  Fixed-point quantum search algorithms succeed at finding one of $M$ target
items among $N$ total items even when the run time of the algorithm is longer
than necessary. While the famous Grover's algorithm can search quadratically
faster than a classical computer, it lacks the fixed-point property --- the
fraction of target items must be known precisely to know when to terminate the
algorithm. Recently, Yoder, Low, and Chuang gave an optimal gate-model search
algorithm with the fixed-point property. Meanwhile, it is known that an
adiabatic quantum algorithm, operating by continuously varying a Hamiltonian,
can reproduce the quadratic speedup of gate-model Grover search. We ask, can an
adiabatic algorithm also reproduce the fixed-point property? We show that the
answer depends on what interpolation schedule is used, so as in the gate model,
there are both fixed-point and non-fixed-point versions of adiabatic search,
only some of which attain the quadratic quantum speedup. Guided by geometric
intuition on the Bloch sphere, we rigorously justify our claims with an
explicit upper bound on the error in the adiabatic approximation. We also show
that the fixed-point adiabatic search algorithm can be simulated in the gate
model with neither loss of the quadratic Grover speedup nor of the fixed-point
property. Finally, we discuss natural uses of fixed-point algorithms such as
preparation of a relatively prime state and oblivious amplitude amplification.
