Quantum Mechanics helps in searching for a needle in a haystack

1. Introduction In 1994 Shor discovered a quantum mechanical algorithm for factorization that was exponentially faster than any known classical algorithm [1]. This paper presents a quantum mechanical algorithm for search that is only polynomially faster than any classical algorithm; however, it does not depend for its impact on the unproven difficulty of the factorization problem. The search problem is this: there is an unsorted database containing N items out of which just one item satisfies a given condition-that one item has to be retrieved. Once an item is examined, it is possible to tell whether or not it satisfies the condition in one step. However, there does not exist any sorting on the database that would aid its selection. The most efficient classical algorithm for this is to examine the items in the database one by one. If an item satisfies the required condition stop; if it does not, keep track of this item so that it is not examined again. It is easily seen that this algorithm will need to examine an average of items before finding the desired item. It is possible for quantum mechanical systems to make interaction-free measurements by using the dual-ity properties of photons [2]. In these the presence (or absence) of an object can be deduced by allowing a small probability of a photon interacting with the object. Therefore, most probably the photon will not interact, however, just allowing a small probability of interaction is enough to make the measurement. Thus in the search problem also, it might be possible to find the object without examining all of the objects, but just by allowing a certain probability of examining the desired object. Indeed, this paper shows that by using the same amount of hardware as in the classical case, but by having the input and output in superpositions of states, we can find an object in quantum mechanical steps instead of classical steps. Each quantum mechanical step consists of an elementary unitary operation (discussed in the following paragraph). 1.1 Quantum Mechanical Algorithms In a quantum computer, the logic circuitry and time steps are essentially classical, only the memory bits that hold the variables are in quantum superpositions (see [1] & [3] for a more detailed introduction to quantum computers). Quantum mechanical operations that can be carried out in a controlled way are unitary operations that act on a small number of bits in …