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arXiv:2207.00218

Interaction Energies on Noisy Intermediate-Scale Quantum Computers

Matthias Loipersberger, Fionn D. Malone, Alicia R. Welden, Robert M. Parrish, Thomas Fox, Matthias Degroote, Elica S. Kyoseva, Nikolaj Moll, Raffaele Santagati, Michael Streif

Published Jul 1, 2022• 2 citations
arXivHTMLPDF

Abstract

The computation of interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, 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, this provides a recipe for complete SAPT(VQE) interaction energies up to second order. 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 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 the quantum computer, simulated through ideal statevectors. The errors on the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions.

DOI
10.48550/arxiv.2207.00218

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The computation of interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, 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, this provides a recipe for complete SAPT(VQE) interaction energies up to second order. 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 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 the quantum computer, simulated through ideal statevectors. The errors on the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions.
The efficient computation of observables beyond the total energy is a key challenge and opportunity for fault-tolerant quantum computing approaches in quantum chemistry. Here, we consider the symmetry-adapted perturbation-theory (SAPT) components of the interaction energy as a prototypical example of such an observable. We provide a guide for calculating this observable on a fault-tolerant quantum computer while optimizing the required computational resources. Specifically, we present a quantum algorithm that estimates interaction energies at the first-order SAPT level with a Heisenberg-limited scaling. To this end, we exploit a high-order tensor-factorization and block-encoding technique that efficiently represents each SAPT observable. To quantify the computational cost of our methodology, we provide resource estimates in terms of the required number of logical qubits and Toffoli gates to execute our algorithm for a range of benchmark molecules, also taking into account the cost of the eigenstate preparation and the cost of block encoding the SAPT observables. Finally, we perform the resource estimation for a heme and artemisinin complex as a representative large-scale system encountered in drug design, highlighting the performance of our algorithm in this new benchmark study and discussing possible bottlenecks that may be improved in future work. Published by the American Physical Society 2024
accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal statevector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A).
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 molecules. Although VQE has a theoretical potential to achieve a higher accuracy than classical approximation methods, it is challenging to achieve it on current NISQ computers due to the significant impact of noise. Density matrix embedding theory (DMET) is a well-known technique that can be used to decompose 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 hardly mitigates it. In this work, we present two proposals to mitigate the noise impact on 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) While VQE is executed on a NISQ computer, reduced density matrices (RDMs), which are used to calculate the energy of fragments in DMET, are calculated accurately by executing quantum circuits using a noise-less quantum computer simulator. The evaluation using a noisy quantum computer simulator shows that our proposals improve the accuracy of the DMET+VQE combination by an order of magnitude.
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.
In resource-efficient near-term quantum computers, development on the variational quantum eigensolver (VQE) algorithm that improves the accuracy of electronic structure calculations while maintaining low-requirement quantum gate counts is indispensable. We integrate Slater-type orbitals within the VQE framework, where the ground-state energy minimization with respect to the Slater exponents is shown as a second-order dependence on these parameters, enhancing accuracy compared with the standard methods while maintaining the same number of qubits. Using the full configuration interaction energy as the cost function, optimized parameters for first- and second-row elements are reported and related to the shielding constant, indicating a need to revise the previously established Slater rules. The active space is improved by substantially contracting the orbital exponents of extended basis sets relative to minimal ones, hence capturing additional electron-correlation effects. By considering the geometric structure of the parameter space, quantum Slater-exponent optimization (Q-SEO) demonstrates accelerated convergence to the target ground state, characterized by distinct parameter trajectories. The integration of a decision tree with Q-SEO effectively mitigates error in noisy environments.
Current noisy intermediate-scale quantum (NISQ) devices remain limited in their ability to perform accurate quantum chemistry simulations because of restricted numbers of high-fidelity qubits and short coherence times. To overcome these challenges, we introduce a reduced-density-matrix (RDM)-based perturbative variational quantum eigensolver (VQE) framework that augments active-space VQE with perturbation theory to recover electron correlation beyond the active space without increasing the qubit count or variational circuit depth. We formulated a fully coupled approach (VQE-PTs) and a diagonal approximation (VQE-PT). The former retains couplings among orthonormalized perturbers, whereas the latter neglects these couplings to simplify the classical postprocessing. Numerical simulations of HF, N2, and F2 show that VQE-PTs provides a robust formulation across different molecular systems, while VQE-PT offers an efficient approximation. We further experimentally implement VQE-PT on the Quafu superconducting quantum processor for F2, achieving a mean absolute error of 1.2 mHa along the potential energy surface after error mitigation. These results demonstrate perturbative VQE as a practical framework for incorporating dynamic correlations in quantum chemistry simulations.