Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster

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.

Automated summary

This work presents a method to relate the UNITARY COPLED CLUSTER approximation to the conventional COUPLED CLUSTER approximation, by manipulating operators. The factorized form of the approximations can be transformed using exponential disentangling identity and Hadamard lemma, and further related to the standard form using the Trotter product formula. The dependence on operators can also be removed by introducing higher-rank operators, yielding the conventional COUPLED CLUSTER method.