# Quantum-Inspired Neural Network Model of Optical Illusions
## 1 Introduction
Such a projective measurement can be visualised using the concept of the Bloch sphere where the qubit is projected on one of the coordinate axes (e.g., \(z\)-axis in Figure 2a). Computational algorithms based on measurements of the states of a qubit are exponentially faster than any possible deterministic classical algorithm [32]. Subsequently, it has been demonstrated that quantum mechanics can explain certain psychological and decision-making processes better than any classical model [9; 31; 33; 34]. A large and growing body of research has provided significant evidence speaking in favour of this hypothesis [8; 21; 22; 34; 35; 36; 37]. In Figure 2b that was rendered using the physical ray-tracing software POV-Ray 3.7, we illustrate how the principle of projective qubit measurement can be generalised to the Necker cube. The two-dimensional (bottom) images in Figure 2b are the shadows cast by the three-dimensional cubes. However, while the three-dimensional cubes are visually different, the shadows cast by them are identical. Yet, the shadows are an ambiguous Necker cube with the alternating left and right faces (this can be seen by observing them for 5-10 s; some observers may also need to blink to notice the optical illusion [48]). Drawing an analogy with the projective measurement pictured in Figure 2a, in Figure 2b we consider the shadows as a qubit-like superposition of the two fundamental perceptual states of the cube and we virtually project these images back to the three-dimensional space to obtain an unambiguous (either \(|0\rangle\) or \(|1\rangle\) basis state) image of the cube. Figure 1: (**a**) The Necker cube. The answer to the question ’Is the shaded face of the cube at the front or at the rear?’ changes suddenly depending on the observer’s perception, giving rise to a series of rectangular pulses corresponding to the front, \(|0\rangle\), and rear, \(|1\rangle\), perceptual states of the cube shown in panel (**b**). It is noteworthy that the application of the concepts of qubit and superposition does not imply the existence of quantum processes in a biological brain. In fact, the analogy with a qubit serves as a mathematical model that can adequately describe the experimental data. At the same time, an ultimate verification of the accuracy of the quantum models is not practicable due to technical immaturity and high cost of quantum computers and adjacent technologies. Yet, from the neurobiological point of view, an idealised experiment would also include measurements conducted with a brain-computer interface that can decipher the human 'thoughts'. Clearly, such complex tests are not yet feasible and they also raise ethics concerns. Subsequently, much of the current research in this area has focused on artificial neural network modelling and digital twins of perception of optical illusions [25; 49; 50; 51; 52; 53; 54; 55]. Some of these works have employed experimental electroencephalogram (EEG) and magnetoencephalography (MEG) data as the signals that are processed using a neural network model and then classified and correlated with experimental perceptual states of ambiguous figures [25; 53; 54; 55]. In turn, works [50; 51; 56] have focused on the analysis of the dynamics of perception of the images of ambiguous figures using neural network architectures that exhibit chaotic behaviour. However, the results obtained in ref. [51] reproduce the results obtained in classical models of bistable perception, i.e., they do not predict any superposition of the two possible perceptual states of the Necker cube (quantum mechanical models of cognition and perception were not widely accepted when paper [51] was published). On the other hand, although paper [56] does not discuss the perception of ambiguous figures, the neural network model proposed in it reveals a possibility of superposition of two states in principle.