Fe b 19 98 A 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. ∗This work was supported in part by The Department of Energy under cooperative agreement DE-FC02-94ER40818 and by the National Science Foundation under grant NSF 95–03322 CCR. †[email protected] [email protected] ‡[email protected] §[email protected]