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Schr\"odingerization based quantum algorithms for the fractional Poisson equation

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Pith's one-line read The paper claims a quantum algorithm for the FEM-discretized fractional Poisson equation with query cost Õ(d 3^{3d/2} h^{-2.5}), independent of dimension in the mesh size, an exponential advantage over classical conjugate gradients in…

desk verdict A legitimate new combination with a real gap: the advertised h^{-2.5} complexity rests on an unconstructed VTAA step that the authors themselves flag as open; the paper's own un-amplified bound still delivers the qualitative exponential advantage. read the letter →

arxiv 2505.01602 v1 pith:BC5I5ZJL submitted 2025-05-02 math.NA cs.NA

Shi Jin , Nana Liu , Yue Yu This is my paper · ORCID
classification math.NAcs.NA MSC 65N3065N2235R1181P68
keywords fractionalLaplacianPoissonequationCaffarelli-SilvestreextensionSchrödingerizationquantumalgorithmfiniteelementmethodblock-encodingcurseofdimensionality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that the high-dimensional fractional Poisson equation, whose nonlocal fractional Laplacian classically brings the curse of dimensionality, can be solved on a quantum computer with a cost that depends on the mesh size $h$ in a dimension-independent way. The route is to apply the Caffarelli–Silvestre extension, converting the nonlocal equation into a local weighted elliptic problem in one extra dimension, then to solve the finite-element linear system by interpreting the solution as the steady state of $du/dt=-Au+b$ and simulating that ODE with the Schrödingerization (warped-phase) method. The quantitative claim is $\widetilde{\mathcal{O}}(d 3^{3d/2} h^{-2.5})$ queries to block-encoding input models, against $\widetilde{\mathcal{O}}(d^{1/2}3^{3d/2} h^{-d-2})$ operations for classical conjugate gradients, an exponential quantum advantage in the dimension $d$ at fixed $h$. A sympathetic reader would care because such a result would make a class of nonlocal high-dimensional PDEs numerically accessible for the first time.

What carries the argument

The carrying mechanism is the pairing of the Caffarelli–Silvestre extension with the Schrödingerization transformation. The extension is the identity $d_s(-\Delta)^s u = -\lim_{y\to 0^+} y^{\alpha}u_y$ for $\alpha=1-2s$, realized as the Dirichlet-to-Neumann map of the weighted elliptic equation $\mathrm{div}(y^{\alpha}\nabla u)=0$ on $\Omega\times(0,\infty)$; it replaces the nonlocal operator by a local problem in $d+1$ dimensions. Schrödingerization is the warped-phase change of variables $v(t,p)=e^{-p}u_f(t)$ that sends the dissipative ODE system $du_f/dt = A_f u_f$ into the Schrödinger-type system $\partial_t v = -H_1\partial_p v + iH_2 v$, whose Fourier discretization in the auxiliary variable $p$ is a Hamiltonian system suitable for quantum simulation. The final ingredient is the block-encoding of $A$ as a sum of Kronecker products $A^{(1)}\otimes S^{(2)}\otimes\cdots\otimes S^{(d+1)}$ plus cyclic permutations, which keeps the input-model cost quadratic in $d$ rather than exponential.

What would settle it

Implement the proposed pipeline for a fixed dimension, say $d=2$ or $d=3$, on the FEM-discretized fractional Poisson equation over a sequence of mesh sizes $h$, and count block-encoding queries needed to prepare an $\epsilon$-accurate solution with success probability $\Omega(1)$. If the empirical exponent of $h^{-1}$ exceeds $2.5$, or if the success probability decays polynomially in $h$, the claimed dimension-independent advantage fails; a direct check of Lemma 2.1 against a stiff positive-definite matrix would settle the same question.

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Extended reading notes

Core claim

The paper's central claim is that a nonlocal fractional problem can be solved through the local extension it inherits. After truncating the Caffarelli–Silvestre cylinder to $C_Y$ and using tensor-product piecewise-linear finite elements on graded meshes, the stiffness matrix $A$ has sparsity $3^{d+1}$ and condition number $\kappa(A)=O((d+1)3^{d+1}h^{-2})$, and it can be block-encoded as a sum of Kronecker products of one-dimensional stiffness and mass matrices, so the exponentially large sparse matrix is never formed explicitly. Treating $Ax=b$ as the steady state of $du/dt=-Au+b$ and applying Schrödingerization produces a Hamiltonian system whose simulation gives an $\epsilon$-approximation of $|x\rangle$; combining the query bound with the trace projection onto the $y=0$ boundary and amplitude amplification yields the headline $\widetilde{\mathcal{O}}(d 3^{3d/2} h^{-2.5})$ query complexity. Without the variable-time amplitude amplification described in Remark 2.2, the paper's own estimate is $\widetilde{\mathcal{O}}(d^2 3^{5d/2} h^{-4.5})$, still independent of $d$ in the $h^{-1}$ factor.

Load-bearing premise

The load-bearing premise is Lemma 2.1, the query-complexity estimate for the Schrödingerization-based linear-system solver, which the paper states as following 'with some modifications' from another work without proof, together with the variable-time amplitude amplification of Remark 2.2 that is invoked to reach the improved exponent but is explicitly not constructed; if either is unsound, the claimed $\widetilde{\mathcal{O}}(h^{-2.5})$ scaling gives way to the paper's own weaker $\widetilde{\mathcal{O}}(d^2 3^{5d/2}h^{-4.5})$ bound.

Editorial extensions

If this is right

  • If the central claim is correct, the fractional Poisson equation loses the classical curse of dimensionality: for fixed mesh size the quantum cost is independent of $d$ in the $h^{-1}$ factor, while classical conjugate gradients scale as $h^{-d-2}$.
  • The Kronecker-product block-encoding construction means the $3^{d+1}$-sparse FEM matrix never needs to be assembled or queried entrywise; block-encodings of one-dimensional stiffness and mass matrices suffice.
  • The same extension-plus-Schrödingerization pipeline is applied to a finite-difference discretization and to the nonlocal variational formulation, with Remark 5.1 observing that block-encoding of the dense nonlocal matrix remains possible on regular domains.
  • Even in the un-amplified version, the cost $\widetilde{\mathcal{O}}(d^2 3^{5d/2}h^{-4.5})$ is still dimension-independent in $h^{-1}$, so the quantum approach would avoid the exponential-in-$d$ mesh dependence even without variable-time amplitude amplification.
  • The explicit ingredients $\kappa(A)=O((d+1)3^{d+1}h^{-2})$ and $g=O(3^{d/2}h^{-1/2})$ for the trace projection are what combine into the reported $3^{3d/2}h^{-2.5}$ exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same localization-plus-Schrödingerization template could be applied to other nonlocal operators admitting a local extension, such as fractional Laplacians with Neumann or Robin boundary conditions or regional fractional operators, but the paper does not treat those cases.
  • Editorial inference: the trace projection is a natural bottleneck, since the cost carries a $3^{d/2}$ factor from recovering the $d$-dimensional solution from the $d+1$-dimensional one; a block-encoding of the trace operator that avoids measurement-and-amplify could improve the dimension dependence.
  • Editorial inference: because the headline bound rests on an unproved lemma and an unconstructed subroutine, a direct proof or a numerical query-count test of Lemma 2.1 on stiff FEM matrices is the minimal next step that would make the exponential-advantage claim conclusive.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 7 minor

Summary. The paper proposes a quantum algorithm for the fractional Poisson equation on bounded domains. It first applies the Caffarelli–Silvestre extension to rewrite the nonlocal fractional Laplacian as a local degenerate/singular elliptic problem in one higher dimension, discretizes the resulting problem with a tensor-product finite element method, and then solves the resulting linear system Au=b by interpreting the solution as the steady state of the ODE du/dt=-Au+b and applying the Schrödingerization technique. The main claimed result is a query complexity of Õ(d3^{3d/2}h^{-2.5}) against block-encoding input models, to be compared with a classical conjugate-gradient cost of Õ(d^{1/2}3^{3d/2}h^{-d-2}), giving an exponential quantum advantage in the spatial dimension d for fixed mesh size h. Numerical experiments in one and two dimensions for both the extension formulation and a nonlocal variational formulation are reported.

Significance. If fully established, an exponential quantum advantage in d for a fractional PDE would be a significant result. The combination of the Caffarelli–Silvestre extension with Schrödingerization is natural and potentially impactful, and the manuscript contains genuinely useful components: a transparent block-encoding construction for the FEM stiffness matrix, an explicit trace-extraction step, and numerical validation of the formulation. I also credit the authors for explicitly stating the un-amplified bound Õ(d^2 3^{5d/2}h^{-4.5}) at the end of Section 4.3; even at that bound, the h-dependence remains dimension-independent, so the qualitative exponential advantage may well survive. However, the headline h^{-2.5} bound depends on an essentially unproved lemma and on a VTAA procedure that the authors themselves describe as an open question, and several supporting estimates are incompletely justified. These gaps are load-bearing for the main complexity claim and must be closed before the result can be accepted as stated.

major comments (5)
  1. [Theorem 4.1 and abstract] The claimed query complexity Õ(d3^{3d/2}h^{-2.5}) in Theorem 4.1 and the abstract is not established. The proof consists of the sentence 'combining Theorem 2.1, the VTAA procedure (see Remark 2.2), Theorem 3.1 and Eq. (4.5)', but Remark 2.2 states that a simple approach to reducing the κ-dependence from quadratic to near-linear remains an open question and that VTAA requires substantial modifications to LCU or QSP algorithms. No VTAA schedule is constructed and no VTAA query count is provided. The paper's own un-amplified bound is Õ(d^2 3^{5d/2}h^{-4.5}) (end of Section 4.3). Please either construct and analyze the VTAA step explicitly, or remove h^{-2.5} from the abstract and Theorem 4.1 and present the un-amplified bound as the main result.
  2. [Lemma 2.1 and Theorem 2.1] Lemma 2.1 is the foundation of Theorem 2.1 but is not proved in this manuscript; it is introduced with 'Following the similar implementation in [36] with some modifications'. Since [36] is the authors' own concurrent work and the parameter dependence in Lemma 2.1, especially the claim η_max = O(log(1/ε)), is crucial for the final complexity, a self-contained proof or a precise statement of the relevant theorem in [36] is needed. Without this, the query-complexity chain lacks a verified base.
  3. [Theorem 3.1] The proof of the condition-number estimate treats only the case 0<s<1/2 (0<α<1) and asserts that s>1/2 'can be deduced in a similar manner'. This is not immediate because for α=1-2s<0 the weight y^α is singular at y=0, and the use of the second integral mean-value theorem to extract Y^α is not justified for a singular weight. In addition, the mesh in the extended direction is graded as in (3.9), not quasi-uniform; the quoted bound λ_min(G_ξ) ≳ (h/3)^{d+1} from [25] may not hold for the actual graded mesh. Since κ(A) enters the quantum complexity bound, a complete proof of Theorem 3.1 for all s∈(0,1) and for the graded mesh used by the algorithm is required.
  4. [Lemma 4.1] The proof of Lemma 4.1 is incomplete. For the 'second inequality' involving S(d+1), the text says only that one can apply the second integral mean-value theorem 'as done in Theorem 3.1', but S(d+1) is a weighted mass matrix and for α<0 the weight is singular, so the asserted bound h/3 ≲ λ(S(d+1)) ≲ h needs a derivation. Moreover, the conclusion g = O(3^{d/2}h^{-1/2}) requires a bound on ∥(uY)_h∥_{L2}/∥u_h∥_{L2}; this trace ratio is not obviously O(1) in the graded-mesh setting and is not discussed. An extra factor here would change the trace-extraction overhead in the final complexity.
  5. [Section 4.1] The block-encoding construction for A is described at the level of combining univariate block-encodings, but the query cost of constructing the block-encoding oracle for A is not quantified. The sentence 'the complexity of block-encoding A ... has only a quadratic dependence on d' is not accompanied by a count. Since Theorem 4.1 counts queries to the block-encoding oracle of A, the cost of implementing that oracle via the proposed LCU of tensor products must either be included in the total cost or stated as a separate input-model assumption. Without this, the claimed d-dependence of the quantum algorithm is not fully established.
minor comments (7)
  1. [Section 3.3] In the paragraph following (3.6), 'the the Dirichlet boundary' should read 'the Dirichlet boundary'.
  2. [Throughout] The spelling of Schrödingerisation/Schrödingerization is inconsistent; please choose one convention and use it consistently.
  3. [References] References [2] and [3] appear to be the same paper with different years; please merge or correct the entries.
  4. [Equation (4.4)] The notation 'ui1,···,id,id+1=0' is ambiguous; it should read 'ui1,...,id,0' or clearly set i_{d+1}=0.
  5. [Lemma 2.1] The condition that the coefficient matrix is 'negative semi-definite over the interval [0,T]' is unclear as stated; if the authors mean that -A in (2.1) is negative definite for positive definite A, the wording should be made precise.
  6. [Figure 6] The caption of Figure 6 refers to a fixed point (x1,x2)=(0,0) but the example is one-dimensional; please adjust the caption to match the plotted quantity.
  7. [Abstract] The phrase 'which can show up to exponential advantage' is awkward; consider rewording to 'which yields an exponential advantage'.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity: the advertised h^{-2.5} query count is not forced by construction, but the proof depends on same-author lemmas and an unconstructed variable-time amplitude amplification step, so the self-citation burden is non-negligible.

full rationale

The derivation is not circular in the strict sense. The Caffarelli-Silvestre extension and its FEM error estimates are taken from external theory (Nochetto et al. [47]), and the new block-encoding construction for the tensor-product stiffness matrix and the trace-projection factor g = O(3^{d/2} h^{-1/2}) are derived in Sections 4.1-4.2 rather than fitted to the desired answer. No parameter is tuned on the fractional Poisson solution to make the h-exponent match a preconceived value. The Schrödingerization framework is imported from the authors' own prior work ([29,36,39,40]), but it is applied as a subroutine to the linear system (3.10); applying a previously developed solver is not a definitional circularity. The main caveat is not circularity: Lemma 2.1 is stated without proof ('Following the similar implementation in [36] with some modifications'), and Remark 2.2 explicitly says the VTAA needed to turn the quadratic kappa-dependence into near-linear 'requires substantial modifications' and that a simple approach 'remains an open question'. Theorem 4.1 nevertheless states the h^{-2.5} bound by combining Theorem 2.1 with that unconstructed VTAA. That is a missing-support/completeness gap, not a fitted input or self-definitional reduction. The un-amplified bound eO(d^2 3^{5d/2} h^{-4.5}) stated in Section 4.3 is the only complexity actually derived from the displayed equations. Because the only circularity-adjacent feature is reuse of same-author lemmas without independent proof, the score is kept at the low end.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Caffarelli-Silvestre localization theory, FEM error estimates from Nochetto et al., and the Schrödingerization framework developed by the authors themselves. No new physical entities or fitted constants are introduced; the free parameters are numerical convergence choices. The most fragile dependencies are the unproved Lemma 2.1 and the assumed VTAA procedure.

free parameters (3)
  • Evolution time T = 15 (numerical experiments)
    Chosen large enough for the steady state; complexity theory prescribes T = Θ(κ/||A|| log(·)). Not fitted to data.
  • Truncated cylinder height Y = 10 (numerical experiments)
    Chosen to make the exponential decay in y small; theoretical threshold Y0 from Lemma 3.3.
  • Number of Fourier modes Np = 2^11 (numerical experiments)
    Grid resolution in the auxiliary p-direction, chosen for accuracy.
assumptions (6)
  • domain assumption Caffarelli-Silvestre extension: the fractional Laplacian is the Dirichlet-to-Neumann map of a degenerate elliptic problem in one higher dimension.
    Invoked in Section 3.1 and is the foundation of the localization step; taken as theorem from [6,47].
  • domain assumption Exponential decay of the extension solution in the extended variable y (Lemma 3.3).
    Used to justify truncating the semi-infinite cylinder to finite height Y; quoted from Nochetto et al. [47].
  • domain assumption FEM error estimates for graded meshes (Lemma 3.4).
    Used to set the target accuracy δ and to justify the graded mesh condition γ>3/(1-α); from [47].
  • standard math Eigenvalue bounds for standard FEM matrices (λmax(G)≲(d+1)h^{d-1}, λmin(G_ξ)≳(h/3)^{d+1}).
    Used in Theorem 3.1 to bound κ(A); partly referenced to [25, Theorem 3.5] and partly asserted.
  • domain assumption Schrödingerization convergence and complexity theorems from the authors' prior work (Lemma 2.1, Theorem 2.1).
    The linear-system solver's complexity is imported from [36] without proof in this paper.
  • standard math Block-encoding and LCU input models are valid and constructible for the 1D FEM matrices.
    Standard in quantum algorithms; the construction is sketched in Section 4.1.

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Cite this review

Pith. "Pith review of Schr\"odingerization based quantum algorithms for the fractional Poisson equation." pith.science (2026). https://pith.science/paper/BC5I5ZJL

@misc{pith2026250501602,
  author       = {Pith},
  title        = {Pith review of: Schr\"odingerization based quantum algorithms for the fractional Poisson equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BC5I5ZJL}},
  note         = {Machine review of arXiv:2505.01602}
}
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 Schr\"odingerization approach from \cite{JLY22SchrShort, JLY22SchrLong, analogPDE}. The Schr\"odingerization technique transforms general linear partial and ordinary differential equations into Schr\"odinger-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

Figures reproduced from arXiv: 2505.01602 by the authors.

Figure 1
Figure 1. Block-encoding of B(1) = A(1) ⊗ S (2) ⊗ · · · ⊗ S (d+1) Although the sparsity s(A) = 3d+1, the complexity of block-encoding A from the sparse query models for A(k) and S (k) has only a quadratic dependence on d. 4.2 Trace projection Let 0 = y0 < y1 < · · · < yN−1 = Y be the grid points in the extended dimension. The corresponding basis functions are denoted by ψ0(y), · · · , ψN−1(y), respectively. Noting that ψ0(0) … view at source ↗
Figure 2
Figure 2. Numerical solutions in 1D with uniform partition ( [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Numerical solutions in 1D with graded partition ( [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Numerical solutions of u based on the Caffarelli-Silvestre extension (s = 0.2) 5.1.2 The finite difference discretization The Schr¨odingerization approach is also applicable to finite difference discretizations for the extension problem. For this test, we use central d…
Figure 5
Figure 5. Figure 5: Solution profile along the extended direction at fixed ( [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Solution profile along the extended direction at fixed ( [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Sparsity pattern for linear systems Let Ω = (−1, 1)2 . The ball B used in the numerical implementation is centered at (0, 0) with a radius of 1.5. The results for s = 0.8 are presented in [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Numerical solutions of u based on nonlocal variational formulation (s = 0.8) Remark 5.1. Ref. [51] proposes a fast Q1 finite element method based on a weighted trapezoidal rule for the nonlocal variational formulation. The resulting matrix is a symmetric block Toeplitz…

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Reviewed August 16, 2026 · model on record in the stance chip above.