Optimal one-shot quantum algorithm for EQUALITY and AND

Abstract: We study the computation complexity of Boolean functions in the quantum black box model. In this model our task is to compute a function $f:{0,1}\to{0,1}$ on an input $x\in{0,1}^n$ that can be accessed by querying the black box. Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries. We show that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.