Measurement

A measurement in the computational basis returns one basis state, with probability given by the squared amplitude, and leaves the qubits in that state. Nielsen and Chuang is the textbook reference for the rule. Algorithms on this site use it in two ways, and they should not be blurred.

Grover's algorithm measures once at the end, after the amplitudes have been steered toward the marked item, and hopes to see that item. VQE measures many times. Each shot estimates one term of a Hamiltonian, and the energy is a weighted average over shots. QAOA measures bit strings and keeps the better ones according to the cost function; the approximation ratio is an expectation over those outcomes. "Reading the answer" is a single shot only when the algorithm has already concentrated the probability. Otherwise the shot count is part of the cost. Quantum phase estimation measures an ancilla register and reads the phase from that outcome. The probability of the closest approximation is part of the theorem. HHL postselects an ancilla and can then estimate an observable on the output state. The samples required for that estimate are part of the cost, as Theorem 6 of the 2009 paper records.