# Quantum-Inspired Neural Network Model of Optical Illusions
## 1 Introduction  
Optical illusions have fascinated humans since the ancient times [1; 2; 3] and served as both object of inspiration in arts [4; 5] and paradigmatic topic of research in the fields of psychology and behavioural science [6; 7; 8; 9; 10; 11; 12; 13]. Nowadays, when artificial intelligence (AI) is all around, a question arises whether a computer or robotic system can recognise optical illusions similarly to a human. Apart from a blue-skies research goal of creating a humanoid robot that both aesthetically resembles a human and can perceive the world as a human, a practicable model of human perception of optical illusions could revolutionise the way video games and spatial computing systems are designed [14], psychiatric illnesses are studied [15] and the effect of gravitation on cognition is investigated [16; 17]. Moreover, establishing the psychological and physiological origin of perception of ambiguous figures promises to unlock the secrets of human decision making [9], also revealing a long-hypothesized but yet elusive link between human mental states and quantum mechanics [8; 9; 18; 19; 20; 21; 22; 23]. We consider the Necker cube [2] shown in Figure 1a. A simple self-examination aimed to answer the question 'Is the shaded face of the cube at the front or at the rear?' results in a series of possible interpretations that randomly switch between 'front' and'rear'. When the sequence of the observed front-rear states, which in the following we denote as \(|0\rangle\) and \(|1\rangle\), is recorded as a function of time as schematically shown in Figure 1b, one obtains a signal consisting of rectangular pulses of random duration. The temporal dynamics of the pulses will vary from one self-examination to another since the perception of optical illusions depends on the observer's age and gender [24]. However, the general trend illustrated in Figure 1b is similar for all observers. On the other hand, electroencephalograms recorded consistently with subjective inputs given by observers of the Necker cube and other ambiguous figures [25; 26; 27] suggest that the perception does not undergo an abrupt switching as sketched in Figure 1b but exhibits a rather continuous oscillation-like behaviour between the \(|0\rangle\) and \(|1\rangle\) states. Data speaking in favour of such a behaviour were also obtained in eye-tracking experiments, where both blink and movement of eye were associated with a perceptual reversal [28; 29; 6]. These experimental results indicate that the perceptual state may not exactly be \(|0\rangle\) or \(|1\rangle\) state but their combination. Mathematically, this scenario can be described as a superposition of \(|0\rangle\) and \(|1\rangle\)[9]. This intriguing observation has motivated the attempts to apply the methods of quantum mechanics and quantum computing to the analysis of human perception [9; 21; 30; 31]. A quantum computer uses a quantum bit (qubit) that can be in states \(|0\rangle=\left[ \begin{smallmatrix}1\\ 0\end{smallmatrix} \right]\) and \(|1\rangle=\left[ \begin{smallmatrix}0\\ 1\end{smallmatrix} \right]\). These states are analogous to the '0' and '1' binary states of a classical digital computer. However, a qubit exists in a continuum of states between \(|0\rangle\) and \(|1\rangle\), i.e., its states are a superposition \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\) with \(|\alpha|^{2}+|\beta|^{2}=1\). When a quantum measurement is performed, a closed qubit system interacts in a controlled way with an external system from which the state of the qubit under measurement can be recovered. For example, using projective measurement operators \(M_{0}=|0\rangle\langle 0|\) and \(M_{1}=|1\rangle\langle 1|\)[32], the measurement probabilities for \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\) are \(P_{|0\rangle}=|\alpha|^{2}\) and \(P_{|1\rangle}=|\beta|^{2}\), which means that the qubit is in one of its basis states.