It was one of first examples of a quantum algorithm, which is a class of algorithms designed for execution on Quantum computer s and have the potential to be more efficient than conventional, classical, algorithms by taking advantage of the quantum superposition and entanglement principles. In Deutsch-Jozsa problem, we are given a black box computing a valued function f x1, x2, The black box takes n bits x1, x2, We know that the function in the black box is either constant 0 on all inputs or 1 on all inputs or balanced returns 1 for half the domain and 0 for the other half.
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We are promised that the function is either constant 0 on all inputs or 1 on all inputs or balanced returns 1 for half of the input domain and 0 for the other half ; the task then is to determine if f is constant or balanced by using the oracle. Motivation The Deutsch—Jozsa problem is specifically designed to be easy for a quantum algorithm and hard for any deterministic classical algorithm. The motivation is to show a black box problem that can be solved efficiently by a quantum computer with no error, whereas a deterministic classical computer would need exponentially many 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. To prove that f is constant, just over half the set of inputs must be evaluated and their outputs found to be identical remembering that the function is guaranteed to be either balanced or constant, not somewhere in between.
The best case occurs where the function is balanced and the first two output values that happen to be selected are different. The Deutsch-Jozsa quantum algorithm produces an answer that is always correct with a single evaluation of f. History The Deutsch—Jozsa Algorithm generalizes earlier work by David Deutsch, which provided a solution for the simple case. The algorithm was successful with a probability of one half.
In , Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes n bits for its input. Further improvements to the Deutsch—Jozsa algorithm were made by Cleve et al.
This algorithm is still referred to as Deutsch—Jozsa algorithm in honour of the groundbreaking techniques they employed. We have the function f implemented as quantum oracle. For each x, f x is either 0 or 1. At this point the last qubit may be ignored. So with certainty we know whether f x is constant or balanced. Cleve, A. Ekert, C. Macchiavello, and M. Mosca Proceedings of the Royal Society of London A — Grover A fast quantum mechanical algorithm for database search.
Shor Algorithms for quantum computation: discrete logarithms and factoring PDF.