Median-amplified spectral estimate

Inputs. An eigenvalue-estimation access instance, positive rational binary inputs $ϵ>0$ and $0<\delta <1$, and certified rational upper enclosures ${\pi }_{+}>\pi$ and $L_{+}\ge \ln (1\mathrm{/}\delta )$ as produced by the finite interval procedures in OP0 of research/quantum-algorithms-planning/operational-contracts.md.

Instructions. Choose the least dyadic $Q=2^m$, $m\ge 1$, satisfying $Qϵ\ge 6{\pi }_{+}$. Choose the least positive odd integer $r$ satisfying $r\ge 8L_{+}$. On each of $r$ fresh trials, invoke the access instance's declared state-preparation circuit on fresh standard-input registers to prepare its normalized state $\psi$, then run Quantum phase-estimation spectral sampler with phase-register size $m$. Decode every result using the same branch convention, then return the median of the $r$ real eigenvalue estimates. Do not reuse the postmeasurement system from one trial as the input to another; each trial begins with a fresh state preparation. A fixed-eigenvalue accuracy guarantee requires the eigenstate promise stated by the corresponding theorem.

The certified enclosures, doubling comparisons, state preparation, each QPE circuit, each controlled-power or controlled-$U$ call, projective readout, decoding, and median computation are charged. If only controlled-$U$ access is supplied, the ideal controlled-call count is $r(Q-1)$; exact supplied powers use their separately declared per-trial costs. No inverse call is used by this procedure. Its accuracy theorem is separate.