Classical sublinear sampling of Fourier coefficients

Construct, in general, a classical data structure capable of sublinear -norm sampling of the Fourier coefficients of a generic smooth signal, in order to determine whether the post-selected HHL framework can be fully dequantised under Fourier-basis sparsity assumptions.

Background

The paper examines whether its post-selected version of the Harrow--Hassidim--Lloyd algorithm can be classically dequantised. Standard dequantisation methods generally assume efficient classical -norm sampling access in the computational basis, whereas the proposed quantum framework exploits concentration of the input state in a Fourier-type eigenbasis. Such spectral sparsity does not generally imply computational-basis sparsity, because Fourier-basis-sparse states can be highly delocalised in the computational basis.

Resolving this problem would require a classical data structure that provides efficient sampling access directly to Fourier coefficients of a generic smooth signal, or would require additional restrictions on the operator and input. The paper explicitly states that the existence of such a general-purpose sublinear sampler is unknown, leaving full dequantisation unresolved.