Boundary-Aware QFT Block-Encoding of Fractional Laplacians

• URL: http://arxiv.org/abs/2605.16749v1
• Date: Sat, 16 May 2026 02:06:47 GMT
• Title: Boundary-Aware QFT Block-Encoding of Fractional Laplacians
• Abstract summary: We study the block-encoding of the semi-discrete fractional Laplacian on bounded domains with open, zero-extension boundary conditions.<n>A finite QFT register diagonalizes circulant matrices rather than Toeplitz truncations.<n>The resulting compressed block satisfies (P_Nto Mdaggerwidetilde A(M)_,hP_Nto M = A(N)_,h+E(M)), where (E(M)) is
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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