[
Verce 1
]:

Unitary operator U with eigenvector psi and eigenvalue e to the i two pi phi, 
Ancillary qubits t in number for precision decree. 
Hadamard on ancilla creates uniform superposition array, 
Controlled powers of U on target apply. 
State evolves to sum over k zero to two to t minus one of ket k tensor U to power k psi, 
Which factors as ket plus psi tensor sum k ket k e to the i two pi phi k. 
Inverse quantum Fourier transform on ancilla extracts phi's binary approximation, 
Measurement yields m over two to t closest to phi's fraction.

[
Chorus
]:

Approximate the phase in controlled cascade, 
Eigenvalues unveiled where classics fade. 
Unitary powers in entangled parade, 
Kitaev's insight: spectral cascade.

[
Verse 2
]:

Phi between zero and one the phase to discern, 
Accuracy epsilon demands t as log one over epsilon term. 
Controlled-U to power two to j on j-th ancilla bit, 
Iterative phase kickback the value commit. 
Like a digital voltmeter sampling wave's crest, 
Binary digits of phi progressively addressed ---
QFT inverse coheres the estimate best, 
Probabilistic success with amplitude test.

[
Chorus
]:

Approximate the phase in controlled cascade, 
Eigenvalues unveiled where classics fade. 
Unitary powers in entangled parade, 
Kitaev's insight: spectral cascade.

[
Bridge
]:

Protocol outline: ancilla in zero initialization, 
Hadamard tensor on ancilla for equal participation. 
Controlled-U to powers on target integration, 
Inverse QFT on ancilla for phase interpolation. 
Core subroutine for Shor's period detection, 
In Hamiltonian simulation's eigen projection. 
Precise quantification of phase accumulation rate, 
No symbolic shroud: explicit binary fraction estimate gate, successive refinement to eigenvalue's accurate state.

[
Breakdown 
]:

Relinquish classical spectral decomposition's burden, 
Harness the coherent overlay's burden. 
No matrix diagonalization's cumbersome burden, 
Controlled evolutions the spectrum unburden.

Eliminate traces of iterative eigen solver strain, 
Reveal the phase continuum's domain.

No extension. No imprecision. Oracle's ordinance ---
Phase disclosure in superposition's ordinance.

[
Verse 3
]:

From nitrogen-vacancy centers to transmon qubit assemblies,
Phase estimation endures in laboratory assemblies.
In quantum metrology's accuracy assemblies,
Gauging frequencies with oracle assemblies.
Integral to operator spectrum interrogation in Hilbert scope,
Eigenvalue resolution at exponential hope.
Theoretical sentinel for quantum superiority's scope,
Reforming dynamical simulation's trope.

[
Chorus
]:

Approximate the phase in controlled cascade,
Eigenvalues unveiled where classics fade.
Unitary powers in entangled parade,
Kitaev's insight: spectral cascade.

[
Outro
]:

Approximate the phase in controlled cascade, 
Eigenvalues unveiled where classics fade. 
Unitary powers in entangled parade, 
Kitaev's insight: spectral cascade.

[
Outro
]:

Estimate --- Coalesce --- Extract --- Elevate --- 
Estimate --- Coalesce --- Extract --- Elevate ---

QUANTUM PHASE ESTIMATION RESOLVE [R1]

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