NISQ Computers Achieve Excited-State Calculations for Challenging Condensed Matter Problems

Calculating excited states is a persistent challenge for quantum computing, yet understanding these states is crucial in fields ranging from materials science to chemistry. Hengzhun Chen, Yingzhou Li, and Bichen Lu, from Fudan University, alongside Jianfeng Lu, now present a detailed analysis of three variations of the variational eigensolver (VQE) algorithm, a promising method for performing these calculations on today’s quantum computers. Their work tackles a key difficulty in excited state calculations, preventing the algorithm from collapsing to the lowest energy ground state, by embedding orthogonality constraints directly into the cost function. Importantly, the team demonstrates that each of these models possesses a unique property, ensuring any solution found by the algorithm is also the best possible solution, thereby simplifying the optimisation process and offering analytical tools with broad applicability to VQE methods. This rigorous landscape analysis provides a crucial step towards reliable and efficient excited state calculations on near-term quantum computers.

VQE for Accurate Excited State Calculations

This research explores the application of the Variational Quantum Eigensolver (VQE) algorithm to a central problem in quantum chemistry: calculating the energies of excited molecular states. Accurate determination of these states is crucial for understanding chemical reactions, material properties, and spectroscopic phenomena. The team focuses on overcoming the inherent difficulties in calculating excited states using near-term quantum computers, devices limited by noise and qubit count. The study centers on VQE, a hybrid quantum-classical algorithm that combines the strengths of both computing paradigms.

Quantum computers prepare and measure trial wavefunctions, while classical computers optimize the parameters of these wavefunctions to minimize the energy. This approach aims to leverage the potential of quantum computation while mitigating the limitations of current hardware. The research addresses the challenges specific to excited state calculations, which are considerably more complex than determining the ground state energy. The work provides a comprehensive review of existing VQE methods for excited states, including techniques like state-specific VQE, subspace search VQE, quantum equation of motion VQE, and linear response VQE.

It also analyzes the challenges posed by complex energy landscapes, including multiple local minima, symmetry breaking, and excited state degeneracy. The team proposes new techniques or improvements to existing methods, such as improved optimization algorithms, more flexible trial wavefunctions, and noise mitigation strategies. The research culminates in numerical benchmarks comparing the performance of different methods on various molecular systems, assessing their accuracy and efficiency. The team also discusses future research directions, including developing more robust and scalable algorithms, exploring new quantum hardware platforms, and applying VQE to increasingly complex molecular systems. This work represents a significant contribution to the field, advancing the potential of quantum computation for solving challenging problems in quantum chemistry.

Orthogonal Variational Quantum Eigenstate Calculations

Scientists have developed novel VQE models for calculating excited states on emerging quantum computers. Recognizing the difficulty of accurately determining excited states, the researchers designed three models that enforce orthogonality between low-lying energy states, a critical algorithmic requirement. These models embed orthogonal constraints directly into the cost functions, simplifying the optimization process and improving the reliability of the results. The team rigorously validated these models through detailed analysis of their stationary points and local minimizers, guaranteeing their favorable properties.

This theoretical work provides analytical tools applicable to a broader range of VQE methods, enhancing the understanding of optimization landscapes. The researchers then conducted a comprehensive comparison of the three models, evaluating their resource requirements and classical optimization complexity. This work employs a hybrid quantum-classical approach, leveraging the strengths of both computing paradigms. Quantum devices compute expectation values, while classical computers minimize the cost function, overcoming limitations of current quantum hardware. The team builds upon the full configuration interaction (FCI) framework, a highly accurate method for solving the many-body Schrödinger equation, adapting it for implementation on quantum computers. This innovative combination of theoretical analysis and practical implementation advances the field of quantum chemistry and opens new possibilities for simulating complex molecular systems.

Global Minima Guarantee for Excited State Calculations

Scientists have achieved a breakthrough in developing VQE algorithms for calculating the low-lying energy states of complex systems on emerging quantum computers. This work focuses on overcoming a key challenge in excited state calculations, which are often more difficult than ground state calculations due to their greater distance from a simple mean-field description. The team rigorously analyzed three distinct VQE models designed to enforce orthogonality between these low-lying energy states, avoiding the need for external enforcement methods. The research demonstrates that all three models possess a remarkable property: any local minimum found during the optimization process is also a global minimum.

This characteristic addresses a common difficulty in optimization algorithms, ensuring that the solution converges to the best possible outcome. Through detailed landscape analysis, scientists characterized the stationary points and local minimizers of each model, providing theoretical guarantees for their favorable properties and establishing analytical tools applicable to a broader range of VQE methods. Specifically, the team proved that local minimizers of one model take the form of a matrix constructed from eigenvectors of the target Hamiltonian, multiplied by an arbitrary unitary matrix. This means the algorithm naturally converges to solutions with mutually orthogonal eigenvectors, eliminating the need for explicit orthogonalization steps during computation. These findings pave the way for more efficient and robust VQE algorithms, potentially accelerating progress in fields like materials science and quantum chemistry.

Guaranteed Global Minima For Excited State VQE

This study provides a comprehensive analysis of three variational quantum eigensolver (VQE) models—qOMM, qTPM, and qL1M—developed to compute excited quantum states, a task that remains particularly challenging for near-term quantum computers. To address the fundamental requirement of maintaining orthogonality between excited states, the researchers incorporate orthogonality constraints directly into the objective functions of the models, eliminating the need for external enforcement techniques. A central result of the work is the proof that, for all three models, any local minimum encountered during optimization is also a global minimum, greatly simplifying the search for correct solutions.

Beyond these theoretical guarantees, the authors introduce analytical tools that extend to a broader class of VQE-based excited-state methods, offering deeper insight into the optimization landscapes of different approaches. The three models differ in how orthogonality is handled: qOMM relies on implicit orthogonalization, while qTPM and qL1M enforce orthogonality through explicit regularization terms. As a result, qOMM and qL1M naturally yield orthogonal solutions at local minima, whereas qTPM requires an additional post-processing step to fully enforce orthogonality.

Selecting between these models involves a trade-off between quantum and classical resources. qOMM demands more quantum measurements, increasing quantum cost, while qTPM and qL1M reduce measurement overhead at the expense of more complex classical optimization. Although qOMM has already been developed into a hybrid quantum–classical algorithm with encouraging numerical performance, this work primarily lays the theoretical foundation for qTPM and qL1M, with extensive numerical benchmarking left for future studies. The authors suggest that designing optimization strategies tailored to the structure of each model could further enhance their effectiveness on noisy intermediate-scale quantum (NISQ) devices.