Geometry-only complexity for oblique eigenprojection

Determine whether an eigenprojection for a non-normal matrix can be implemented at a cost depending only on the geometry of the target spectral region, rather than on the spectral gap between the input spectrum and the region boundary, as occurs for Hermitian operators.

Background

For Hermitian operators, quantum singular value transformation can implement spectral window functions with a cost determined by the function's transition width and independent of the particular input matrix. The paper contrasts this with its oblique eigenprojection algorithm for non-normal matrices, whose resolvent-based analysis requires the spectrum to be separated from the boundary of the target region by a gap.

The unresolved issue is whether this dependence on an input-specific spectral gap is intrinsic or whether a quantum algorithm could instead achieve complexity governed only by the geometry of the target region. Such a result would extend the input-independent complexity behavior known in the Hermitian setting to non-normal oblique eigenprojections.