Most literature in the Variational Quantum Eigensolver (VQE) algorithm focuses on finding the ground state of a physical system, by minimizing a quantum-computed cost-function. When excited states are required, the cost-function is usually modified to include additional terms ensuring orthogonality with the ground state. This generally requires additional quantum circuit executions and measurements, increasing algorithmic complexity. Here we present a design strategy for the variational ansatz which enforces orthogonality in candidate excited states while still fully exploring the remaining subset of Hilbert space. The result is an excited-state VQE solver which trades increasing measurement complexity for increasing circuit complexity. The latter is anticipated to become preferable as quantum error mitigation and correction become more refined. We demonstrate our approach with three distinct ansatze, beginning with a simple single-body example, before generalizing to accommodate the full Hilbert space spanned by all qubits, and a constrained Hilbert space obeying particle number conservation.
Most literature in the Variational Quantum Eigensolver (VQE) algorithm focuses on finding the ground state of a physical system, by minimizing a quantum-computed cost-function. When excited states are required, the cost-function is usually modified to include additional terms ensuring orthogonality with the ground state. This generally requires additional quantum circuit executions and measurements, increasing algorithmic complexity. Here we present a design strategy for the variational ansatz which enforces orthogonality in candidate excited states while still fully exploring the remaining subset of Hilbert space. The result is an excited-state VQE solver which trades increasing measurement complexity for increasing circuit complexity. The latter is anticipated to become preferable as quantum error mitigation and correction become more refined. We demonstrate our approach with three distinct ansatze, beginning with a simple single-body example, before generalizing to accommodate the full Hilbert space spanned by all qubits, and a constrained Hilbert space obeying particle number conservation.
The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.
We introduce the orbital-optimized multistate contracted variational quantum eigensolver (oo-MC-VQE) method with spin-adapted operators for the computation of ground and excited states, as well as state-specific and transition properties. The use of spin-adapted operators ensures that the spin symmetry of the reference states is conserved throughout the VQE optimization. In multistate variational approaches, achieving a balanced description of an increasing number of electronic states places growing demands on the expressibility of the underlying ansatz, thereby introducing a fundamental trade-off between accuracy and circuit complexity. We consider the effects of this trade-off explicitly and find that the number of circuit parameters required to obtain accurate results is reported to scale approximately linearly in the number of states. We further present an explicit quantum-circuit implementation of the oo-MC-VQE method and demonstrate its integration with quantum error mitigation techniques. Finally, we execute the method on real quantum devices to compute absorption spectra for two benchmark molecular systems.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
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.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.
Variational Quantum Algorithms (VQAs) are iterative algorithms suited to implementation on current-era quantum devices. VQAs employ classical optimization to minimize cost functions evaluated on quantum circuits. However, the extent to which VQAs manage noise is often insufficient for quantum chemistry applications. One method of improving VQAs is through accurate ansatz initialization. The CAFQA (Clifford Ansatz For Quantum Accuracy) protocol runs a discrete search through a classically simulatable subset of the entire state space to find a desirable initialization. Prior work has evaluated CAFQA applied to the Variational Quantum Eigensolver (VQE), a VQA that computes grounds states of a Hamiltonian. Motivated by CAFQA's success, we propose Excited-CAFQA initialization for Variational Quantum Deflation (VQD), a quantum algorithm that extends VQE by allowing the computation of excited states. VQD recursively computes excited states, by constraining the kth state to be orthogonal to the previous k-1 computed energy states via a penalty term appended to the standard VQE cost function. Just as with VQE, the VQD cost function can be efficiently computed classically for the states considered in the discrete CAFQA search, allowing for the discrete CAFQA optimizer to find good initial parameters for each energy level computation. Preliminary evaluation shows that Excited-CAFQA achieves 90 to 99+% accuracy across a variety of bond lengths and excited states for H2 and HeH+ molecular systems.