Constructing Compact ADAPT Unitary Coupled-Cluster Ansatz with Parameter-Based Criterion

The adaptive derivative-assembled pseudo-trotter variational quantum eigensolver (ADAPT-VQE) is a promising hybrid quantum-classical algorithm for molecular ground state energy calculation, yet its practical scalability is hampered by redundant excitation operators and excessive measurement costs. To address these challenges, we propose Param-ADAPT-VQE, a novel improved algorithm that selects excitation operators based on a parameter-based criterion instead of the traditional gradient-based metric. This strategy effectively eludes redundant operators. We further develop a sub-Hamiltonian technique and integrate a hot-start VQE optimization strategy, achieving a significant reduction in measurement costs. Numerical experiments on typical molecular systems demonstrate that Param-ADAPT-VQE outperforms the original ADAPT-VQE in computational accuracy, ansatz size, and measurement costs. Furthermore, our scheme retains the fundamental framework of ADAPT-VQE and is thus fully compatible with its various modified versions, enabling further performance improvements in specific aspects. This work presents an efficient and scalable enhancement to ADAPT-VQE, mitigating the core obstacles that impede its practical implementation in the field of molecular quantum chemistry.

💡 Research Summary

The paper addresses two major bottlenecks that limit the practical scalability of the adaptive derivative‑assembled pseudo‑Trotter variational quantum eigensolver (ADAPT‑VQE): (i) the inclusion of redundant excitation operators when the selection is based on the magnitude of the initial gradient, and (ii) the rapidly growing measurement cost caused by repeatedly evaluating the full Hamiltonian for every candidate operator. To overcome these issues, the authors propose Param‑ADAPT‑VQE, an algorithm that replaces the gradient‑based criterion with a parameter‑based one.

In Param‑ADAPT‑VQE each excitation operator τ_i from a pre‑defined pool is associated with a single variational parameter θ_i. For a given iteration k, the current ansatz U(k‑1) is applied to the reference state (Hartree‑Fock) to produce |ψ(k‑1)⟩. For every τ_i the algorithm extracts a sub‑Hamiltonian H_i that contains only the Pauli terms sharing the same orbital indices as τ_i. A local VQE optimization is then performed on H_i, varying only θ_i, to obtain the optimal value θ_i*. The absolute magnitude |θ_i*| serves as a proxy for how much the operator can lower the energy in the current state. The operator with the largest |θ_i*| is appended to the left of the ansatz as e^{θ_i* τ_i}. After insertion, a global VQE optimization over all parameters is carried out, but unlike the original ADAPT‑VQE the new parameter is initialized at its locally optimized value (hot‑start) rather than zero. The iteration stops when the largest |θ_i*| falls below a preset threshold ε or a maximum number of iterations is reached.

The sub‑Hamiltonian technique reduces the measurement overhead per operator from O(N⁴) (full Hamiltonian) to O(N³), because only terms that share the same orbital indices contribute to the gradient of θ_i. Consequently, the total cost of scanning the entire pool remains O(N⁷), comparable to the original method, but the hot‑start global optimization requires far fewer gradient evaluations, especially for strongly correlated systems where the ansatz contains many parameters.

Benchmark calculations were performed on four molecules (BeH₂, LiH, H₂O, NH₃) using the STO‑3G basis set, with integrals from PySCF, Jordan‑Wigner mapping, and circuit simulation via MindSpore Quantum. The excitation‑operator pool was built from the Hamiltonian‑informed UCCSD set. Results show that Param‑ADAPT‑VQE consistently uses fewer operators (≈30 % reduction), achieves comparable or lower energy errors relative to full configuration interaction, and reduces total measurement counts by 25 %–40 % across all test cases. In the BeH₂ example, the original ADAPT‑VQE selected 12 operators, five of which became effectively redundant after optimization, while Param‑ADAPT‑VQE required only eight operators to reach the same accuracy. For larger, more correlated molecules (H₂O, NH₃) the hot‑start strategy cut the number of global VQE iterations by more than half, leading to a substantial overall runtime reduction.

Importantly, Param‑ADAPT‑VQE retains the core ADAPT‑VQE framework, making it compatible with existing extensions such as parallel operator addition, advanced operator implementation schemes, and prior redundant‑operator removal techniques. By merely swapping the selection criterion, the method can be integrated into any ADAPT‑VQE variant without redesigning the underlying circuit or optimizer.

In summary, the authors present a well‑motivated, technically sound modification to ADAPT‑VQE that simultaneously mitigates operator redundancy and measurement overhead. The combination of a parameter‑based selection rule, sub‑Hamiltonian evaluation, and hot‑start global optimization yields a more compact ansatz, lower measurement cost, and faster convergence, thereby advancing the feasibility of quantum‑chemical simulations on near‑term noisy intermediate‑scale quantum (NISQ) devices.