Quantum Algorithms Efficiently Compute Molecular Eigenvalues and Spectroscopy Data.

Researchers present a quantum algorithm estimating eigenvalues and singular values of matrices, crucial for modelling molecular systems and spectroscopy. It utilises amplitude amplification and phase estimation to locate eigenvalues via singular value minimisation, offering improved scaling up to (N) compared to classical methods for certain potentials and avoiding numerical instability.

The accurate determination of energy levels within complex quantum systems remains a significant computational challenge, particularly when investigating excited states and dense spectra. Classical algorithms struggle with the exponential increase in computational demand as system size grows, limiting the scope of simulations in fields such as chemistry, materials science, and spectroscopy. Researchers from BEIT sp. z o.o., namely Grzegorz Rajchel-Mieldzioć, Szymon Pliś, and Emil Zak, address this limitation in their work, entitled ‘Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation’. They present a novel quantum algorithm designed to efficiently estimate eigenvalues and singular values of parameterized matrix families, offering a potential advantage over classical methods when solving generalized eigenvalue problems, a common requirement in modelling quantum mechanical systems.

Their approach leverages quantum techniques like amplitude amplification and phase estimation to locate eigenvalues by identifying minima within the singular value spectrum, and proposes a quantum formulation of the pseudospectral collocation method, a numerical technique used to solve differential equations like the Schrödinger equation.

Calculating eigenvalues, a fundamental operation across numerous scientific and engineering disciplines, presents a significant computational challenge for large, complex systems. Classical algorithms typically exhibit a computational cost that scales cubically with system size, denoted as O(N³), rendering the analysis of high-dimensional problems intractable. This limitation impacts fields such as quantum chemistry, materials science, and spectroscopy, where accurate eigenvalue determination is crucial for understanding molecular properties and material behaviour.

A novel quantum algorithm addresses this computational bottleneck by reformulating the eigenvalue problem as a singular value minimisation task. Singular value decomposition (SVD) is a factorisation of a matrix, revealing information about its linear transformations and providing insights into its rank and dimensionality. By focusing on minimising the singular values of a related matrix, the algorithm indirectly determines the eigenvalues of the original system, offering a potentially more efficient computational pathway.

The algorithm leverages core quantum computational techniques to accelerate this process. Amplitude amplification, a quantum algorithm that enhances the probability of measuring a desired outcome, is employed to expedite the search for minimal singular values. Complementing this is quantum phase estimation, a quantum algorithm that estimates the eigenvalues of a unitary operator, providing a precise determination of the energy levels associated with the system. These quantum routines, when combined, offer a speedup over their classical counterparts.

A key innovation lies in the algorithm’s circumvention of the computationally intensive matrix inversion step inherent in traditional pseudospectral collocation methods. Pseudospectral collocation is a numerical technique used to solve differential equations, often requiring the inversion of large matrices, which contributes significantly to the O(N³) scaling. By avoiding this step, the algorithm substantially reduces computational overhead and improves efficiency.

While a precise scaling analysis remains an area of ongoing research, preliminary results suggest a potential improvement over the classical O(N³) scaling. This improvement is particularly relevant for high-dimensional molecular systems characterised by dense spectra and highly excited states, where classical methods struggle due to the exponential growth of computational demands. The algorithm’s versatility extends beyond molecular systems, offering potential applications to a broader range of eigenvalue problem formulations across diverse scientific disciplines.

Current research focuses on mitigating the practical challenges associated with implementing the algorithm on near-term quantum computers. A primary concern is fault tolerance, the ability of a quantum computer to maintain the integrity of quantum information despite environmental noise and imperfections. Reducing the resource requirements, such as the number of qubits and circuit depth, is crucial for making the algorithm feasible on currently available hardware.

Further development explores extending the algorithm’s applicability to a wider range of potential energy surfaces, which define the energy landscape of a molecular system. Hybrid quantum-classical approaches, combining the strengths of both computational paradigms, are also being investigated to optimise performance and reduce computational overhead. Ultimately, the goal is to apply this algorithm to simulate dynamic processes, providing insights into the time evolution of complex systems and unlocking new possibilities for scientific discovery.