Boundary-Aware QFT Block-Encoding of Fractional Laplacians

Abstract: We study the quantum Fourier transform (QFT) block-encoding of the semi-discrete fractional Laplacian on bounded domains with open, zero-extension boundary conditions. In the notation of the main construction, the target operator is the finite Toeplitz truncation (A^{(N)}_{α,h}) obtained from the full-lattice semi-discrete operator with symbol (|ξ|^α). A finite QFT register, however, diagonalizes circulant matrices rather than Toeplitz truncations. The native QFT circuit therefore implements a periodic surrogate (\widetilde A^{(N)}_{α,h}), not the open-boundary operator. We identify this mismatch through an exact Toeplitz-to-circulant aliasing identity. To recover the open-boundary action, we zero-pad the state into a larger (M)-point QFT register, apply the same Fourier-symbol block-encoding, and compress back to the physical subspace. The resulting compressed block satisfies (P_{N\to M}^{\dagger}\widetilde A^{(M)}{α,h}P{N\to M} = A^{(N)}_{α,h}+E^{(M)}), where (E^{(M)}) is controlled by the tail of the semi-discrete convolution kernel. Thus, the QFT layer implements the fractional symbol, while zero-padding supplies the open-boundary geometry. The construction is an operator-compilation primitive for boundary-aware quantum simulation rather than a complete PDE solver.

• The paper introduces a zero-padding method to transform QFT's circulant encoding into a boundary-aware Toeplitz block-encoding for open-boundary fractional Laplacians.
• It provides an error analysis showing that the residual decays polynomially with register size, ensuring controlled accuracy in quantum simulations.
• The approach generalizes to multidimensional grids, enabling boundary-sensitive quantum simulation of nonlocal PDEs and anomalous diffusion models.

Boundary-Aware QFT Block-Encoding of Fractional Laplacians

Overview

The paper "Boundary-Aware QFT Block-Encoding of Fractional Laplacians" (2605.16749) systematically addresses the algebraic mismatch between quantum Fourier transform (QFT)-diagonalizable operators and the open-boundary fractional Laplacian, elucidating the construction required for boundary-sensitive block-encodings on quantum computers. The main contribution is the introduction of a zero-padding approach that enables QFT circuits to encode Toeplitz truncations, rather than native circulant operators, thus facilitating quantum simulation of nonlocal bounded-domain fractional models with controlled boundary error.

Operator Distinction and Boundary Obstruction

Quantum algorithms natively exploit the diagonal structure of periodic operators via the QFT, primarily resulting in circulant matrices. However, bounded-domain fractional Laplacians, relevant for anomalous diffusion and bounded fractional PDEs, manifest as Toeplitz truncations stemming from the semi-discrete convolution kernel with symbol $|\xi|^\alpha$ [DuoWykZhang2018, ZhouZhang2023]. The paper formalizes the distinction:

• QFT-native Operator: Circulant matrix $\widetilde{A}_\alpha^{(N)}$ implementing periodic wrap-around couplings.
• Open Boundary Target: Toeplitz matrix $A_\alpha^{(N)}$ reflecting zero-extension boundary conditions.

An exact Toeplitz-to-circulant aliasing identity is stated:

$$(\widetilde{A}_\alpha^{(N)})_{ij} = \sum_{\ell \in \mathbb{Z}} c_{i-j+\ell N}$$

This identity quantifies the artificial wrap-around created by periodic geometry, which becomes significant near boundaries and is suppressed by the kernel tail for bulk-localized states.

Figure 1: Illustrative decomposition of a four-qubit QFT; this spectral layer imposes periodic geometry intrinsically.

Figure 2: Native QFT block-encoding realizes a circulant operator, not the open-boundary Toeplitz target.

Compressed Block-Encoding via Zero-Padding

To resolve the boundary mismatch, the authors apply zero-padding to embed the state into a larger QFT register ($M \geq 2N$) and post-compress back to the physical $N$-site domain. The result is a block-encoding whose ancilla block implements

$$P^{\dagger}_{N \to M} \widetilde{A}_\alpha^{(M)} P_{N \to M} = A_\alpha^{(N)} + E^{(M)}$$

where $E^{(M)}$ encodes the kernel-tail residual explicitly. The residual, constructed from periodic images outside the physical block, decays polynomially with $M-N$ according to the kernel's analytic properties.

Figure 3: Compression mechanism: periodic QFT circuit on padded register, compressed back to yield Toeplitz plus tail.

Error Analysis and Resource Scaling

The norm of the residual $E^{(M)}$ is bounded as

$\widetilde{A}_\alpha^{(N)}$0

with $\widetilde{A}_\alpha^{(N)}$1 for $\widetilde{A}_\alpha^{(N)}$2. This yields sufficient scaling

$\widetilde{A}_\alpha^{(N)}$3

for target accuracy $\widetilde{A}_\alpha^{(N)}$4. The block-encoding gate complexity per call is $\widetilde{A}_\alpha^{(N)}$5 for $\widetilde{A}_\alpha^{(N)}$6 and arithmetic precision $\widetilde{A}_\alpha^{(N)}$7.

Multidimensional Extension

The mechanism generalizes to tensor-product grids, where zero-padding and compression are applied separately in each spatial direction. The multidimensional kernel tail governs the precision, and the block-encoding cost scales as $\widetilde{A}_\alpha^{(N)}$8 for three-dimensional grids.

Numerical Diagnostics

The paper offers matrix-level and functional benchmarks to substantiate the boundary correction. For $\widetilde{A}_\alpha^{(N)}$9, $A_\alpha^{(N)}$0, $A_\alpha^{(N)}$1, diagnostic plots illustrate:

• The aliasing structure; discrepancies near the matrix corners from wrap-around artifacts.
• Functional comparisons: the padded construction mitigates boundary error evident in QFT-circulant action.
• Spectral norm decay of the zero-padding residual, confirming theoretical expectations.

  Figure 4: Matrix visualization for $A_\alpha^{(N)}$2, $A_\alpha^{(N)}$3, $A_\alpha^{(N)}$4; logarithmic aliasing error highlights corners.

  

  Figure 5: Functional benchmark—comparison of operator actions on test functions; boundary artifacts are eliminated by zero-padding.

  

  Figure 6: Zero-padding residual decay normalized by $A_\alpha^{(N)}$5; theoretical tail-bound slope realized.

Further State Diagnostics

Gaussian-state inputs localized near boundaries demonstrate heightened sensitivity to boundary condition; the zero-padded construction suppresses the artificial periodic coupling, as shown in the error sweep across domain positions.

Figure 7: Gaussian-state diagnostics comparing open-boundary, QFT-circulant, and zero-padded compressed operators.

Figure 8: Relative error as the Gaussian center is displaced from the boundary; boundary artifacts diminish away from edges.

Implications and Outlook

This operator-level primitive advances boundary-aware quantum simulation of fractional Laplacians, with explicit error control via kernel tail decay. The QFT-based construction complements higher-level routines—quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum linear-system solvers—by providing a modular, boundary-adapted block-encoding.

Practically, this architecture facilitates fractional PDE and anomalous diffusion modeling on quantum platforms, provided boundary artifacts are appropriately suppressed. Further theoretical development may target coefficient-decoupled and variable-coefficient generalizations [javanmard2026coefficient], integration with tensor-network protocols, and optimization of block-encoding arithmetic. Potential expansion to other boundary settings (e.g., Dirichlet, mixed, or nonlinear extensions) requires tailored embedding and compression strategies.

Conclusion

The paper rigorously solves the native QFT block-encoding boundary mismatch for fractional Laplacians by zero-padding and compression, transforming circulant QFT representations into boundary-aware Toeplitz forms with analytically controlled residuals. The construction is applicable across quantum simulation and operator compilation workflows, forming a foundation for boundary-sensitive quantum algorithms in nonlocal PDE settings.