Synthesis of Quantum Superpositions by Quantum Computation

The quantum search algorithm can be looked at as a technique for synthesizing a particular kind of superposition-one whose amplitude is concentrated in a single basis state. This basis state is defined by a binary function f() that is nonzero in this desired basis state and zero everywhere else. This paper extends the quantum search algorithm to an algorithm that can create an arbitrarily specified superposition on a space of size N in O(sqrtN ) steps. The superposition is specified by a complex valued function f() that specifies the desired amplitude of the system in basis state .