## Connected Moments Expansions Allow for Efficient Quantum Computing

*Mathematical technique decreases time and power needed for simulations*

## The Science

Advancing quantum computing requires models that can solve multi-body problems quickly and accurately. This research proposes a new algorithm for performing quantum calculations on chemical systems that reduces the effect of noise on calculations. The approach uses a mathematical tool called “connected moments,” which was first described 40 years ago. When applied to quantum calculations, the connected moments expansions require fewer qubits in quantum circuits to reach a desired level of accuracy in calculated energies of many-body quantum systems. The researchers used their method to describe relatively simple models, which allowed them to compare the results of their approach with previously validated full-scale computing models to confirm their accuracy.

## The Impact

Using quantum simulations for chemical systems will significantly advance our understanding of the chemical processes that occur in catalysis, photochemistry, biochemistry, and materials science. This work is a step toward creating simulations that can be performed with high accuracy, regardless of the complexity of the chemical system. The increase in efficiency from the connected moments approach provides a novel route to the effective use of quantum computers for modeling chemical systems as the technology increases in power.

## Summary

Quantum computing has the potential to accurately describe the quantum behavior of chemical processes, a task that is, from a practical perspective, impossible for medium to large systems on classical computers. However, current quantum computing devices are inherently noisy and error-prone. Researchers are developing algorithms for quantum simulations that reduce the effect of noise. But the circuit depth and number of qubits needed for these algorithms leave room for error to enter the calculations.

To this end, researchers developed a quantum computing approach that employs finite-order connected moment expansions and computationally affordable procedures for preparing the system’s initial state. They used a H_{2} molecule potential energy surface and the Anderson model with a broad range of correlation strengths as test cases to evaluate the performance of this new approach. That the results of the quantum calculations agree with results from established classical computing establishes this new approach as robust, flexible, and with comparable accuracy, a promising first step that demonstrates minimal computational error. Good agreement with the exact solutions can be maintained even at the dissociation and strong correlation limits, providing further evidence of the broad utility of this approach.

## PNNL Contact

Karol Kowalski, Pacific Northwest National Laboratory, karol.kowalski@pnnl.gov

## Funding

This work was supported by the “Embedding QC into Many-body Frameworks for Strongly Correlated Molecular and Materials Systems” project, which is funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, the Division of Chemical Sciences, Geosciences, and Biosciences. All calculations have been performed at the Pacific Northwest National Laboratory (PNNL). PNNL is operated for the U.S. Department of Energy by the Battelle Memorial Institute under Contract No. DE-AC06-76RLO-1830.

*
Published: March 9, 2021*

K. Kowalski and B. Peng. “Quantum simulations employing connected moments expansions*” **J. Chem. Phys.* **153,** 201102, (2020).