Coupled cluster equations for superconducting systems

Brian D. Josephson

Bo Peng, Karol Kowalski

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 $\ensuremath{\sim}40$ 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.

H. Mitter, K. Yamazaki

Jacob Townsend, Konstantinos D. Vogiatzis

Solving the coupled-cluster (CC) equations is a cost-prohibitive process that exhibits poor scaling with system size. These equations are solved by determining the set of amplitudes ( t ) that minimize the system energy with respect to the coupled-cluster equations at the selected level of truncation. Here, a novel approach to predict the converged coupled-cluster singles and doubles (CCSD) amplitudes, thus the coupled-cluster wave function, is explored by using machine learning and electronic structure properties inherent to the MP2 level. Features are collected from quantum chemical data, such as orbital energies, one-electron Hamiltonian, Coulomb, and exchange terms. The data-driven CCSD (DDCCSD) is not an alchemical method because the actual iterative coupled-cluster equations are solved. However, accurate energetics can also be obtained by bypassing solving the CC equations entirely. Our preliminary data show that it is possible to achieve remarkable speedups in solving the CCSD equations, especially when the correct physics are encoded and used for training of machine learning models.

Alexandre M. Zagoskin

Piotr Piecuch B, Jeffrey R. Gour, Marta Włoch

Abstract The energy expansion defining the biorthogonal method of moments of coupled‐cluster equations (MMCC) [Piecuch and Włoch, J Chem Phys, 2005, 123, 224105 and Piecuch et al., Chem Phys Lett 2006, 418, 467], which leads to the size extensive completely renormalized (CR) coupled‐cluster (CC) approach with singles, doubles, and noniterative triples employing the left eigenstates of the similarity‐transformed Hamiltonian, termed CR‐CC(2,3), is overviewed and rederived. The rederivation of the biorthogonal MMCC expansion presented in this work is based on a direct resummation and subsequent elimination of the many‐body components of the exponential wave operator of CC theory that appear at individual moment contributions in the original MMCC energy expansion [Kowalski and Piecuch, J Chem Phys, 2000, 113, 18; Kowalski and Piecuch, J Chem Phys 2001, 115, 2966], enabling one to understand why the CR‐CC(2,3) method using the biorthogonal MMCC theory is more accurate than the earlier CR‐CCSD(T) approach. The superiority of the CR‐CC(2,3) method over the CR‐CCSD(T) and other previously developed single‐reference CC methods with a noniterative treatment of triply excited clusters, including the widely used CCSD(T) approach and the triples corrections defining the CCSD(2) schemes, is illustrated by examining the singlet–triplet gap of the (HFH)− magnetic system in which two paramagnetic centers are linked via a polarizable diamagnetic bridge. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008