*3.2. Quantum Representation*

The representation of the AQDE [45] is used in the proposed algorithm, where each quantum individual *q* corresponds to a phase vector *qθ*, which is a string of phase angles *θi* (1 ≤ *i* ≤ *m*), which can be given by

$$\eta\_{\theta} = [\theta\_1, \theta\_2, \dots, \theta\_m], \theta\_i \in [0, 2\pi] \tag{3}$$

where *m* is the length of the quantum bit (qubit) individual. Each quantum individual *q* is a string of qubits:

$$q = \begin{bmatrix} \cos(\theta\_1) & \cos(\theta\_2) & \dots \\ \sin(\theta\_1) & \sin(\theta\_2) \end{bmatrix} \dots \begin{bmatrix} \cos(\theta\_m) \\ \sin(\theta\_m) \end{bmatrix} \tag{4}$$

The probability amplitudes of a quantum bit are expressed as a pair of numbers (cos(*θi*), sin(*θi*)). |sin(*θi*)|<sup>2</sup> represents the probability of selecting item *xi*, and |cos(*θi*)|<sup>2</sup> represents the probability of rejecting item *xi*.

The quantum population *Q*(*t*) = *qt* 1, *qt* 2,..., *qt n* is made up of the quantum individuals at the *t*th generation, where *n* is the size of the population.
