Quantum phase-estimation spectral sampler

Inputs. An instance of Eigenvalue-estimation access and aliasing convention, a phase-register size $m\ge 1$ with $Q=2^m$, and an input system register already prepared in the normalized state $\psi$ declared by the access instance. The caller invokes and charges $A_{\psi }$ for that preparation; this procedure does not prepare the system again. The access instance fixes the phase encoding $\theta (\lambda )$ and decoder.

Instructions. Initialize the phase register to $\mathrm{\mid }{0}^m⟩$. Apply the coherent QPE gate list through the inverse QFT from Quantum phase estimation to this prepared system, using the supplied controlled powers for $j=0,\dots ,m-1$, or synthesize each required power from the declared controlled-$U$ interface. Perform its specified computational-basis Lüders measurement once on the phase register and record $y\in \{0,\dots ,Q-1\}$. Return $y\mathrm{/}Q$ as a circular phase sample and apply the specified decoder to return an eigenvalue estimate when the injective encoding promise holds. If $U=e^{iH}$ and $\sigma (H)\subseteq [-1,1]$, use the principal representative in $(-1\mathrm{/}2,1\mathrm{/}2]$ and output $2\pi \mathrm{unwrap}(y\mathrm{/}Q)$.

The phase register, controlled powers, gates, decoder arithmetic, measurement, and output costs follow the access record. In particular, with only a controlled-$U$ oracle, implementing the binary powers uses ${\sum }_{j=0}^{m-1}{2}^j=Q-1$ controlled-$U$ calls.

This definition asserts no error probability. The exact mixture law, accuracy, and resource guarantees are established separately by the spectral-law, additive spectral-estimation, and matrix-function error-cost theorem carriers.