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)_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→ M}^daggerwidetilde A^(M)_α,hP_{N→ 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.