Schrödingerization based quantum algorithms for the fractional Poisson equation

TL;DR

The paper tackles solving the high-dimensional fractional Poisson equation $(-\Delta)^s u=f$ by first localizing the problem through the Caffarelli-Silvestre extension to a $(d{+}1)$-D weighted elliptic PDE, then solving the resulting linear system with a Schrödingerization-based quantum algorithm that recasts the FEM discretization as a time-evolution system. The main contribution is a detailed complexity analysis under block-encoding input models, showing a quantum cost of roughly $\tilde{O}(d\,3^{\frac{3d}{2}}\,h^{-2.5})$ versus a classical $\tilde{O}(d^{1/2}\,3^{\frac{3d}{2}}\,h^{-d-2})$, implying exponential speedup in the inverse mesh size for large dimension $d$. The work also covers truncation of the extended domain, graded anisotropic FEM discretization, and the trace projection required to extract the physical solution, along with numerical demonstrations in 1D and 2D that validate the method and its numerical stability. Overall, the combination of CS extension with Schrödingerization provides a pathway for efficient quantum simulation of nonlocal PDEs in high dimensions, with practical validation and clear guidelines for block-encoding and ODE-to-Schrödinger mappings.

Abstract

We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the $d$-dimensional fractional equation is reformulated as a local partial differential equation in $d+1$ dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schrödingerization approach from \cite{JLY22SchrShort, JLY22SchrLong, analogPDE}. The Schrödingerization technique transforms general linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage -- with respect to the inverse of the mesh size in high dimensions -- compared to its classical counterpart. Specifically, while the classical method requires $\widetilde{\mathcal{O}}(d^{1/2} 3^{3d/2} h^{-d-2})$ operations, the quantum counterpart requires $\widetilde{\mathcal{O}}(d 3^{3d/2} h^{-2.5})$ queries to the block-encoding input models, with the quantum complexity being independent of the dimension $d$ in terms of the inverse mesh size $h^{-1}$. Numerical experiments are conducted to verify the validity of our formulation.

Figures (8)

• Figure 1: Block-encoding of $B^{(1)} = A^{(1)} \otimes S^{(2)} \otimes \cdots \otimes S^{(d+1)}$
• Figure 2: Numerical solutions in 1D with uniform partition ($s=0.2$)
• Figure 3: Numerical solutions in 1D with graded partition ($s=0.8$)
• Figure 4: Numerical solutions of $u$ based on the Caffarelli-Silvestre extension ($s=0.2$)
• Figure 5: Solution profile along the extended direction at fixed $(x_1,x_2) = (0,0)$

Theorems & Definitions (14)

• Definition 2.1
• Lemma 2.1
• Theorem 2.1
• proof
• Lemma 3.1: The Caffarelli-Silvestre extension, Caffarelli2007FractionalNochetto2015Fractional
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.1
• proof