The general computational process

A suitably programmed quantum computer should act on a number x to produce another number f(x) for some specified function f. Appropriately interpreted, with an accuracy that increases with increasing k, we can treat such numbers as non-negative integers less than 2k. Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits.

If we specify the numbers x as n-bit integers and the numbers f(x) as m-bit integers, then we shall need at least n + m Qbits: a set of n-Qbits, called the input register, to represent x, and another set of m-Qbits, called the output register, to represent f(x). Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f(x) assigns the same value to different values of x, as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f(x). Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall find that this dual-register architecture can also be usefully exploited by a quantum computer in some strikingly nonclassical ways.