 The initial state is denoted by formula_94.The unitary transformations are denoted by the map formula_95,

so that

As of 2019, most quantum computers are implementations of measure-once quantum finite automata, and the software systems for programming them expose the state-preparation of formula_22, measurement formula_36 and a choice of unitary transformations formula_7, such the controlled NOT gate, the Hadamard transform and other quantum logic gates, directly to the programmer.The primary difference between real-world quantum computers and the theoretical framework presented above is that the initial state preparation cannot ever result in a point-like pure state, nor can the unitary operators be precisely applied.