Boundary-Aware QFT Block-Encoding of Fractional Laplacians

TL;DR

This paper develops a boundary-aware quantum Fourier transform block-encoding method for fractional Laplacians, addressing boundary condition mismatches via zero-padding and operator compression.

Contribution

It introduces a novel zero-padding technique to accurately encode open-boundary fractional Laplacians in quantum algorithms, improving boundary condition handling.

Findings

Identifies the mismatch between circulant and Toeplitz matrices in QFT encodings.

Proposes a zero-padding and compression method to recover open-boundary operators.

Provides bounds on the error introduced by the approximation.

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 \(A^{(N)}_{\alpha,h}\) 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.