In this work, we solve differential equations using quantum Chebyshev feature maps. We propose a tensor product over a summation of Pauli-Z operators as a change in the measurement observables resulting in improved accuracy and reduced computation time for initial value problems processed by floating boundary handling. This idea has been tested on solving the complex dynamics of a Riccati equation as well as on a system of differential equations. Furthermore, a second-order differential equation is investigated in which we propose adding entangling layers to improve accuracy without increasing the variational parameters. Additionally, a modified self-adaptivity approach of physics-informed neural networks is incorporated to balance the multi-objective loss function. Finally, a new quantum circuit structure is proposed to approximate multivariable functions, tested on solving a 2D Poisson's equation.

翻译：本文利用量子切比雪夫特征映射求解微分方程。我们提出将泡利-Z算符求和上的张量积作为测量可观测量的变化，从而在浮动边界处理下提高初始值问题的精度并减少计算时间。该方法已通过求解Riccati方程的复杂动力学及微分方程组得到验证。此外，针对二阶微分方程，我们提出通过添加纠缠层来提升精度，同时不增加变分参数。进一步，结合物理启发的神经网络改进的自适应方法，以平衡多目标损失函数。最后，提出一种新的量子电路结构用于逼近多变量函数，并通过求解二维泊松方程进行了测试。