Variational quantum algorithms, particularly the Variational Quantum Eigensolver (VQE), are crucial for near-term quantum devices (NISQ). VQE has been widely applied to quantum chemistry, extending from ground state simulations to excited states. Building upon imaginary time evolution (ITE), which aims to converge an initial state to the ground state, several quantum algorithms have emerged. Variational ITE (VITE) uses a fixed ansatz, optimizing parameters via McLachlan's variational principle. Probabilistic ITE (PITE) uses measurements for probabilistic ITE, but its success probability decays exponentially. Quantum ITE (QITE) approximates non-unitary ITE with a unitary evolution, solving linear equations to avoid high-dimensional optimizations. While QITE is effective for ground states, excited state calculations remain challenging. Quantum Lanczos diagonalization (QLanczos) is often used, but excited state components vanish exponentially with imaginary time, especially for large energy gaps. Folded-spectrum QITE (FSQITE) addresses this by using a folded-spectrum propagator, but requires prior knowledge of the target energy. This paper introduces MSQITE to overcome these limitations, providing stable and accurate solutions for excited states by evolving an orthogonal model space to the complete subspace, simulating multiple states simultaneously.

The authors reviewed existing variational quantum algorithms and their applications in quantum chemistry, highlighting the limitations of previous approaches to excited state calculations. They discussed Variational Quantum Eigensolver (VQE), Variational Imaginary Time Evolution (VITE), Probabilistic Imaginary Time Evolution (PITE), and Quantum Imaginary Time Evolution (QITE), comparing their strengths and weaknesses. The limitations of QITE in accurately capturing excited states and the challenges posed by spin contamination were also addressed. The authors examined existing classical and quantum approaches to obtaining excited states from imaginary time evolution, specifically mentioning the folded-spectrum method. The review sets the stage for the proposed MSQITE algorithm, which is designed to address the limitations of existing methods.

MSQITE prepares an orthogonal subspace of zeroth-order ground and excited states and evolves this subspace using the propagator e⁻ⁱβH (Trotterized). The imaginary time evolution makes basis states non-orthogonal, necessitating orthonormalization. The authors employ a unitary approximation, introducing an energy shift Eₗ = ⟨Φₗ⁽ᵉ⁾|H|Φₗ⁽ᵉ⁾⟩ for convenience. The Löwdin symmetric orthonormalization is used to determine the transformation matrix *d*, minimizing the distance in Hilbert space and ensuring near orthogonality of the model space. The resultant model space basis is used to express physical states as a linear combination, solving the generalized eigenvalue problem Hc = Sc. Two approaches for determining the unitary e⁻ⁱΔβA are presented: state-specific (different *a*ₗ and Aₗ for each state) and state-averaged (same *a* and A for all states). The state-specific approach minimizes the function Fˡ(aˡ) to second order in Δβ, resulting in a linear equation. The state-averaged approach uses a single linear equation for all states. The quantum circuit for state-specific MSQITE is implemented using the Hadamard test, with controlled gates for e⁻ⁱθA and e⁻ⁱθA'. The circuit is optimized to reduce gate complexity by leveraging the commutation of controlled gates. MS-QLanczos, an extension of QLanczos to model space, is introduced to accelerate convergence. The algorithm expands the Krylov model subspace and diagonalizes the effective Hamiltonian, generating a wave function as a linear combination of time-evolved states. A global energy shift E₀ is introduced to ensure that the propagator is independent of the state and imaginary time, while avoiding the vanishing norm. The overlap matrix and Hamiltonian matrix elements are calculated, and the generalized eigenvalue problem is solved. Noiseless simulations utilize the UCCGSD ansatz. To address spin contamination, a spin-shift is added to the propagator, projecting out states with undesired spin symmetry. For noisy simulations, the authors utilize Qiskit, modeling noise with depolarizing errors and employing zero-noise extrapolation (ZNE) for error mitigation. A simplified UCCGSD ansatz is used for reduced circuit complexity, and a reduced Hamiltonian is obtained using a tapering-off technique.

MSQITE demonstrates significantly faster convergence to excited states compared to QITE and FSQITE in noiseless simulations of BeH₂ and H₄. The convergence profile is state-dependent, with strongly correlated states requiring more imaginary time steps. MS-QLanczos further accelerates convergence by incorporating additional configurations automatically. MSQITE shows advantages for strongly correlated ground states. The spin-shift in the propagator effectively removes spin contamination in excited state calculations, preventing convergence to incorrect spin states. The state-specific MSQITE approach yields accurate energies for various molecular systems, while the state-averaged approach is less accurate, especially for larger systems, highlighting the limitations of using a single unitary for multiple states. Noise simulations show that error accumulation in both QITE and MSQITE increases with imaginary time. However, error mitigation techniques, such as ZNE, significantly improve the results, making MSQITE as stable as QITE even with noise. The improved stability of MSQITE with noise is attributed to the relatively smaller magnitude of errors in the off-diagonal elements compared to the diagonal ones, as well as similar error magnitude in diagonal and off-diagonal elements after error mitigation.

The findings demonstrate the effectiveness of MSQITE in addressing the challenges associated with simulating excited states on quantum computers. The superior performance of MSQITE compared to QITE and FSQITE, particularly in handling spin contamination and strongly correlated systems, highlights its potential for advancing quantum computational chemistry. The state-specific approach proves more robust than the state-averaged approach due to the limitations of approximating multiple states with a single unitary. The results of the noise simulations underscore the importance of error mitigation techniques and suggest MSQITE's practicality for NISQ devices. The observed balanced error propagation between diagonal and off-diagonal elements implies that the overall noise impact on MSQITE energies is similar to that on QITE energies. The versatility of MSQITE and its potential for extension to adaptive algorithms and variational algorithms opens avenues for further improvements and broader applications beyond quantum chemistry.

MSQITE offers a significant improvement over existing quantum imaginary time evolution methods for simulating both ground and excited states. Its ability to handle multiple states simultaneously, effectively mitigate spin contamination, and maintain stability under noise makes it a promising tool for quantum computational chemistry and other fields. Future research directions include combining MSQITE with adaptive algorithms and variational algorithms to further reduce circuit depth and improve efficiency.

The study primarily focuses on relatively small molecular systems. The scalability of the state-specific MSQITE approach to larger systems needs further investigation. The computational cost of the state-specific MSQITE increases with the number of states in the model space. While error mitigation techniques enhance the accuracy of noisy simulations, they also increase the computational overhead. The effectiveness of the spin-shift method might be limited in cases with strong spin-orbit coupling.