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 (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.

• The paper introduces a quantum algorithm that leverages rational approximation to transform the nonlocal fractional Poisson equation into a series of sparse, local PDEs.
• It uses block-encoding and Schrödingerization techniques to explicitly construct quantum circuits, achieving a dimension-independent quantum query complexity.
• Rigorous complexity bounds and error controls demonstrate scalable quantum speedup over classical methods for high-dimensional and ill-posed problems.

Quantum Algorithms for the Fractional Poisson Equation via Rational Approximation

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Problem Motivation and Background

The paper addresses the quantum solution of the spectral fractional Poisson equation, $(-\Delta)^s u = f$, for $s \in (0,1)$, on bounded domains. Fractional Laplacians, as nonlocal operators, underlie a broad class of PDEs describing anomalous diffusion, memory effects, and long-range correlations in fields ranging from physics to computational finance. The spectral and integral definitions of the fractional Laplacian, while distinct, both introduce dense system matrices upon discretization, undermining the scalability and tractability of classical numerical solvers in high dimensions.

Traditional algorithms—finite elements, finite differences, spectral approaches—succumb to the curse of dimensionality, with computational complexity scaling exponentially in dimension $d$ and inverse discretization size $h^{-1}$. Specialized classical strategies such as the Caffarelli-Silvestre extension and Monte Carlo schemes mitigate but do not eliminate these bottlenecks. As quantum computing matures, there is potential to exploit its exponential-memory compression and Hamiltonian simulation techniques for large-scale scientific computing.

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Rational Approximation of the Inverse Fractional Laplacian

Central to the approach is the rational approximation of $x^{-s}$—corresponding to the inverse of the fractional Laplacian. The method represents $(-\Delta)^{-s}$ through a rational function expansion: $$x^{-s} \approx \sum_{\ell=1}^{N_r} \frac{c_{\ell}}{x + b_{\ell}} + c_{\infty}$$ using barycentric (e.g., AAA) rational approximation algorithms for stability and efficiency. Applying this expansion to the operator yields the solution representation: $$u \approx \left( \sum_{\ell=1}^{N_r} c_{\ell} (-\Delta + b_{\ell} I)^{-1} + c_{\infty} I \right) f$$ This reduces the nonlocal problem to a weighted sum of solutions for shifted elliptic PDEs, each with a sparse system matrix after discretization.

By consolidating all shifted problems into a single, larger block-structured linear system, the approach streamlines quantum implementation and obviates the overhead of a full LCU (linear combination of unitaries) step on independent solves.

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Quantum Linear System Framework

The block-encoding formalism is leveraged for inputting sparse matrices onto a quantum register. The critical coefficient block matrix, encompassing all rational-shifted resolvents, is encoded with complexity scaling only polylogarithmically in the problem size. Preparation of the right-hand side is similarly optimized using oracles for vector coefficients and source terms.

For solution recovery, the work adopts advanced quantum linear system algorithms (QLSA) capable of preparing the normalized quantum state $|u\rangle$ approximating the discrete solution, with query complexity scaling as $\mathcal{O}(\kappa \log(1/\varepsilon))$, where $s \in (0,1)$0 is the condition number of the system matrix and $s \in (0,1)$1 the desired solution precision.

A crucial numerical result established is that, under the finite-difference mesh, the quantum query complexity in the inverse spacing $s \in (0,1)$2 is independent of spatial dimension $s \in (0,1)$3, in stark contrast to the exponential scaling of best-known classical algorithms.

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Explicit Quantum Circuit Construction via Schrödingerization

To implement the required non-unitary linear system solve on a quantum computer, the Schrödingerization technique is employed. The original positive-definite linear system is recast as the steady state of a first-order ODE, which is embedded in a higher-dimensional homogeneous Hamiltonian system: $s \in (0,1)$4 By splitting $s \in (0,1)$5 into Hermitian and anti-Hermitian components and introducing a momentum-like auxiliary variable, the method reduces non-unitary evolution to Hamiltonian simulation. Discretization in the auxiliary direction proceeds via a Fourier spectral method. Block-encodings of the shift operators composing the discrete Laplacian are used for explicit circuit realization, and controlled gates (select oracles) are constructed for selecting among rational approximation shifts. Single-qubit rotations and sparse Hamiltonian simulation subroutines suffice for the assembled circuit, as detailed in the as-provided quantum circuit diagrams.

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Complexity Analysis and Numerical Guarantees

The authors provide rigorous bounds on discretization error and quantum solution approximation error in the discrete $s \in (0,1)$6-norm, demonstrating that both can be controlled to $s \in (0,1)$7 for mesh size $s \in (0,1)$8 and prescribed quantum precision. The total quantum query cost to the Laplacian block-encoding oracle scales as: $s \in (0,1)$9 with $d$0 the number of rational approximation terms and $d$1, $d$2 the rational coefficients and shifts, respectively.

The dominant term—$d$3—is dimension-independent, granting an exponential quantum speedup over classical methods as $d$4 increases. The authors highlight that the method is readily extensible to arbitrary dimensions without algorithmic modification, making it especially valuable for high-dimensional scientific computing.

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Algorithmic and Theoretical Implications

The presented framework provides several critical theoretical and practical advancements:

• Nonlocal-to-Local Transformation: The rational approximation approach effectively translates the nonlocal fractional Laplacian problem into a tractable system of sparse local PDEs, facilitating efficient quantum encoding.
• Scalable Quantum Solution: By consolidating all rational approximation terms into a single scalable linear system with a specially-modified right-hand side, the need for postprocessing linear combination of independent solves is removed, simplifying quantum resource requirements.
• Practical Circuit Realization: The Schrödingerization procedure produces explicit quantum circuits, resolving prior challenges in simulating non-unitary dynamical systems with quantum resources. All essential operations—block encoding, select oracles, and Hamiltonian simulation—are compatible with current quantum algorithmic primitives.

A potential limitation is that while the rational approximation can be made arbitrarily accurate, the overhead in the number of shifts $d$5 and the condition number $d$6 may increase with $d$7 and the required mesh resolution.

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Future Directions

Several avenues for further development are evident:

• Extension to Non-spectral Fractional Operators: Exploration of the rational approximation framework for alternative (e.g., integral) definitions of the fractional Laplacian.
• Generalization to Other Fractional PDEs: Application to broader classes of time-fractional and space-fractional evolution equations, possibly requiring higher-order rational or pole-residue expansions.
• Quantum Preconditioning: Investigation into preconditioning strategies to further suppress the condition number for especially ill-posed systems.
• Alternative Discretizations: Integration with finite element or spectral element discretizations, and compatibility with nonuniform or adaptive meshes.

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Conclusion

The paper establishes a dimension-independent quantum algorithm for the fractional Poisson equation, overcoming the curse of dimensionality that limits classical approaches. It combines rational approximation, block-encoding, and the Schrödingerization technique to construct scalable quantum circuits for practical simulation of high-dimensional nonlocal PDEs. The presented methodology paves the way for efficient quantum solvers of fractional and other nonlocal operators in scientific computing and warrants further exploration into generalized nonlocal quantum PDE solvers.

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For further details and proofs, see "Quantum algorithms for the fractional Poisson equation via rational approximation" (2604.00603).