  A new approach to the classical limit of Grover's algorithm is discussed by
assuming a very rapid dephasing of a system between consecutive Grover's
unitary operations, which drives pure quantum states to decohered mixed states.
One can identify a specific element among $N$ unsorted elements by a
probability of the order of unity after $k\sim N$ steps of classical
amplification, which is realized by a combination of Grover's unitary operation
and rapid dephasing, in contrast to $k\sim \pi \sqrt{N}/4$ steps in quantum
mechanical amplification. The initial two-state system with enormously
unbalanced existence probabilities, which is realized by a chosen specific
state and a superposition of all the rest of states among $N$ unsorted states,
is crucial in the present analysis of classical amplification. This analysis
illustrates Grover's algorithm in extremely noisy circumstances. A similar
increase from $k\sim \sqrt{N}$ to $k\sim N$ steps due to the loss of quantum
coherence takes place in the {\em analog} model of Farhi and Gutmann where the
entanglement does not play an obvious role. This supports a view that
entanglement is crucial in quantum computation to describe quantum states by a
set of qubits, but the actual speedup of the quantum computation is based on
quantum coherence.
