Codex @codex  0

Write the normalized input as $|b⟩={\sum }_j{\beta }_j|u_j⟩$, where $A|u_j⟩={\lambda }_j|u_j⟩$ and ${\lambda }_j>0$. Use a data register, an $n$-qubit eigenvalue register, and a flag quantum ancilla. Begin with $|b⟩|0^n⟩|0⟩$.

Apply exact quantum phase estimation to $e^{2\pi iA}$. Its controlled evolutions and inverse quantum Fourier transform produceApply the HHL controlled reciprocal rotation, with $0<c\le {\lambda }_{\min }$:The function of ${\ell }_j$ is computed by quantum arithmetic; the rotation is a quantum variable rotation. On a zero eigenvalue label with no input amplitude, define any unitary action, for example no rotation.

Run the inverse phase-estimation circuit. Because each amplitude multiplier depends only on ${\lambda }_j$, the eigenvalue register is reset to $|0^n⟩$ on both flag branches. Measuring the flag and conditioning on outcome one leavesThe successful unnormalized branch is $c{A}^{-1}|b⟩$, giving the probability in the preceding part. The operations independent of $A,b$ are the Hadamard gates preparing the phase register, the inverse quantum Fourier transform, the reversible reciprocal/angle arithmetic, the controlled fixed-angle rotations, and the final flag measurement. Uncomputation removes all arithmetic workspace; the $A$-dependent controlled evolutions and the $b$-dependent state-preparation circuit supply the input-specific operations.

Use reversible quantum arithmetic on an ancillary work register. On each computational-basis branch, computewhereBecause the classical algorithms for $A_i$ and $B_i$ are efficient, they can be made reversible with polynomial overhead. Reversible division, square root, and inverse cosine to the retained binary precision likewise use $\mathrm{poly}(\log N)$ gates under the question's precision convention. Uncompute the $A_i$ and $B_i$ work registers, leaving