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arXiv:2409.05939 · Communications Physics

Fast gradient-free optimization of excitations in variational quantum eigensolvers

Jonas Jäger, Thierry N. Kaldenbach, Max Haas, Erik Schultheis

Published Oct 30, 2025• 7 citations
arXivHTMLPDF

Abstract

Abstract Finding molecular ground states and energies with variational quantum eigensolvers is central to chemistry applications on quantum computers. Physically motivated ansätze based on excitation operators respect physical symmetries, but existing quantum-aware optimizers, such as Rotosolve, have been limited to simpler operator types. To fill this gap, we introduce ExcitationSolve, a fast quantum-aware optimizer that is globally-informed, gradient-free, and hyperparameter-free. ExcitationSolve extends these optimizers to parameterized unitaries with generators G of the form G 3 = G exhibited by excitation operators in approaches such as unitary coupled cluster. ExcitationSolve determines the global optimum along each variational parameter using the same quantum resources that gradient-based optimizers require for one update step. We provide optimization strategies for both fixed and adaptive variational ansätze, along with generalizations for simultaneously selecting and optimizing multiple excitations. On molecular ground state energy benchmarks, ExcitationSolve outperforms state-of-the-art optimizers by converging faster, achieving chemical accuracy for equilibrium geometries in a single parameter sweep, yielding shallower adaptive ansätze and remaining robust to real hardware noise. By uniting physical insight with efficient optimization, ExcitationSolve paves the way for scalable quantum chemistry calculations.

DOI
10.1038/s42005-025-02375-9

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Abstract Finding molecular ground states and energies with variational quantum eigensolvers is central to chemistry applications on quantum computers. Physically motivated ansätze based on excitation operators respect physical symmetries, but existing quantum-aware optimizers, such as Rotosolve, have been limited to simpler operator types. To fill this gap, we introduce ExcitationSolve, a fast quantum-aware optimizer that is globally-informed, gradient-free, and hyperparameter-free. ExcitationSolve extends these optimizers to parameterized unitaries with generators G of the form G 3 = G exhibited by excitation operators in approaches such as unitary coupled cluster. ExcitationSolve determines the global optimum along each variational parameter using the same quantum resources that gradient-based optimizers require for one update step. We provide optimization strategies for both fixed and adaptive variational ansätze, along with generalizations for simultaneously selecting and optimizing multiple excitations. On molecular ground state energy benchmarks, ExcitationSolve outperforms state-of-the-art optimizers by converging faster, achieving chemical accuracy for equilibrium geometries in a single parameter sweep, yielding shallower adaptive ansätze and remaining robust to real hardware noise. By uniting physical insight with efficient optimization, ExcitationSolve paves the way for scalable quantum chemistry calculations.
In recent years, the variational quantum eigensolver (VQE)[1] has carried the hope of the quantum computing community for achieving quantum utility. It aims to find the ground state and corresponding energy of a molecular hamiltonian by classically optimizing the parameters θj\theta_jθj​ of a variational quantum circuit. In order to guarantee the conservation of qunatities such as particle number or spin, physically inspired ansätze such as the UCCSD ansatz are most promising. ExcitationSolve provides a quantum-aware optimizer for such physically inspired ansätze, making best use of prior knowlege about the operators used in the ansatz.
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
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.
computations of the ground and excited electronic states. While substantial progress has been made in designing physically informed quantum algorithms to address the ground state under the constraints of near-term quantum hardware, relatively fewer developments exist for reliably computing excited state energetics, presumably due to the growing complexity of the quantum circuits for excited eigenroots. In this work, we exploit the inherent point-group and spin symmetries of molecular systems to construct symmetry-adapted sets of orthogonal reference states which are evolved with a common symmetry-scalar unitary, enabling the simultaneous determination of ground and excited eigenroots. By minimizing the energy functional over a restricted symmetry-adapted correlated manifold, the weighted subspace-search variational quantum eigensolver enables us to directly target multiple eigenroots within each spatial and spin symmetry sector. This physically motivated selection of references and the unitary confer substantial flexibility to the weighted energy landscape, largely allowing the optimization to avoid numerical traps and converge directly to the targeted eigenstates. More crucially, symmetry resolution also enables access to a larger number of eigenstates without substantially increasing the complexity of the cost function. Through benchmarking on moderate to strongly correlated molecular systems, we demonstrate the accuracy and advantages of our approach relative to several established excited state algorithms on quantum computers.
We present a novel approach for efficient preparation of electronic ground states, leveraging the optimizer ExcitationSolve [Jäger et al., Comm. Phys. (2025)] and established variational quantum eigensolver-based operator selection methods, such as Energy Sorting (ES). By combining these tools, we demonstrate a computationally efficient protocol that enables the construction of an approximate ground state from a unitary coupled cluster ansatz via a single sweep over the operator pool. Utilizing efficient classical pre-processing to select the majority of relevant operators, this approach reduces the computational complexity associated with traditional variational quantum eigensolver (VQE) optimization methods. We further show that second-order Epstein-Nesbet (EN2) perturbation theory emerges as the first-order Taylor expansion of our protocol in terms of a correlation measure, clarifying why our approach provides a more robust initial guess for the ground state in strongly correlated regimes. We also find that second-order Møller-Plesset perturbation (MP2) theory, which is widely used for unitary coupled cluster (UCC) initialization, performs worse than both EN2 and our protocol. Furthermore, we show that our method can be seamlessly integrated with one-variational-parameter couple exchange operators, thereby further reducing the number of required CNOT operations. Overall, we empirically observe a quadratic convergence speedup beyond state-of-the-art methods, advancing the preparation of high-fidelity electronic ground states - one of the cornerstones of meaningful electronic structure calculations in the noisy intermediate-scale quantum computing (NISQ) era, and a prerequisite for fault-tolerant quantum computing (FTQC) algorithms such as quantum phase estimation.