Quantum spectral filtering ($h(A)$)

Let a Hermitian operator $A$ have dyadic eigenvalues ${\lambda }_j=c_j/2^t$ in $[0,1)$ for known $t$, with controlled access to the unitary operator $U=e^{2\pi iA}$. This interval ensures that different eigenvalues have different eigenphases; exact representability alone would not exclude phase aliasing, as $0$ and $1$ both give phase zero. Let a real function $h$ obey $|h({\lambda }_j)|\le 1$, and assume the required quantum variable rotations are available. Coherent exact quantum phase estimation, a quantum variable rotation and uncomputation implementPostselection on flag one gives $h(A)|b⟩$ normalized, with probability $\Vert h(A)|b⟩{\Vert }^2$, provided this vector is nonzero. Erasing the eigenvalue label by uncomputation is essential to preserve coherence between different eigenvectors. On this dyadic spectrum, $U^{2^t}=I$ gives $U^{-1}=U^{2^t-1}$, so reversing the phase-estimation gates is possible using forward controlled-$U$ calls. The choice $h(\lambda )=\lambda$ multiplies by $A$; a scaled reciprocal gives the different filter used in the HHL algorithm.

If $|b⟩$ lies in the kernel of $A$, then $A|b⟩=0$ has no normalized quantum state. A request to produce that normalized vector with positive probability is consequently undefined. For example, $A=\mathrm{diag}(0,1/2)$ on one qubit has distinct dyadic eigenvalues, but annihilates $|0⟩$. The corrected filtering assertion requires $A|b⟩\ne 0$; it cannot be repaired by assigning a positive success probability to this input.

For $0\le A<I$ and normalized $|b⟩={\sum }_j{\beta }_j|u_j⟩$, filtering by $h(\lambda )=\lambda$ succeeds withThe inequality follows by replacing each nonnegative ${\lambda }_j^2$ by its minimum and using ${\sum }_j|{\beta }_j{|}^2=1$. Equality holds exactly when all occupied eigenvectors belong to the minimum-eigenvalue eigenspace. If ${\lambda }_{\min }=0$, success is positive exactly for inputs outside the kernel of $A$.