Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H 2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.
Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H 2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.
We develop a resource-efficient step-merged quantum imaginary time evolution approach (smQITE) to solve for the ground state of a Hamiltonian on quantum computers. This heuristic method features a fixed shallow quantum circuit depth along the state evolution path. We use this algorithm to determine the binding energy curves of a set of molecules, including H 2, H 4, H 6, LiH, HF, H 2 O, and BeH 2, and find highly accurate results. The required quantum resources of smQITE calculations can be further reduced by adopting the circuit form of the variational quantum eigensolver (VQE) technique, such as the unitary coupled cluster ansatz. We demonstrate that smQITE achieves a similar computational accuracy as VQE at the same fixed-circuit ansatz, without requiring a generally complicated high-dimensional nonconvex optimization. Finally, smQITE calculations are carried out on Rigetti quantum processing units, demonstrating that the approach is readily applicable on current noisy intermediate-scale quantum devices.
Quantum simulation is one of the key applications of quantum computing, which accelerates research and development in the fields such as chemistry and material science. The recent development of noisy intermediate-scale quantum (NISQ) devices urges the exploration of applications without the necessity of quantum error correction. In this paper, we propose an efficient method to simulate quantum dynamics driven by a static Hamiltonian on NISQ devices, named subspace variational quantum simulator (SVQS). SVQS employs the subspace-search variational quantum eigensolver (SSVQE) [Phys. Rev. Res. 1, 033062 (2019)] to find a low-lying eigensubspace and extends it to simulate dynamics within the subspace with lower overhead compared to the existing schemes. We experimentally simulate the time-evolution operator in a low-lying eigensubspace of a hydrogen molecule. We also define the subspace process fidelity as a measure between two quantum processes in a subspace. The subspace time evolution mimicked by SVQS shows the subspace process fidelity of 0.896–0.989.
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.
Preparing ground states of correlated molecules on near-term quantum computers requires algorithms that are both accurate and resource-efficient. We introduce the qubit-adaptive variational quantum imaginary time evolution (QAITE) algorithm, which synthesizes the systematic convergence of imaginary-time evolution with hardware-efficient, adaptive ansatz construction. By integrating the linear-scaling qubit operator pool into the adaptive variational quantum imaginary time evolution framework, QAITE deterministically constructs a compact ansatz without costly classical optimization loops. Numerical simulations for molecular systems (H4, LiH, H2O, BeH2), benchmarked against our own implementations of established adaptive algorithms under identical conditions, show that QAITE achieves chemical accuracy with a 15-45% reduction in CNOT gates compared to the standard adaptive variational quantum imaginary time evolution (AVQITE) method. For hydrogen chains (RHH = 1.5 Å, N = 6-16 qubits), QAITE's circuit complexity grows approximately quadratically and remains the lowest of the four methods at every system size, reaching 31% fewer CNOT gates than AVQITE at 16 qubits─a saving of 1322 gates. This efficiency extends to heterogeneous (LiH)n and (HBe)n chains, where QAITE maintains a CNOT gate ratio below 1 for all sizes studied (e.g., 0.81 for (LiH)2). We further show that the deterministic parameter update is numerically robust: the results are insensitive to the regularization of the quantum geometric tensor over at least 4 orders of magnitude, and chemical accuracy is retained under realistic shot noise applied to both the quantum geometric tensor and the force vector. By outperforming established adaptive algorithms in both absolute gate count and scaling, QAITE establishes a promising, scalable pathway for quantum molecular simulations on noisy intermediate-scale quantum devices.
Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of LiH. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizer, in obtaining LiH's first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three H4's excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near −1.0 Ha with only one optimizing procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with individual optimizing procedures, the optimizing procedure converges faster.
Calculating the energy spectrum of a quantum system is an important task, for example to analyse reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique, we employ the low depth swap test to energetically penalise the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates towards the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimisation problems of up to 18 qubits, and the electronic structure of the Lithium Hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.