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

TL;DR

The paper tackles the difficulty of achieving chemical accuracy with conventional UCC approaches on NISQ devices by introducing a fixed, non-iterative disentangled UCC (NI-DUCC) built in qubit space using Lie-algebraic structure. It constructs a symmetry-preserving, minimal complete pool (MCP) of Pauli-string excitations with size $2n-2$ and forms a $k$-layer NI-DUCC ansatz that scales as $\mathcal{O}(knp)$ in CNOTs, significantly reducing circuit depth while maintaining expressivity. Numerical results on LiH, H$_6$, and BeH$_2$ show that strong symmetric excitations with Lie algebraic closure yield rapid convergence to the FCI energy, achieving chemical accuracy and sometimes exact FCI with modest layers (e.g., $k=8$); NI-DUCC also compares favorably against fixed fermionic and adaptive VQE methods in gate counts and optimization effort. The method’s main trade-off is a classical MCP-generation cost that grows exponentially with qubits, currently limiting scalability beyond ~20 qubits, but ongoing optimizations and potential ML-assisted MCP screening may extend its feasibility. Overall, NI-DUCC offers a hardware-efficient, non-iterative pathway to accurately capture strong correlation on NISQ devices, with promising implications for excited-state and open-shell extensions as well as integration with Lie-algebraic techniques in variational quantum computing.

Abstract

Due to their non-iterative nature, fixed Unitary Coupled-Cluster (UCC) ansätze 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.

Figures (4)

• Figure 1: Schematic of NI-DUCC-VQE algorithm. (1) The necessary conditions for constructing a robust symmetry-preserving minimal complete pool (MCP) of size $2n-2$. (1.c) The initial qubit excitations (starters) are strong, symmetric operators, selected via a fermionic pre-screening criterion based on a threshold $\epsilon$, ensuring only dominant excitations are included while weaker excitations are filtered out. (1.d) Verifying the completeness of the set of the chosen starters (More details can be found in section 4 of the Supplementary Materials
• Figure 2: Energy convergence plots for ground states of H$_6$ (12 qubits) and BeH$_2$ (14 qubits), using the STO-3G basis set, at bond distances ( $r_{\text{H-H}}$ = 1.0 Å,) and ($r_{\text{Be-H}}$ = 3.5 Å), respectively. The plots are obtained using the NI-DUCC-VQE algorithm: Top panel (a, b), compares the quality of excitation set, insuring the significance of strong symmetric excitations when they are combined with the closure Lie algebraic properties. In Figure 2(a), the weak symmetric excitations with or without the Lie algebraic closure are overlapping. Panel (c, d), illustrates the rapid convergence as the number of layers, $k$, increases in the circuit. Bottom Panel (e, f), shows the fidelity property, which is calculated by overlapping $\langle\Psi_j(\vec{\theta^*}) |\Psi_{g}\rangle$ between the computed NI-DUCC state at each optimization step "$j$" and the theoretical eigenvector ground state $|\Psi_{g}\rangle$ of Hamiltonian ($\hat{H}$).
• Figure 3: Dissociation curves performance of NI-DUCC-VQE: for LiH (12 qubits), H$_6$ (12 qubits) and BeH$_2$ (14 qubits) molecules, in the STO-3G orbital basis set. The energy error is the difference between the obtained energy and FCI solution in (Hartree). The plots compare the NI-DUCC- VQE ($k=8$) (red), the UsCCSDTQ-VQE (yellow), the UCCSD-VQE (black) and the UsCCD-VQE (purple) algorithms. All convergence plots related to UsCCD and UsCCSDTQ are terminated at an energy threshold equal to $\epsilon = 10^{-8}$ Hartree.
• Figure 4: Resource comparison of the QEB-ADAPT-VQE, the fermionic-ADAPT-VQE, the qubit-ADAPT-VQE and NI-DUCC-VQE. The tested molecules (STO-3G) are LiH ($r_{\text{Li-H}}$ = 1.546 Å), BeH$_2$ ($r_{\text{Be-H}}$ = 1.316 Å) , and H$_6$ ( $r_{\text{H-H}}$ = 1.5 Å,). (a) CNOT counts, (b) Parameter counts, (c,d) function evaluation for H$_6$ (c) and BeH$_2$ (d), respectively. Note that the number of operators are equal to the number of parameters in each algorithm. Also, the function evaluations are the number of optimization steps in the BFGS optimizer.