Deutsch–Jozsa algorithm

The Deutsch–Jozsa algorithm is a deterministic quantum algorithm proposed by David Deutsch and Richard Jozsa in 1992 with improvements by Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca in 1998. Although of little practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. The Deutsch–Jozsa problem is specifically designed to be easy for a quantum algorithm and hard for any deterministic classical algorithm. It is a black box problem that can be solved efficiently by a quantum computer with no error, whereas a deterministic classical computer would need a exponential number of queries to the black box to solve the problem. More formally, it yields an oracle relative to which EQP, the class of problems that can be solved exactly in polynomial time on a quantum computer, and P are different. Since the problem is easy to solve on a probabilistic classical computer, it does not yield an oracle separation with BPP, the class of problems that can be solved with bounded error in polynomial time on a probabilistic classical computer. Simon's problem is an example of a problem that yields an oracle separation between BQP and BPP. In the Deutsch–Jozsa problem, we are given a black box quantum computer known as an oracle that implements some function: The function takes n-bit binary values as input and produces either a 0 or a 1 as output for each such value. We are promised that the function is either constant (0 on all inputs or 1 on all inputs) or balanced (1 for exactly half of the input domain and 0 for the other half). The task then is to determine if is constant or balanced by using the oracle. For a conventional deterministic algorithm where is the number of bits, evaluations of will be required in the worst case. To prove that is constant, just over half the set of inputs must be evaluated and their outputs found to be identical (because the function is guaranteed to be either balanced or constant, not somewhere in between).