New Algorithm Solves Non-Hermitian Eigenvalue Problems With Quantum Devices

The efficient calculation of eigenvalues and eigenvectors constitutes a fundamental problem across numerous scientific and engineering disciplines, from quantum mechanics to structural analysis. However, most established computational methods assume that the systems under investigation adhere to the constraints of Hermitian symmetry, which limits their applicability to a broader range of physical scenarios. Jiaxin Li, Zhaobing Fan, and colleagues, from Harbin Engineering University and Capital Normal University, alongside researchers from Origin Quantum Computing Technology Co., Ltd., now present a novel variational quantum algorithm designed to address generalized eigenvalue problems specifically within non-Hermitian systems. Their work, detailed in the article “Variational quantum algorithm for generalized eigenvalue problems of non-Hermitian systems”, leverages the generalized Schur decomposition to transform the problem into a search for appropriate unitary transformation matrices, and importantly, outlines a practical method for implementation on currently available quantum hardware. Numerical simulations, including applications to ocean acoustics and robustness testing via noise simulations, validate the algorithm’s performance.

Generalized eigenvalue problems are fundamental to diverse scientific and engineering disciplines, including structural mechanics and quantum mechanics. Classical computational methods struggle with these problems as system complexity increases, experiencing exponential growth in both memory requirements and computational time. Quantum computing offers a potential solution, leveraging quantum mechanical principles to tackle computations intractable for classical computers. Algorithms such as the quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) show promise in solving eigenvalue problems, particularly within quantum chemistry, by efficiently manipulating exponentially large Hilbert spaces.

However, existing quantum algorithms largely focus on Hermitian systems, where matrices possess specific symmetry properties. This limits their applicability to the broader class of non-Hermitian systems frequently encountered in real-world scenarios, including open quantum systems and problems in ocean acoustics and mechanical engineering. Consequently, developing quantum algorithms capable of addressing generalized eigenvalue problems in these non-Hermitian systems presents a significant challenge.

This research addresses this gap by presenting a variational quantum generalized eigensolver (VQGE) specifically designed for non-Hermitian systems. Building upon variational quantum algorithms, which employ parameterized quantum circuits to minimize or maximize the expectation value of observables, the VQGE transforms the generalized eigenvalue problem into a search for unitary transformation matrices. This novel approach enables solutions on both near-term and fault-tolerant quantum devices. Numerical simulations validate the algorithm’s efficacy on generalized eigenvalue problems relevant to ocean acoustics, and simulations incorporating noise assess its robustness, a crucial consideration for practical implementation on current quantum hardware. This work contributes to the growing field of quantum algorithms for linear algebra and extends the capabilities of quantum computation beyond traditional Hermitian systems.

Researchers have developed a novel variational quantum algorithm to solve generalized eigenvalue problems, particularly those arising in non-Hermitian systems. Current methodologies largely concentrate on Hermitian systems, limiting their applicability to a broader range of physical and engineering challenges. The authors address this gap by formulating a solution based on the generalized Schur decomposition, effectively transforming the problem into a search for appropriate unitary transformation matrices. This circumvents limitations inherent in traditional methods when dealing with non-Hermitian matrices, which frequently appear in areas like ocean acoustics and open quantum systems. The algorithm leverages variational quantum circuits, employing a hybrid quantum-classical approach, and efficiently evaluates both the loss function, quantifying solution accuracy, and its gradients, crucial for optimising the quantum circuit parameters.

This optimisation process utilises classical optimisers, iteratively refining the quantum circuit to converge on the desired eigenvalues and eigenvectors. A key innovation lies in the algorithm’s design for implementation on near-term quantum devices, acknowledging the limitations of current hardware in terms of qubit count and coherence. Researchers validate the algorithm’s performance through numerical simulations, demonstrating its ability to accurately solve generalized eigenvalue problems, and specifically apply it to a problem in ocean acoustics, showcasing its practical relevance. Furthermore, the study rigorously assesses the algorithm’s robustness by subjecting it to noise simulations, mimicking imperfections present in real quantum hardware. These simulations confirm the algorithm’s resilience and potential for reliable operation even in noisy environments. The work establishes a viable pathway for utilising near-term quantum computers to tackle generalized eigenvalue problems, extending beyond the scope of traditional Hermitian systems.

The algorithm employs a specific type of quantum circuit known as an ansatz, a trial solution iteratively refined during the optimisation process. This ansatz is constructed using a linear combination of unitary operations, allowing for a flexible and expressive representation of the solution space. The choice of ansatz is critical for the algorithm’s performance, as it determines the range of solutions that can be effectively explored. Researchers meticulously designed this ansatz to balance expressiveness with computational cost, ensuring accurate representation while remaining feasible for implementation on near-term devices. Extensive numerical simulations tested the algorithm on a variety of generalized eigenvalue problems, demonstrating its accuracy even in the presence of noise. Furthermore, its practical utility was showcased by applying it to a specific problem in ocean acoustics, a field where generalized eigenvalue problems frequently arise in the analysis of underwater sound propagation.

Future work will focus on scaling the algorithm to handle larger matrices, a significant hurdle for quantum computations, and exploring alternative ansatz designs and optimisation techniques represents a key avenue for improving scalability and performance. A direct comparison with state-of-the-art classical algorithms, benchmarked on identical test cases, will provide a clearer understanding of the quantum algorithm’s advantages and limitations, and further investigation into error mitigation strategies is also warranted. A detailed error analysis, examining the algorithm’s sensitivity to different noise types, will be crucial for enhancing its reliability and accuracy, and expanding on the specific matrices arising in applications like seismology and ocean acoustics, and detailing the benefits over existing classical methods, will strengthen the practical relevance of this research. The availability of the code and data used in the research, openly shared via platforms like Figshare, promotes reproducibility and encourages further investigation by the wider scientific community.