Accurate and Efficient Quantum Computations of Molecular Properties Using Daubechies Wavelet Molecular Orbitals: A Benchmark Study against Experimental Data

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Accurate and Efficient Quantum Computations of Molecular Properties Using Daubechies Wavelet Molecular Orbitals: A Benchmark Study against Experimental Data

Cheng-Lin Hong, Ting Tsai, Jyh-Pin Chou, Peng-Jen Chen, Pei-Kai Tsai, Yu-Cheng Chen, En-Jui Kuo, David Srolovitz, Alice Hu, Yuan-Chung Cheng, and Hsi-Sheng Goan

PRX Quantum 3, 020360 – Published 22 June 2022

Abstract

Although quantum computation 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 quantum bits (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 quantum computation 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 can not provide useful predictions. Herein, we propose to adopt Daubechies wavelet functions as an accurate and efficient method for quantum computations 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 quantum computations 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.

Received 27 December 2021
Accepted 13 May 2022

DOI:https://doi.org/10.1103/PRXQuantum.3.020360

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Cheng-Lin Hong ^1,†, Ting Tsai ^1,†, Jyh-Pin Chou ^2,3,†, Peng-Jen Chen^2,4, Pei-Kai Tsai¹, Yu-Cheng Chen ², En-Jui Kuo^1,5, David Srolovitz ^4,6, Alice Hu^2,7, Yuan-Chung Cheng^8,9,10, and Hsi-Sheng Goan ^1,9,10,*

¹Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan
²Department of Mechanical Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR 999077, China
³Department of Physics, National Changhua University of Education, Changhua 50007, Taiwan
⁴Hong Kong Institute for Advanced Study, City University of Hong Kong, Kowloon, Hong Kong SAR 999077, China
⁵Department of Physics and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
⁶Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR 999077, China
⁷Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR 999077, China
⁸Department of Chemistry, National Taiwan University, Taipei 10617, Taiwan
⁹Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan
¹⁰Physics Division, National Center for Theoretical Sciences, Taipei, 10617, Taiwan

^*goan@phys.ntu.edu.tw
^†Contributed equally to this work.

Popular Summary

Quantum chemistry calculation is consideredone of the most compelling application for quantum computing. However, this is technically limited to only small molecules due to the limitations on the number of qubits and the depth and complexity of computational circuits available in nowadays quantum computers. Consequently, reducing the number of required qubits is necessary to make the quantum computation of molecular systems practical. Currently, the minimal contracted Gaussian basis set is commonly used in benchmark studies because it requires the minimum number of spin orbitals and thus the minimal number of qubits; nonetheless, the accuracy is generally low and thus cannot provide useful predictions.

We demonstrate that a minimal basis set constructed from Daubechies wavelet functions for quantum computing can yield accurate results for H2 and LiH in excellent agreement with experimental data. This is an unprecedented demonstration of quantum computation with accuracy comparable with that of the full configuration interaction (FCI) method using a large basis set, whereas the computational cost is merely the same as that of a minimal basis set calculation. We also perform numerical experiments on a quantum simulator with a noise model implemented from a real quantum machine. We demonstrate that most of the error-mitigated data agree well with the exact FCI results within chemical accuracy. Thus, our work provides an efficient and accurate scheme for quantum computations 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.

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Vol. 3, Iss. 2 — June - August 2022

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Figure 1
Potential energy curves in (a),(c), and (e) and corresponding errors in (b),(d), and (f) for $H_{2}$ , $LiH$ , and ${BeH}_{2}$ , respectively. The potential energies are computed with VQE on a quantum computer simulator or by exact diagonalization (FCI) on a classical computer. All curves are denoted with basis $(N_{e}, N_{orb}) /$ method, where $N_{e}$ and $N_{orb}$ represent the numbers of electrons and spin orbitals, respectively. The error is defined as the absolute energy difference with respect to the ground-state energy value from the exact diagonalization (FCI) in the largest affordable Gaussian basis set of cc-PVTZ (in black curve) considered here. The position of the minimum energy and the curvature near the minimum of the potential energy curve give the equilibrium bond length and vibrational frequency of the molecule, respectively. Note that we use the same number of spin orbitals as the STO-3G minimal basis set for the minimal DW basis and the cc-PVTZ* basis with unoccupied orbitals frozen.
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Figure 2
Two-dimensional DW/UCCSD potential energy surface of $H_{2} O$ . Color representation of the energy is depicted by the color bar on the right and the energy of the optimized structure, marked by the red triangle, is set to zero.
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Figure 3
Two-qubit heuristic circuit ansatz for $H_{2}$ in the VQE algorithm.
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Figure 4
Four-qubit heuristic circuit ansatz for $LiH$ in the VQE algorithm.
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Figure 5
Potential energy curves and computational errors of $H_{2}$ and $LiH$ obtained from calculations under noisy environments. The errors presented for the Daubechies wavelet and the STO-3G methods are deviations from their respective classical exact FCI calculation values. (a) $H_{2}$ . Results calculated using the minimal basis Daubechies wavelet molecular orbitals (blue dots) as well as the classical FCI (solid line) are shown. (b) $LiH$ . The blue (pink) dots depict the raw data calculated from the STO-3G (DW) basis set. The green (red) dots denote the extrapolated energies over three different cnot gate counts from the STO-3G (DW) basis set. These simulations with noise are averaged over ten sets of VQE experiments with $10^{4}$ shots for each iteration. The solid lines are the respective classical FCI results. (c) $H_{2}$ errors. The color scheme is the same as in the legend of (a). (d) $LiH$ errors. The color scheme is the same as in the legend of (b). The error bars represent the 95% confidence uncertainties that take the standard error of experiments at each cnot gate count and the residues of linear regression into account. Most of the errors after mitigation are within the chemical accuracy defined as 1 kcal/mol.
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