Numerical Analysis on a Quantum Computer

Stefan Heinrich

We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.

Rüdiger Schack

Colin P. Williams

Jincheng Zhang

Dynamic Matrix Geometry represents a novel approach to analyzing the geometric properties of dynamic matrices, utilizing self-correlation within the matrix and quantum computing principles. This paper proposes a method based on dynamically-transformed matrices and quantum computation to investigate the inherent geometric characteristics of dynamic matrices and subsequently apply this framework to numerical analysis and quantum computing. The core mechanism centers on the identification of geometric properties through dynamic matrix transformations, enabling the derivation of their behavior and facilitating accurate numerical simulations and quantum computations. This work represents a significant advancement in the field, offering a new perspective on the geometric analysis of dynamic systems.

Katsuhiro Endo, Kazuaki Z. Takahashi

Nonlinear structural analysis, which can account for geometrical or material nonlinearities beyond the scope of linear methods, typically incurs high computational costs. Recent advances have shown that quantum computers can accelerate various numerical simulations. Although quantum acceleration of linear structural analysis via the Quantum Linear System Algorithm (QLSA) has been studied, its application to nonlinear structural analysis remains unexplored. Here, we introduce our proposed quantum algorithm for accelerating nonlinear structural analysis, summarizing its key features and performance results from our recent study.

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