Quantum Algorithm for Element Distinctness

Years and Authors of Summarized Original Work

• 2004; Ambainis

Problem Definition

In the element distinctness problem, one is given a list of N elements x₁, …, x _N ∈ {1, …, m} and one must determine if the list contains two equal elements. Access to the list is granted by submitting queries to a black box, and there are two possible types of query.

Value Queries. In this type of query, the input to the black box is an index i. The black box outputs x _i as the answer. In the quantum version of this model, the input is a quantum state that may be entangled with the workspace of the algorithm. The joint state of the query, the answer register, and the workspace may be represented as $\sum_{i,y,z} a_{i,y,z}|i,y,z \rangle$, with y being an extra register which will contain the answer to the query and z being the workspace of the algorithm. The black box transforms this state into$\sum\limits_{i,y,z} {a_{i,y,z} } |i,(y + x_i )\bmod m,z\rangle$. The simplest particular case is if the input...