Today I read a paper titled “Lower Bounds for Quantum Search and Derandomization”
The abstract is:
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes.
In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant.
If we want error <1/2^N then we need T=Omega(N) queries.
We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine.
A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k.
Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.