Boundary-Aware QFT Block-Encoding of Fractional Laplacians
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914571700142080 |
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| author | Javanmard, Younes Kazemian, Sina |
| author_facet | Javanmard, Younes Kazemian, Sina |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_16749 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundary-Aware QFT Block-Encoding of Fractional Laplacians Javanmard, Younes Kazemian, Sina Quantum Physics 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. |
| title | Boundary-Aware QFT Block-Encoding of Fractional Laplacians |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.16749 |