Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster

James K Freericks

doi:10.3390/sym14030494

Abstract

The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems.

Highlights

The coupled cluster (CC) approach [1] is regarded as the gold standard for quantum chemistry, especially as it is applied to weakly correlated molecular systems
We have shown how one can relate a single-reference UCC ansatz in factorized form to its corresponding single-reference CC ansatz in a factorized form
By using the Trotter product formula, this approach can be extended to include the traditional UCC ansatz, and by adding additional higher-rank terms, the factorized form of the CC ansatz can be converted to the traditional form

Summary

Introduction

The coupled cluster (CC) approach [1] is regarded as the gold standard for quantum chemistry, especially as it is applied to weakly correlated molecular systems. There are a number of innovative keys to the CC approximation in quantum chemistry It provides a low-rank representation of a many-body quantum state that is size-consistent for closed-shell fragments (unlike many configuration interaction approximations), meaning it reduces to the closed shell atomic systems when the molecule is pulled apart by stretching. It is size extensive, implying it has a linked-cluster-like expansion in terms of diagrams, so it scales the energy properly in the thermodynamic limit. Because quantum computers work most efficiently with unitary operations, the UCC ansatz is the only approach that can be practically implemented on a quantum computer

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Conclusion