A quantum computing algorithm is proposed to obtain low-lying excited states in many-body interacting systems. The approximate eigenstates are obtained by using a quantum space diagonalization method in a subspace of states selected from the convergence path of the ADAPT-VQE (adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver) towards the ground state of the many-body problem. This method is shown to be accurate with only a small overhead in terms of quantum resources required to get the ground state. We also show that the quantum algorithm might be used to facilitate the convergence of the ADAPT-VQE method itself. Successful applications of the technique are made to like-particle pairing as well as neutron-proton pairing. Finally, the H 4 molecule’s dissociation also illustrates the technique, demonstrating its accuracy and versatility.
A quantum computing algorithm is proposed to obtain low-lying excited states in many-body interacting systems. The approximate eigenstates are obtained by using a quantum space diagonalization method in a subspace of states selected from the convergence path of the ADAPT-VQE (adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver) towards the ground state of the many-body problem. This method is shown to be accurate with only a small overhead in terms of quantum resources required to get the ground state. We also show that the quantum algorithm might be used to facilitate the convergence of the ADAPT-VQE method itself. Successful applications of the technique are made to like-particle pairing as well as neutron-proton pairing. Finally, the H 4 molecule’s dissociation also illustrates the technique, demonstrating its accuracy and versatility.
The ADAPT-VQE (adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver) approach is used to solve the neutron-proton pairing problem in atomic nuclei. This variational approach is considered today to be one of the most powerful methods to iteratively find the ground state of a many-body problem, provided a performing set of operators, called the pool of operators, is used to explore the Hilbert space of many-body wave-functions. Three different pools of operators, which might eventually break one or several symmetries of the Hamiltonian during the descent to the ground state, are tested for the neutron-proton pairing problem. We observe that the breaking of some symmetries during the optimization of the trial wave function might, in general, help to speed up the convergence towards the ground state. Still, we rejected the pool of operators that might explicitly break the total particle number because they become uncontrollable during the optimization process. Overall, we observed that the iterative optimization process rapidly becomes a delicate problem when the number of parameters to build the Ansatz increases, and the energy might get stuck at energies higher than the ground state energy. To improve the convergence in this case, several techniques have been proposed, with some better controlling the symmetries during the energy minimization. Among the proposed methods, two have proved effective: one based on an embedding technique and the other on a randomized preparation of the initial state. We conclude that the ADAPT-VQE, complemented by these techniques, can provide a very accurate description of the neutron-proton pairing problem, and can outperform other standardly used techniques that break the particle number symmetry and restore afterwards.
Quantum computing offers a scalable approach to solving the nuclear shell model, a highly complex and exponentially scaled many-body problem. This work presents a numerical simulation of the subspace search variational quantum eigensolver (SSVQE) combined with an adaptive derivative-assembles pseudo-trotter (ADAPT) ansatz to obtain the low-lying states of any nuclear system in a single optimization run. As an example, we apply this method in this work to a trivial identical nucleon system, two nucleons in the 0p3/2 orbital, mapped to 4 qubits depicting m-scheme single-particle states including a surface delta effective interaction using the Jordan-Wigner transformation. The ADAPT-SSVQE algorithm, by utilizing a symmetry-preserving double-excitation ADAPT operator pool, uniquely optimizes a weighted energy sum, forcing the simultaneous convergence of two lowest states within the total angular momentum MJ=0 subspace. We demonstrate the accuracy of the method by benchmarking against the exact diagonalization, confirming its potential for probing nuclear structure and pairing phenomena on current and near-future quantum devices without requiring multi-step procedure for excited states.
Designing compact ansätze in Variational Quantum Eigensolver (VQE) is crucial for solving energetic problems of practical molecules on near-term quantum devices. However, existing Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansätze face two challenges: improper operator selection and accumulation of degraded operators. In this paper, we propose the Hamiltonian-Aware (HA) ADAPT-VQE algorithm to address these issues. First, we establish a novel excitation operator selection criterion. It breaks the local constraint of existing criteria by incorporating Hamiltonian information, prioritizes physically meaningful excitation operators, and incurs no extra classical or quantum computational overhead. Furthermore, we develop a problem-adaptive method for discriminating and pruning redundant excitation operators stemming from improper selection and inevitable degradation. This method balances redundant operator pruning and convergence guarantee, and is applicable to ansätze with arbitrary scales. Systematic numerical experiments on typical strongly correlated molecular systems demonstrate that our HA-ADAPT-VQE avoids energy plateaus and outperforms baseline algorithms in terms of energy error, ansatz size, and measurement cost. This work offers an efficient, robust ansatz construction paradigm, facilitating the development and practical deployment of large-scale VQE in quantum chemistry.
By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H2, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.
Abstract The prediction of electronic structure for strongly correlated molecules represents a promising application for near-term quantum computers. Significant attention has been paid to ground state wavefunctions, but excited states of molecules are relatively unexplored. In this work, we consider the adaptive, problem-tailored (ADAPT)-variational quantum eigensolver (VQE) algorithm, a single-reference approach for obtaining ground states, and its state-averaged generalization for computing multiple states at once. We demonstrate for both rectangular and linear H4, as well as for BeH2, that this approach, which we call multistate-objective, Ritz-eigenspectral (MORE)-ADAPT-VQE, can make better use of small excitation manifolds than an analogous method based on a single-reference ADAPT-VQE calculation, q-sc-EOM. In particular, MORE-ADAPT-VQE is able to accurately describe both avoided crossings and crossings between states of different symmetries. In addition to more accurate excited state energies, MORE-ADAPT-VQE can recover accurate transition dipole moments in situations where traditional ADAPT-VQE and q-sc-EOM struggle. These improvements suggest a promising direction toward the use of quantum computers for difficult excited state problems.
The adaptive derivative-assembled pseudo-trotter (ADAPT) variational quantum eigensolver (VQE) is an appealing framework for iteratively constructing accurate representations of ground- and excited-state wave functions as products of exponentialized one- and two-body operators acting on a reference state. In this work, we propose a CNOT-efficient circuit for implementing exponentialized two-body qubit-excitation-based (QEB) operators, enabling the construction of the low-depth wave function Ansätze. Our proposed circuit architecture employs a 2-qubit-controlled rotation gate flanked by two layers of CNOT gates, requiring only 9 CNOT gates per two-body QEB operator while preserving essential symmetries, including particle number and the Z -component of spin. By integrating these optimized circuits into the ADAPT–VQE framework, we achieve a reduction in circuit depth of approximately 28% compared to the original QEB ADAPT–VQE method, without sacrificing accuracy in simulating ground- and excited-state properties of small molecules. Numerical simulations validate the improved performance of our approach, demonstrating its potential as an efficient tool for quantum simulations of electronic structures.