Limit on the Speed of Quantum Computation in Determining Parity

Consider a function f which is defined on the integers from 1 to N and takes the values -1 and +1 . The parity of f is the product over all x from 1 to N of f(x) . With no further information about f , to classically determine the parity of f requires N calls of the function f . We show that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f . Thus, for this problem, quantum computers cannot outperform classical computers. copyright 1998 The American Physical Society