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Suppose an oracle is known to hold one of a given set of D two-valued functions. To successfully identify which function the oracle holds with k classical queries, it must be the case that D is at most 2 k . In this paper we derive a bound for how many functions can be distinguished with k quantum queries.

Abstract.A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.

Consider a function f which is defined on the integers from 1 to N and takes the values 21 and 11. The parity of f is the product over all x from 1 to N of f͑x͒. 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 N͞2 applications of the unitary operator which evaluates f. Thus, for this problem, quantum computers cannot outperform classical computers. [S0031-9007(98)07850-8] PACS numbers: 03.67.Lx, 02.70. -c, 07.05.TpIf a quantum computer is ever built, it could be used to solve certain problems in less time than a classical computer. Simon found a problem that can be solved exponentially faster by a quantum computer than by the provably best classical algorithm [1]. The Shor algorithm for factoring on a quantum computer gives an exponential speedup over the best known classical algorithm [2]. The Grover algorithm gives a speedup for the following problem [3]. Suppose you are given a function f͑x͒ with x an integer and 1 # x # N. Furthermore you know that f is either identically equal to 1 or it is 1 for N 2 1 of the x's and equal to 21 at one unknown value of x. The task is to determine which type of f you have. Without any additional information about f, classically this takes of order N calls of f, whereas the quantum algorithm runs in time of order p N. In fact, this p N speedup can be shown to be optimal [4].It is of great interest to understand the circumstances under which quantum speedup is possible. Recently Ozhigov has shown that there is a situation where a quantum computer cannot outperform a classical computer [5]. Consider a function g͑t͒, defined on the integers from 1 to L, which takes integer values from 1 to L. We wish to find the Mth iterate of some input, say, 1, that is, g ͓M͔ ͑1͒.[Here g ͓n͔ ͑t͒ g͑ ͑ ͑g ͓n21͔ ͑t͒͒ ͒ ͒ and g ͓0͔ ͑t͒ t.] Ozhigov's result is that if L grows at least as fast as M 7 then any quantum algorithm for evaluating the Mth iterate takes of order M calls of the unitary operator which evaluates g; of course, the classical algorithm requires M calls. Later we will show that our result, in fact, implies a stronger version of Ozhigov's with L 2M.In this paper we show that a quantum computer cannot outperform a classical computer in determining the parity of a function; similar and additional results are obtained in [6] and [7]. Let f͑x͒ 61 for x 1, . . . , N .(1)Define the parity of f by par͑ f͒so that the parity of f can be either 11 or 21. The parity of f always depends on the value of f at every point in its domain so classically it requires N function calls to determine the parity. The Grover problem, as described above, is a special case of the parity problem where additional restrictions have been placed on the function. Although the Grover problem can be solved in time of order p N on a quantum computer, the parity problem has no comparable quantum speedup.Preliminaries.-We imagine that the function f whose parity we...

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