Boundary-Aware QFT Block-Encoding of Fractional Laplacians

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 $A^{(N)}_{\alpha,h}$ obtained from the full-lattice semi-discrete operator with symbol $|\xi|^\alpha$. A finite QFT register, however, diagonalizes circulant matrices rather than Toeplitz truncations. The native QFT circuit therefore implements a periodic surrogate $\widetilde A^{(N)}_{\alpha,h}$, 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 $P_{N\to M}^{\dagger}\widetilde A^{(M)}_{\alpha,h}P_{N\to M} = A^{(N)}_{\alpha,h}+E^{(M)}$, 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.