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Let the spectral decomposition of the Hermitian matrix beThe HHL algorithm proceeds as follows.

Efficiently prepare the normalized amplitude encoding $|b⟩$.
Apply quantum phase estimation to the simulated evolution $e^{iAt}$, producing an approximation of each eigenvalue: Add one ancilla and perform an eigenvalue-controlled rotationwhere $0<C\le {\min }_j|{\lambda }_j|$.
Uncompute the eigenvalue register. Conditional on measuring the ancilla as $1$, the system register isThe postselection probability can be increased with amplitude amplification.

The ingredients used here are efficient sparse Hamiltonian simulation of $e^{iAt}$ and quantum phase estimation, which converts an eigenphase of that evolution into a binary approximation of ${\lambda }_j$. For sparsity $s$, condition number $\kappa$, and error $ϵ$, the cost is polynomial in $s$, $\kappa$, $1/ϵ$, and $\log N$ under the stated access assumptions.

Finally estimate $⟨x|M|x⟩$ by repeated measurement of an efficient observable decomposition of $M$, or by a Hadamard test when $M$ is unitary. A general efficiently block-encoded Hermitian $M$ can similarly be measured through its block encoding. Repetition and a concentration inequality give additive sampling error $O(1/\sqrt{R})$ after $R$ independent preparations.

After the first Hadamard gate and the controlled unitary gate, the state isThe final Hadamard gate givesHence the Hadamard test returns one with probabilityIf $U|\psi ⟩=|\psi ⟩$, then $\theta =0$ and the outcome is zero with certainty. If $U|\psi ⟩=-|\psi ⟩$, then $\theta =\pi$ modulo $2\pi$ and the outcome is one with certainty. The single measurement therefore distinguishes the two promised cases exactly.

Both $Z\otimes Z$ and $X\otimes I$ are unitary and Hermitian, so their expectations are real. Run the real-part Hadamard test twice, once with controlled $Z\otimes Z$ and once with controlled $X\otimes I$. If the respective ancilla outcome-one probabilities are $p_{ZZ}$ and $p_{XI}$, thenThe identity in part (b) implements each controlled tensor product by gates on its two target wires with the common control. By linearity,