Schrödingerisation based computationally stable algorithms for ill-posed problems in partial differential equations

TL;DR

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, optimizes the complexity of the Schr\"odingerization based quantum algorithms for any non-unitary dynamical system.

Abstract

We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, 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 Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized 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 Schrödingerized 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 Schrödingerization 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].

Figures (6)

• Figure 1: Left: $\|w_h^de^{p}-u\|_{L^2(\Omega_x)}$ with respect to $p$, with $w_h^d$ computed by \ref{['eq:tilde w']}-\ref{['eq:whd(t,x,p)']}. Right: the recovery from Schrödingerisation by choosing $p>p^{\Diamond}= \pi^2/4$.
• Figure 2: Left: $\|w_h^de^{p}-u\|_{L^2(\Omega_x)}$ with respect to $p$, with $w_h^d$ computed by \ref{['eq:tilde w']}-\ref{['eq:whd(t,x,p)']}. Right: the recovery from Schrödingerisation by choosing $p>p^{\Diamond} = 9\pi^2$.
• Figure 3: The exact solution in the legend is the numerical solution at time $t=0.5$, $t=0.25$, and $t=0$, respectively, obtained by solving the forward heat conduction equation using the finite difference method. The results of Schrödingerisation are shown with $\eta_{\max} = (\frac{1}{\triangle x})^{1/2} = \sqrt{3.2},(\frac{1}{\triangle x})^{1/2} = \sqrt{3.2}, \frac{1}{\triangle x} =3.2$, respectively.
• Figure 4: The first row: input data with different noise levels from left to right. The noise levels are $6\times 10^{-2}$, $6\times 10^{-3}$ and $6\times 10^{-4}$. The second row: results of Schrödingerisation computed by \ref{['eq:tilde w']}-\ref{['eq:uhd']} using the input data above with $\eta_{\max} = 1.5,2,2.5$,respectively.
• Figure 5: Left: $\|w_h^de^{p}-v\|_{L^2(\Omega_x)}$ with respect to $p$, where $w_h^d$ is from \ref{['eq:tilde w']}. Right: the recovery from Schrödingerisation by choosing $p>p^{\Diamond}= 10$, where the numerical solution is obtained by $v_d^*=\frac{\int_{10}^{15}w_h^d\;\mathrm{d} p}{e^{-10}-e^{-15}}$, $p\in [-20,20]$, $\triangle p = \frac{10}{2^8}$ and $\triangle x = \frac{1}{2^6}$.

Theorems & Definitions (17)

• Theorem 2.1
• Definition 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Lemma 4.1
• proof
• Lemma 4.2