-qubit relativistic VQE simulations for calculating the permanent electric dipole moments (PDMs) of BeH to RaH molecules on traditional computers, and 6- and 12-qubit PDM computations for SrH on IonQ quantum devices. To achieve high precision on current noisy intermediate scale era quantum hardware, we apply various resource reduction methods, including Reinforcement Learning and causal flow preserving ZX-Calculus routines, along with error mitigation and post-selection techniques. Our approach reduces the two-qubit gate count in our 12-qubit circuit by 99.71%, with only a 2.35% trade-off in precision for PDM when evaluated classically within a suitably chosen active space. On the current generation IonQ Forte-I hardware, the error in PDM is -1.17% relative to classical calculations and only 1.21% compared to the unoptimized circuit.
The quantum--classical hybrid variational quantum eigensolver (VQE) algorithm is among the most actively studied topics in atomic and molecular calculations on quantum computers, yet few studies address properties other than energies or account for relativistic effects. This work presents high-precision 18-qubit relativistic VQE simulations for calculating the permanent electric dipole moments (PDMs) of alkaline-earth metal monohydride molecules (BeH to RaH) on traditional computers, and 6- and 12-qubit PDM computations for SrH on IonQ quantum devices. To achieve high precision on current noisy intermediate-scale era quantum hardware, we apply various resource-reduction methods, including reinforcement learning-based and causal flow preserving ZX--calculus routines, along with error mitigation and postselection techniques. Our approach reduces the two-qubit gate count in our 12-qubit circuit by 99.71, with only a 2.35 trade-off in precision for PDM when evaluated classically within a suitably chosen active space. On the current generation IonQ Forte-I hardware, the error in PDM is $\ensuremath{-}1.17%$ relative to classical calculations and only 1.21 compared with the unoptimized circuit.
The quantum-classical hybrid Variational Quantum Eigensolver (VQE) algorithm is recognized to be the most suitable approach to obtain ground state energies of quantum many-body systems in the noisy intermediate scale quantum era. In this work, we extend the VQE algorithm to the relativistic regime and carry out quantum simulations to obtain ground state energies as well as molecular permanent electric dipole moments of single-valence diatomic molecules, beginning with the light BeH molecule and all the way to the heavy radioactive RaH molecule. We study the correlation trends in these systems as well as assess the precision in our results within our active space of 12 qubits.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
New approaches into computational quantum chemistry can be developed through the use of quantum computing. While universal, fault-tolerant quantum computers are still not available, and we want to utilize today's noisy quantum processors. One of their flagship applications is the variational quantum eigensolver (VQE)-an algorithm for calculating the minimum energy of a physical Hamiltonian. In this study, we investigate how various types of errors affect the VQE and how to efficiently use the available resources to produce precise computational results. We utilize a simulator of a noisy quantum device, an exact statevector simulator, and physical quantum hardware to study the VQE algorithm for molecular hydrogen. We find that the optimal method of running the hybrid classical-quantum optimization is to: (i) allow some noise in intermediate energy evaluations, using fewer shots per step and fewer optimization iterations, but ensure a high final readout precision; (ii) emphasize efficient problem encoding and ansatz parametrization; and (iii) run all experiments within a short time-frame, avoiding parameter drift with time. Nevertheless, current publicly available quantum resources are still very noisy and scarce/expensive, and even when using them efficiently, it is quite difficult to perform trustworthy calculations of molecular energies.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of a target molecule. Although VQE has a potential to achieve a higher accuracy than classical approximation methods in QCC, it is challenging to achieve it on current NISQ computers due to the significant impact of noises. Density matrix embedding theory (DMET) is a well-known technique to divide a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE does not outperform a gold standard classical method. In this work, we present three approaches to mitigate the noise impact for the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) Reduced density matrices (RDMs), which are used to calculate a molecular energy in DMET, are calculated accurately based on expectation values obtained by executing quantum circuits using a noise-less quantum computer simulator. (3) The parameters of a quantum circuit optimized by VQE are refined with mathematical post-processing. The evaluation using a noisy quantum computer simulator shows that our approaches significantly improve the accuracy of the DMET+VQE combination. Moreover, we demonstrate that on a real NISQ device, the DMET+VQE combination applying our three approaches achieves a higher accuracy than the gold standard classical method.
Understanding complex chemical systems—such as biomolecules, catalysts, and novel materials—is a central goal of quantum simulations. Near-term strategies hinge on the use of variational quantum eigensolver (VQE) algorithms combined with a suitable ansatz. However, straightforward application of many chemically-inspired ansatze yields prohibitively deep circuits. In this work, we employ several circuit optimization methods tailored for trapped-ion quantum devices to enhance the feasibility of intricate chemical simulations. The techniques aim to lessen the depth of the unitary coupled cluster with singles and doubles (uCCSD) ansatz's circuit compilation, a considerable challenge on current noisy quantum devices. Furthermore, we use symmetry-inspired classical post-selection methods to further refine the outcomes and minimize errors in energy measurements, without adding quantum overhead. Our strategies encompass optimal mapping from orbital to qubit, term reordering to minimize entangling gates, and the exploitation of molecular spin and point group symmetry to eliminate redundant parameters. The inclusion of error mitigation via post-selection based on known molecular symmetries improves the results to near milli-Hartree accuracy. These methods, when applied to a benzene molecule simulation, enabled the construction of an 8-qubit circuit with 69 two-qubit entangling operations, pushing the limits for variational quantum eigensolver (VQE) circuits executed on quantum hardware to date.11This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-000R22725 with the US Department of Energy (DOE). The publisher acknowledges the US government license to provide public access under the DOE Public Access Plan(https://energy.gov/doe-public-access-plan).