We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schr\"odinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schr\"odingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schr\"odingerized equation which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost $\log\log {1/\varepsilon}$ where $\varepsilon$ is the desired precision. This optimizes the complexity of the Schr\"odingerization based quantum algorithms for any non-unitary dynamical system introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603].

翻译：我们针对不适定偏微分方程问题提出了一种简单且稳定的计算方法。该方法基于[S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603]中提出的薛定谔化技术，该技术将所有线性偏微分方程映射至高一个维度的薛定谔型方程，以实现这些偏微分方程的量子模拟。尽管原始问题是不适定的，但薛定谔化后的方程是哈密顿系统且具有时间可逆性，允许在时间上向前和向后进行稳定计算。原始变量可以通过扩展维度中适当选取区域的数据进行恢复。我们将以反向热方程和具有虚波速的线性对流方程为例进行说明。对这些算法进行了误差分析并进行了数值验证。这些方法既适用于经典计算机也适用于量子计算机，我们还为这些方法设计了量子算法。此外，我们为薛定谔化方程引入了光滑初始化方法，若采用谱方法，将使得扩展空间中的近似达到本质上的谱精度。因此，对于基于量子比特的量子算法，无论是适定问题还是不适定问题，由于额外维度所需增加的量子比特数几乎仅为$\log\log {1/\varepsilon}$，其中$\varepsilon$为目标精度。这优化了[S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603]中提出的针对任何非幺正动力系统的基于薛定谔化的量子算法的复杂度。