arXiv subjects

Kazuaki Z. Takahashi

Publications and source records attributed to Kazuaki Z. Takahashi.

3 recordsLinked to original sources

arXiv2026

Quantum-Accelerated Solution of Nonlinear Equations from Variational Principles

Nonlinear equilibrium problems derived from variational principles arise throughout physics and engineering, including structural mechanics, fluid dynamics, and electromagnetism. While fault-tolerant quantum algorithms have shown promising advantages for linear systems and linear dynamical simulations, extending quantum acceleration to nonlinear equilibrium problems remains a major challenge. Here we introduce a quantum algorithmic framework for nonlinear equilibrium analysis based on gradient-flow linearization. The key idea is to reformulate equilibrium conditions as nonlinear gradient-flow dynamics and transform the resulting evolution into a linear dynamical system using exact linearization techniques such as Carleman and Pivot Switching Carleman (PSC) linearization. This construction enables nonlinear equilibrium and energy-minimization problems to be addressed using quantum algorithms for linear dynamical simulation. We demonstrate the framework for nonlinear elasticity problems ranging from single nonlinear springs and chain-spring systems to two-dimensional truss structures. The resulting truncated linearized dynamics accurately reproduce nonlinear equilibrium states, while PSC linearization substantially improves stability in regimes where conventional Carleman linearization becomes unreliable. More broadly, our work establishes a connection between variational principles, nonlinear energy minimization, exact linearization, and quantum dynamical simulation. This perspective opens a route toward quantum algorithms for nonlinear physical systems beyond the scope of existing linear-system-based approaches.

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arXiv2024

Divergence-free algorithms for solving nonlinear differential equations on quantum computers

From weather to neural networks, modeling is not only useful for understanding various phenomena, but also has a wide range of potential applications. Although nonlinear differential equations are extremely useful tools in modeling, their solutions are difficult to obtain. Based on the expectation of quantum transcendence, quantum algorithms for efficiently solving nonlinear differential equations continue to be developed. However, even the latest promising algorithms have been pointed out to have an evolution time limit. This limit is the theoretically predestined divergence of solutions. We propose algorithms of divergence-free simulation for nonlinear differential equations in quantum computers. For Hamiltonian simulations, a pivot state $\bf{s}$ in the neighborhood of state $\bf{x}$ is introduced. Divergence of the solutions is prevented by moving $\bf{s}$ to a neighborhood of $\bf{x}$ whenever $\bf{x}$ leaves the neighborhood of $\bf{s}$. Since updating $\bf{s}$ is directly related to computational cost, to minimize the number of updates, the nonlinear differential equations are approximated by nonlinear polynomials around $\bf{s}$, which are then Carleman linearized. Hamiltonian simulations of nonlinear differential equations based on several representative models are performed to show that the proposed method breaks through the theoretical evolution time limit. The solution of nonlinear differential equations free from evolution time constraints opens the door to practical applications of quantum computers.

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arXiv2024

Function Smoothing Regularization for Precision Factorization Machine Annealing in Continuous Variable Optimization Problems

Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by factorization machine (FM) have been overlooked. This study shows that in the widely common case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable due to the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.

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