Mapping renormalized coupled cluster methods to quantum computers through a compact unitary representation of nonunitary operators

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Mapping renormalized coupled cluster methods to quantum computers through a compact unitary representation of nonunitary operators

Bo Peng and Karol Kowalski

Phys. Rev. Research 4, 043172 – Published 8 December 2022

Abstract

Nonunitary theories are commonly seen in the classical simulations of quantum systems. Among these theories, the method of moments of coupled cluster equations (MMCCs) and the ensuing classes of the renormalized coupled cluster (CC) approaches have evolved into one of the most accurate approaches to describe correlation effects in various quantum systems. The MMCC formalism provides an effective way for correcting the energies of approximate CC formulations (parent theories) using moments, or CC equations, that are not used to determine approximate cluster amplitudes. In this paper, we propose a quantum algorithm for computing MMCC ground-state energies that provides two main advantages over classical computing or other quantum algorithms: (i) the possibility of forming superpositions of CC moments of arbitrary ranks in the entire Hilbert space and using an arbitrary form of the parent cluster operator for MMCC expansion, and (ii) significant reduction in the number of measurements in quantum simulation through a compact unitary representation for a generally nonunitary operator. We illustrate the robustness of our approach over a broad class of test cases, including $\sim 40$ molecular systems with varying basis sets encoded using 4–40 qubits, and we exhibit the detailed MMCC analysis for the 8-qubit half-filled, four-site, single impurity Anderson model and the 12-qubit hydrogen fluoride molecular system from the corresponding noise-free and noisy MMCC quantum computations. We also outline the extension of the MMCC formalism to the case of a unitary CC wave-function Ansatz.

Received 21 June 2022
Accepted 10 November 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.043172

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)

Electron correlation calculations for atoms & ions Electronic structure of atoms & molecules Potential energy surfaces Quantum algorithms Quantum computation Quantum simulation

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Bo Peng ^* and Karol Kowalski ^†

Physical Sciences and Computational Division, Pacific Northwest National Laboratory, Richland, Washington 99354, USA

^*peng398@pnnl.gov
^†karol.kowalski@pnnl.gov

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Vol. 4, Iss. 4 — December - December 2022

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Figure 1
(a) Schematic workflow of the unitary partitioning approach proposed in Ref. [139]. (b) Numbers of Pauli terms ( $N_{Paulis}$ ) and corresponding numbers of unitary groups ( $N_{Unitaries}$ ) determined from the proposed anticommuting rule for the Hamiltonians of 1D, 2D, and 3D hydrogen systems encoded using up to 20 qubits. The geometries of these hydrogen systems are given in Ref. [155], and the nearest $H − H$ distance in these geometries is 1.5 Å. The fitting curves (orange and green dashed lines) are line functions, $y = k x + c$ , where $k$ and $c$ are fitting coefficients. (c) The $N_{Unitaries} / N_{Paulis}$ ratios for the Hamiltonians of a wider class of quantum molecular systems encoded using up to 40 qubits. Ratios for a series of 20-qubit systems studied are shown in histograms in the inset. The fitting curves take the expression, ${log}_{10} y = a {log}_{10} x + b$ , where $a$ and $b$ are fitting coefficients.
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Figure 2
(a),(b) Numbers of Pauli terms ( $N_{Paulis}$ ) and corresponding numbers of unitary groups ( $N_{Unitaries}$ ) determined from the SR-guided unitary partitioning approach for the approximate expansion of $〈 Ψ_{CCSD} |$ and $| Ψ_{CCSD} 〉$ of 1D, 2D, and 3D hydrogen systems encoded using up to 20 qubits. The expansions of the left and right CCSD wave functions are shown in Eqs. (23) and (24). In these hydrogen systems, the nearest $H − H$ distance is 1.5 Å. The fitting curves (orange and green dashed lines) are line functions, $y = k x + c$ , where $k$ and $c$ are fitting coefficients. (c),(d) The $N_{Unitaries} / N_{Paulis}$ ratios for $〈 Ψ_{CCSD} |$ and $| Ψ_{CCSD} 〉$ of a wider class of quantum molecular systems encoded using up to 40 qubits. Ratios for a series of 20-qubit systems studied are shown in histograms in the insets. The fitting curves take the expression $log y = a log x + b$ , where $a$ and $b$ are fitting coefficients.
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Figure 3
MMCC energy estimates from noise-free (a),(c) and noisy (b),(d) quantum simulations of the ground-state energy of the half-filling, four-site SIAM at different $U / V$ ratios (here we set $V = 1.0$ and vary $U$ values to get different $U / V$ ratios). The noisy quantum simulations using direct and indirect measurements are performed with IBM Qiskit employing the Fake_Toronto quantum simulator that mimics the simulation on the real ibmq_toronto quantum device. Each noisy data point and the associated error bar in (c) represents an average measurement outcome and the associated standard deviation from 10 experiments with 10 000 shots for each experiment. Tabulated data are provided in Appendix pp2.
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Figure 4
The PES of the HF molecule computed from noise-free simulation (a),(c) and noisy simulation (b),(d) employing different CC approaches. The double-zeta basis is employed for all quantum simulations in which the active space $(6 o, 6 e)$ is chosen to map to 12 qubits. For the HF molecule with the (6o,6e) active space, the CCSDTQ results are very close to the exact full configuration-interaction solution. The noisy quantum simulations are performed using IBM Qiskit employing the Fake_Toronto quantum simulator that mimics the simulation on the real ibmq_toronto quantum device. Each CC data point and the associated error bar in (b),(d) represents an average measurement outcome and the associated standard deviation from 10 experiments with 1,024 shots for each experiment. Tabulated data are provided in Appendix pp2.
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