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Quantum algorithms for the fractional Poisson equation via rational approximation

Yin Yang, Yue Yu, Long Zhang, Ming Zhou

Abstract

This paper presents a quantum algorithm for solving the fractional Poisson equation \((-Δ)^s u = f\) with \(s \in (0,1)\) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size \(h^{-1}\) that is independent of the spatial dimension \(d\), in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.

Quantum algorithms for the fractional Poisson equation via rational approximation

Abstract

This paper presents a quantum algorithm for solving the fractional Poisson equation \((-Δ)^s u = f\) with \(s \in (0,1)\) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size that is independent of the spatial dimension , in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.

Paper Structure

This paper contains 12 sections, 9 theorems, 107 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rational approximation of fractional Poisson equation
  3. Review of the rational approximation
  4. Solution given by rational approximation
  5. Quantum algorithm for the fractional Poisson equation
  6. Block-encoding of the coefficient matrix
  7. Recover the solution
  8. Time complexity analysis
  9. Explicit quantum circuit via Schrödingerization
  10. The Schrödingerization approach
  11. Quantum circuit for the Hamiltonian evolution
  12. Conclusions

Key Result

Lemma 2.1

For any $f \in \mathscr{L}_1(\widetilde{D})$, there exists an algorithm that outputs $b_{\ell} \ge 0, c_{\ell} > 0$ for ${\ell} = 1, \cdots, N_r$ and $c_{\infty} \ge 0$, such that where $K$ is an constant with $K > 1$, independent of $f$ and $N_r$, $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Quantum circuit for the Schrödingerization method, where $\bm{\psi} = \sum_{k=0}^{N_p-1} \psi(p_k)| k \rangle$.
  • Figure 2: Quantum circuit for $V_1(\Delta t)$. The first gate $W^{-2^{n_p-1}}$ corresponds to $(\mathrm{e}^{\mathrm{i} r H_1})^{-2^{n_p-1}}$, and the subsequent controlled-$W^{2^j}$ gates implement the select oracle $\text{SEL}(\mathrm{e}^{\mathrm{i} r H_1})$.
  • Figure 3: Quantum circuit for $\mathrm{e}^{-\mathrm{i} r \tilde{H}} = | 0 \rangle\langle 0 | \otimes \mathrm{e}^{-\mathrm{i} r H} + | 1 \rangle\langle 1 | \otimes \mathrm{e}^{-\mathrm{i} r}I^{\otimes n_H}$.

Theorems & Definitions (16)

  • Lemma 2.1: rational approximation, E.q (6.1) in Berrut2004BLI
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1: Block-encoding of sparse-access matrices
  • Lemma 3.2
  • Theorem 3.1
  • ...and 6 more