Non-Iterative Disentangled Unitary Coupled-Cluster based on Lie-algebraic structure

Due to their non-iterative nature, fixed Unitary Coupled-Cluster (UCC) ans\"atze are attractive for performing quantum chemistry Variational Quantum Eigensolver (VQE) computations as they avoid pre-circuit measurements on a quantum computer. However, achieving chemical accuracy for strongly correlated systems with UCC requires further inclusion of higher-order fermionic excitations beyond triples increasing circuit depth. We introduce $k$-NI-DUCC, a fixed and Non-iterative Disentangled Unitary Coupled-Cluster compact ansatz, based on specific $"k"$ sets of "qubit" excitations, eliminating the needs for fermionic-type excitations. These elements scale linearly ($\mathcal{O}(n)$) by leveraging Lie algebraic structures, with $n$ being the number of qubits. The key excitations are screened through specific selection criteria, including the enforcement of all symmetries, to ensure the construction of a robust set of generators. NI-DUCC employs $"k"$ products of the exponential of $\mathcal{O}(n)$- anti-Hermitian Pauli operators, where each operator has a length $p$. This results in a fewer two-qubit CNOT gates circuit, $\mathcal{O}(knp)$, suitable for hardware implementations. Tested on LiH, H$_6$ and BeH$_2$, NI-DUCC-VQE achieves both chemical accuracy and rapid convergence even for molecules deviating significantly from equilibrium. It is hardware-efficient, reaching the exact Full Configuration Interaction energy solution at specific layers, while reducing significantly the VQE optimization steps. While NI-DUCC-VQE effectively addresses the gradient measurement bottleneck of ADAPT-VQE-like iterative algorithms, the classical computational cost of constructing the $\mathcal{O}(n)$ set of excitations increases exponentially with the number of qubits. We provide a first implementation for constructing the generators' set able to handle up to 20 qubits and discuss the efficiency perspectives.