Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>N</a:mi><a:mi>tot</a:mi></a:msub></a:math> times, the variance of the estimates can scale as <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>O</b:mi><b:mo>(</b:mo><b…