## Increasing the Accuracy of Quantum Algorithms

*Utilizing the Peeters-Devreese-Soldatov formulation improves the accuracy of calculations for quantum chemistry*

## The Science ** **

Quantum computing has exciting potential applications in the field of quantum chemistry because Hamiltonians can effectively be mapped to qubit registers. However, it is unclear which quantum algorithm will yield the most efficient encoding of electron correlation effects needed to describe molecular systems. Researchers developed a new variational quantum solver, PDS(K)-VQS, based on the Peeters-Devreese-Soldatov (PDS) formulation to improve the accuracy and efficiency of the quantum simulations. From this, a new form of energy functionals can be efficiently optimized using low-depth quantum circuits over a range of approximation orders (K). This approach outperforms usual variational quantum solver (VQS) methods and static PDS calculations when finding the ground state energy for H_{2} and H_{4} molecules and the four-site 2D Heisenberg model, even at the lowest order (K=2). Additionally, preliminary simulations on the four-site 2D Heisenberg model have been performed using four qubits of the IBMQ Toronto quantum computer.

## The Impact

Calculating electron correlation effects with high accuracy remains a challenge for classical computing. Quantum computing has the potential to overcome this challenge by efficiently mapping Hamiltonians to qubits, though current quantum algorithms fall short of this challenge in terms of both efficiency and accuracy. Researchers developed the PDS(K)-VQS approach to produce more accurate calculations. This research lays the groundwork for incorporating the PDS formulation into calculations for ground and excited state energies for more accurate simulations. Researchers also demonstrated the applicability of their algorithm on quantum hardware. Preliminary research indicates that this methodology may be applied to systems whose true ground and excited states are challenging to obtain classically.

## Summary

Researchers developed a new class of variational quantum solver, PDS(K)-VQS, that combines the PDS formalism with minimization procedures based on the quantum gradient approach. In comparison with the static PDS(K) expansions, PDS(K)-VQS guides the rotation of the trial wave function of modest quality and can achieve high accuracy at the expense of low order PDS(K) expansions. The researchers demonstrated the capability of the PDS(K)-VQS approach at finding the ground state and its energy for H_{2} and H_{4} molecules and the four-site 2D Heisenberg model. In these case studies, the PDS(K)-VQS outperforms both the standalone VQS and static PDS(K) calculations.

PDS(K)-VQS improves the accuracy of quantum calculations by identifying an upper bound energy surface, thus freeing the dynamics of the iterative process from getting trapped in the local minima that correspond to different states. From this, a new form of energy functionals can be efficiently optimized using low-depth quantum circuits. Additionally, excited state energies can be directly estimated without additional measurements required by the usual VQS.

## Contact

Bo Peng

Physical and Computational Sciences Directorate

peng398@pnnl.gov

Karol Kowalski

Physical and Computational Sciences Directorate

karol.kowalski@pnnl.gov

## Funding

This work is supported by the “Embedding QC into Many-body Frameworks for Strongly Correlated Molecular and Materials Systems” project, which is funded by the Department of Energy, Office of Science, Office of Basic Energy Sciences (BES), the Division of Chemical Sciences, Geosciences, and Biosciences. The authors also acknowledge the use of the IBMQ quantum hardware for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBMQ team.

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Published: March 8, 2022*

Bo Peng, Karol Kowalski. “Variational quantum solver employing the PDS energy functional”. Quantum 5, 473 (2021). DOI: 10.22331/q-2021-06-10-473