Unitary Maps and Quantum Artificial Neural Networks

Abstract

Unitary quantum maps provide a bridge between classical and quantum dynamical systems theories, having been applied within the context of quantum chaos research. When applied to quantum artificial neural networks, as models of networked quantum computation, unitary quantum maps allow one to address these networks as quantum networked dynamical systems. In this chapter, we address the application of these maps to quantum artificial neural networks, specifically studying the simulation and implementation of these maps for quantum recurrent neural networks, simulating these networks as dynamical computational systems and researching the topological properties of the series of neural firing operators’ quantum averages for nonstationary network states. We also research the results of a simulation of one of these networks on a quantum computer by cloud-based access to IBM Q Experience resources. The results show the emergence of complex dynamics, fitting into similar classes as those of classical cellular automata and coupled maps, including topological markers of chaos, edge of chaos and fractal attractors in the sequences of quantum averages. The implications for quantum complexity research, quantum chaos theory and quantum computing are addressed.

1. Introduction

Complexity research deals with complexity in different contexts including physical systems, biological systems, social, economic, financial and even technological systems. Despite this diversity of contexts, there is a core disciplinary basis in complexity research that intersects computer science, dynamical systems science, statistical mechanics and network science [1, 2, 3, 4, 5, 6, 7]. The dialog between these research fields that form the core disciplinary basis of complexity research has led to fundamental research developed around general families of models of computing systems based on networked computation, these include: cellular automata, coupled maps, random Boolean networks and neural networks [2, 5, 6, 7].

Several of these models have their quantum counterparts and form an important research direction within quantum complexity research, associated with quantum networked computation and artificial intelligence (AI) [8, 9, 10, 11, 12].

Early work in quantum complexity has included the contributions of Ilya Prigogine and Hermann Haken’s groups, focusing on far-from-equilibrium dynamics, irreversibility, dissipative structures and synergetics [3, 4].

In the case of Ilya Prigogine’s group, the relevance of quantum complexity research led to the transformation of The Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems into The Center for Complex Quantum Systems, which includes research into quantum chaos [3, 13, 14].

Other quantum complexity research groups have since developed, an example being the Richter Group on Complex Quantum Systems, from the University of Regensburg, which has also worked on many bodies’ quantum systems, quantum chaos theory and, also, control of quantum chaos [15, 16, 17].

Of the different research lines in quantum complexity, the research on the dynamics of quantum artificial neural networks, or quantum neural networks (QNNs) for short, and on quantum cellular automata (QCAs) leads to a direct bridge between fundamental research within complexity science and quantum computer science, with implications for quantum technologies, linked to quantum networked computation with relevance for distributed AI systems and automation in quantum networks.

The current chapter addresses QNNs. The QNNs are models of networked quantum computation and have been studied by different authors [11, 12, 18, 19, 20, 21, 22]. They can be employed both in the context of quantum machine learning [18, 19, 20, 21, 22] and as models of quantum networked dynamical systems [12, 22], in this case, for artificial neurons with two-level neural firing patterns, a network with n-neurons corresponds to an n-qubit quantum computing system. In this context, neural firing patterns can be represented by neural firing operators, which correspond to the projectors onto the basic binary computational basis that spans an n-qubit system.

The difference with respect to a general n-qubit quantum computing system is that the network’s unitary computation is comprised of conditional unitary operators, that is, different local unitary operations are applied for different neural firing patterns, in this way, the sequence of unitary transformations associated with a QNN, can be placed in the form of general quantum circuits in terms of representation of the conditional unitary operators, which establishes a bridge to standard quantum computation [20, 21].

While these works address specifically QNNs as running quantum algorithms that end at a specific step, another type of application involves a bridge to quantum dynamical systems, leading to the need to work with the concept of a quantum map.

Indeed, a feature of classical artificial neural networks is that they can be used to simulate complex networked dynamical systems, the quantum extension of this type of application, involves performing the conditional unitary operations of the network running the full activation cycle, leading to a new state of the network at the end of each computational cycle, as addressed in [12, 22].

In this way, each computational cycle corresponds to an iteration of the network and a sequence of iterations of the network can be represented by the concept of unitary quantum map [12].

The concept of unitary quantum map was introduced in quantum chaos research to deal with systems like the kicked top and has become a standard research tool of quantum chaos research [23].

This concept is an extension of the concept of a classical map, from classical dynamics, that maps a classical phase space onto itself and can be used to express the classical transformation of a state into another state leading to a discrete-time sequence of iterations [23].

A unitary quantum map is, in turn, a unitary operator defined on a quantum system’s Hilbert space that maps a state vector into another state vector, leading to a sequence of state vectors, this sequence can be researched, using the quantum theory’s formalism, by calculating the quantum averages of observables and researching this sequence of quantum averages. A major feature of a unitary quantum map is that, being unitary, it maps pure states into pure states.

In the case of QNNs, when addressed as basic quantum dynamical systems, the concept of unitary quantum map becomes relevant, since we are not running the network a single time with a measurement at the end of a computational cycle, instead one runs the sequence of iterations of the quantum map that corresponds to a full cycle of computations of the network [12].

It turns out that, using this methodology, specific complex dynamics have been identified in these networks, including complex emergent attractors in the sequences of quantum averages linked to neural firing patterns as well as topological features of chaos-like dynamics [12, 22].

In order to study the network’s dynamics, one can use basic quantum formalism, like in quantum chaos theory, and employ matrix calculus to calculate, at the end of each iteration of the map, the quantum averages for relevant observables on the network’s activity such as neural firing patterns.

In the case of neural firing patterns, since they are represented by the basic projectors onto the network’s computational basis, the sequence of quantum averages has a specific interpretation, namely, the quantum averages of these projectors onto the computational basis, at the end of t iterations of the network provide the probability distribution associated with the network’s alternative firing pattern if a measurement was performed at the end of those t iterations.

This last property allows, as we show in the present chapter, for an empirical testing of the theoretical sequence of quantum averages obtained from the sequence of unitary state vectors using the formalism of quantum mechanics.

By running the network for a sequence of iterations, which is a unitary stage, and then extracting frequencies obtained by measuring each quantum register’s state at the end, the results, when run for multiple repeated experiments should match the theoretical quantum averages. This can be done for different iteration indexes t which allows for an extraction of the sequence of quantum averages which can be compared with the theoretical sequence obtained from matrix calculus, using the basic formalism of quantum mechanics.

In order to characterize the network’s dynamics one can apply topological analysis methods, namely, recurrence analysis to the sequences of quantum averages. This approach was followed in [12, 22] to characterize these networks’ dynamics, both in the case of a dissipative quantum map [22] and in the case of a unitary quantum map [12].

When considering recurrence analysis, there are four main classes of topological signatures, as we will review in the present chapter, that mirror four main dynamical classes identified in cellular automata [6], and that also occur in coupled map lattices [5] and in random Boolean networks [7], it turns out that these four dynamical classes are shown to also occur in QNNs [12, 22]. These classes are:

• Class 1: The fixed point, which is a stationary state, corresponds to the eigenvectors of the quantum map for the network, which usually involves entangled states [12];

• Class 2: The periodic or quasiperiodic dynamics;

• Class 3: The random dynamics, which can occur in deterministic systems including in cellular automata and in deterministic chaos [5, 6], and that also occurs in QNNs as we will see in the present chapter, especially when the number of neurons is increased;

• Class 4: Finally, there is the fourth class of dynamics which includes a mixture of class 2 and class 3, and that has been called the edge of chaos [6, 7, 24, 25].

Class 4 dynamics can occur in chaotic dynamical systems near the onset of chaos, in cellular automata, in random Boolean networks, as well as in coupled map lattices, as propagating localized order that intermixes with random fluctuations characterizing phenomena such as spatiotemporal intermittency [5, 6, 7, 24, 25], and also occurs in the sequences of quantum averages of neural activity operators for QNNs being identifiable through the application of recurrence analysis methods as researched in detail in [12, 22, 26], showing the effectiveness of topological analysis methods [27, 28], which have now been expanded and adapted to quantum chaos research [12, 22, 26, 29].

In the current chapter, we illustrate these dynamics for different QNNs, focusing on class 4 (edge of chaos) and class 3 (random) dynamics.

In Section 2, we review the concept of unitary quantum map and its application to QNNs, this section is divided into multiple subsections.

In Section 2.1, we address the concept of unitary quantum map and how it can be studied both theoretically and experimentally in general.

In Section 2.2, we address the example of a unitary quantum map’s dynamics for a single qubit and illustrate the basic application of recurrence analysis methods on the theoretical simulation of the map.

In Section 2.3, we review a two-neuron recurrent neural network that has been researched in [12, 26] and in Section 2.4, we address the comparison between the theoretical simulation of the model and the empirical implementation on one of IBM’s quantum computers, accessed by cloud using IBM Q Experience resources.

In Section 3, we address the dynamics of a QNN with a ring topology which includes elements of a feedforward network with a recurrent connection at the end, also providing for a bridge to quantum cellular automata and quantum deep neural networks. We simulate this network for different numbers of neurons showing the transition from class 4 to class 3.

In the case of the ring topology network and for the IBM quantum computer simulation, we also uncover the emergence of fractal attractors, a point that is also discussed in Section 3.

In Section 4, we conclude with a final reflection on the chapter’s findings and its implications for quantum complexity research, quantum technologies and quantum chaos theory.

2. Quantum artificial neural networks as dynamical quantum computing systems: theory and experiments

2.1 Unitary quantum maps and quantum dynamics

In classical dynamical systems, a map is a function that maps the phase space onto itself, leading to a discrete-time sequence of states. In quantum theory’s formalism, a quantum extension of a classical map is a quantum unitary map, defined on a quantum system’s Hilbert space $\mathbb{H}$, that maps quantum state vectors into quantum state vectors [23]:

$$\mid \psi \left(t+1\right)>=F\mid \psi \left(t\right)>$$E1

In this context, $t=0,1,2\dots$ is an iteration index, leading to a sequence of quantum states $\mid \psi \left(t\right)>$.

To research quantum dynamics with respect to dynamical classes, the sequences of states themselves are not the most useful since the concept of trajectory breaks down, however, when an observable is used, like a position, momentum, Hamiltonian operator and spin operator, one can effectively study the sequence of quantum averages for that observable by employing basic quantum matrix calculus, so that, given the sequence of quantum states one can extract a form of trajectory for the observable:

$${⟨O⟩}_t=⟨\psi \left(t\right)\left|O\right|\psi \left(t\right)⟩$$E2

This is a well-known basic scheme used in both the Schrödinger and Heisenberg picture in quantum complexity research [30]. However, it should be stressed that, in this scheme, which has been extensively applied to the study of quantum dynamics [30], one is not measuring the observable, instead one is using quantum matrix calculus to extract a sequence of quantum averages that provides a picture of a quantum trajectory in terms of an observable.

Of course, the sequence of quantum averages provides an important information in what regards the empirical implications, namely, ${⟨O⟩}_t$ corresponds to the theoretical prediction of the expected value of the observable if the iterations were to be stopped at step t and the observable was measured at the end of that step.

Indeed, if one were to perform the measurement on multiple identical copies of the system, the sample average should approximate the quantum average for a large number of experiments, this is a basic point that comes out of the early empirical testing of quantum theory.

Of course, if a measurement is performed at the t-th iteration, the unitarity will break down at that point, so we are effectively ending the iteration sequence at step t. This point has an important implication for the theoretical versus empirical research on the dynamics resulting from quantum unitary maps.

In the theoretical research, the sequence of quantum averages for an observable resulting from the sequence of quantum states can be calculated and dealt with as a form of trajectory using the basic formalism of quantum mechanics, as stated above, providing another picture of the unitary evolution itself in regards to the system in question and a relevant observable, a point that is extensively addressed in the quantum noise research [30].

The theoretical value of the quantum average at iteration step t provides, as stated above, for a reference on what the quantum average would be for the observable if we performed a quantum measurement of that observable, for a large number of identically prepared systems, at that iteration step, ending the iterations at that step.

In this way, while the theoretical simulation of the map can either be extracted as a closed formula or numerically using standard matrix calculus, in order to obtain the physical empirical testing of the quantum map’s iterations, one must use an experimental loop comprised of the following steps that we explain next:

1. For $t=1,2,\dots ,T$:

   1. For $n=1,2,\dots ,N$:

      1. Set the initial quantum state for: $\mid \psi \left(0\right)>$.

      2. Perform the unitary evolution defined by applying the map s times ($\mid \psi \left(s\right)>=F^t\mid \psi \left(0\right)>$)

      3. Perform a quantum measurement of the observable at the end of iteration t and store the result.

   2. Calculate the average of the results stored at the end of each step 1.1.2.2. and keep it.

   3. Return to step 1.1.

The above general experimental setup means that, for each sequence of iterations t, we perform the unitary evolution t times on an ensemble of N identical copies of the system in question for the same initial condition and then measure the observable at the end of the iteration step t, calculating the empirical average for that ensemble for that iteration step, in this way we get a sequence of quantum averages that will closely match the theoretical sequence, given quantum theory’s predictions. This is a basic extension of the standard empirical testing of quantum mechanics applied to the general context involving unitary maps.

Now, considering the specific case of an observable, with a discrete eigenvalue spectrum, such that its eigenvectors span the Hilbert space $\mathbb{H}$:

$$O\mid v_k>=v_k\mid v_k>$$E3

We can introduce the complete set of projectors onto the observable’s basis:

$$P_O=\left\{P_k=\left|v_k><v_k\right|:k=0,1,\dots \right\}$$E4

In this case, given the sequence of state vectors $\mid \psi \left(t\right)>$, employing the formalism of quantum mechanics [30], we can calculate the sequence of quantum averages for the observable and research the corresponding dynamical sequence:

$${⟨O⟩}_t=⟨\psi \left(t\right)\left|O\right|\psi \left(t\right)⟩$$E5

We can also study the sequence of quantum averages for each projector:

$${⟨P_k⟩}_t=⟨\psi \left(t\right)\left|P_k\right|\psi \left(t\right)⟩$$E6

In the case of a finite-dimensional basis, which leads to a finite-dimensional projector set and corresponding Hilbert space, we can embed the quantum averages of the projectors into a d-dimensional Euclidean space:

$$P\left(t\right)=\left(⟨\psi \left(t\right)\left|P_0\right|\psi \left(t\right)⟩,⟨\psi \left(t\right)\left|P_1\right|\psi \left(t\right)⟩,\dots ,⟨\psi \left(t\right)\left|P_{d-1}\right|\psi \left(t\right)⟩\right)$$E7

The above tuple is a probability tuple containing the probability distribution associated with each alternative eigenstate, calculated for the t-th iteration step, in this way, the sum of the elements in the tuple is equal to 1. Since the sequence of values for P(t) is a d-dimensional tuple of real numbers, the sequence can be represented and studied in a d-dimensional Euclidean space.

Again, as for an observable’s quantum average, the above tuple’s values can be extracted by applying basic matrix calculus employing quantum mechanics’ formalism and can be simulated in any software that has linear algebra features, including Python’s NumPy, MATLAB or Mathematica.

The theoretical simulation of the tuple sequence provides for a sequence of values of the quantum probability distribution, again, as in the case of the quantum average of an observable, the value of the tuple at iteration step t has an interpretation as the probability distribution associated with interrupting the iterations at that iteration step and performing a quantum measurement of the observable, at that step.

For repeated identically prepared experiments, one can extract the statistical frequencies for the different eigenstates at the end of the t iterations, stopping the iterations at that point, so that the frequency distribution for a large number of experiments will match the theoretical results, this is a basic approach that has been followed since the beginning of quantum mechanics and that carries over to the experimental implementation of quantum dynamics of quantum maps.

In order to study the dynamics of the quantum probability tuple P(t) or of an observable’s sequence of quantum averages, one can apply topological data analysis methods, in particular, one can apply recurrence analysis methods, that are effective in distinguishing between dynamical regimes [27, 28].

The analysis is based on calculating a distance matrix between each dynamical point in Euclidean space, this matrix is symmetric and the main diagonal is comprised only of zeros. From the distance matrix, one can estimate a binary recurrence matrix that stores the value 1 when two dynamical points in a sequence are distant from each other by at most a radius of $\epsilon$ and 0 when two points are distant from each other by more than $\epsilon$.

The recurrence matrix can also be worked with as a transition matrix, such that each dynamical point is linked to all the neighboring points in a closed Euclidean $\epsilon$-neighborhood.

The use of the closed neighborhood, instead of the open neighborhood, allows the distinction between periodic and nonperiodic orbits, in the case of a periodic orbit, a closed neighborhood is non-empty for a radius of zero between periodic points, while for nonperiodic orbits, if the radius is set to zero, the recurrence matrix will be a null matrix.

Periodic dynamics appear when the radius is set to zero, as evenly spaced diagonals parallel to the main diagonal that are completely filled with the value of 1. This is no longer the case for nonperiodic dynamics, in this last case, when the dynamics is quasiperiodic, long unevenly spaced diagonals filled with the value of 1 appear for a sufficiently high radius.

Topological markers of chaos appear as interrupted diagonals and isolated dots, these interrupted unevenly spaced diagonals result from the exponential divergence associated with positive Lyapunov exponents.

When considering the types of dynamics, there are four main recurrence classes that occur in dynamical systems that closely mirror cellular automata’s four classes reviewed in the introduction [6, 12, 22], these are:

• Class 1 (fixed point): the recurrence matrix is a unit matrix, all points are recurrence points.

• Class 2 (periodic or quasiperiodic dynamics): the dynamics is either periodic with evenly spaced diagonals matching the period or quasiperiodic, which shows up as unevenly spaced diagonals.

• Class 3 (random): the random dynamics can either result from external noise or be endogenously generated in a deterministic system, as it happens in the case of deterministic chaos and class 3 cellular automata, in these cases, the dynamics is actually pseudorandom and its topological markers can include broken diagonals, linked to an unstable periodic orbit skeleton and disconnected points in the recurrence matrix.

• Class 4 (edge of chaos): this is a mixture of class 2 and class 3, its major characteristic in black and white recurrence plots include very long resilient unbroken diagonal lines that appear with different distances along with broken diagonals and isolated dots that represent random-like signatures, QNNs tend to produce this dynamical class.

In the case of chaotic dynamics for shift maps, the randomness is associated with the randomness of the binary expansion of the initial condition, which differs from class 3 random cellular automata that can produce random patterns even for non-complex initial conditions, this is a point stressed in [6].

In networks of chaotic oscillators, such as coupled map lattices, class 3 and class 4 behavior also occur in these networks’ dynamics, with class 4 type dynamics occurring, for instance, as spatiotemporal intermittence [5].

It is important to stress that there is a continuum between class 3 and class 4, in the case of classical chaotic dynamical systems, in the sense that, in the chaotic phase, but near the onset of chaos, with lower values of positive Lyapunov exponents, pockets of unstable cycles can be mirrored for a longer time and lead to class 4-type topological markers, these markers, as stated, also occur in networks of chaotic oscillators [5].

In this way, the concept of edge of chaos that was addressed in both the analysis of cellular automata and random Boolean networks [6, 7] can also apply to deterministic chaos near the onset of chaos and networks of chaotic oscillators [5].

Now, in order to illustrate the application of the concept of unitary quantum map, in the context of quantum computation, we begin with a single qubit example and a Walsh-Hadamard transform.

2.2 Single qubit dynamics: the example of the Walsh-Hadamard transform

Returning to the quantum dynamics, as an example of a quantum map applied to a single qubit computational system, let us consider the Walsh-Hadamard transform:

$$\mid \psi \left(t+1\right)>=W\mid \psi \left(t\right)>$$E8

$$W=\frac{\mid 0><0\mid -\mid 1><1\mid +\mid 0><1\mid +\mid 1><0\mid }{\sqrt{2}}$$E9

Starting with $\mid \psi \left(0\right)>=\mid 0>$, and working with the projectors $P_0=\mid 0><0\mid$ and $P_1=\mid 1><1\mid$, the sequence P(t), for the above initial condition, is periodic cycling through the tuples $\left(0.5,0.5\right)$ and $\left(1,0\right)$, with the first tuple holding for even iterations and the second for the odd iterations. This means that if we stop the iterations at an odd t and were to measure the qubit at that step, then, one would obtain the logical state 0, with probability equal to 1, on the other hand, if were one to stop the iterations at an even t and were to measure the qubit at that step then one would obtain the logical state 0, with probability 0.5, and the logical state 1, with probability 0.5.

In Figure 1 (right), we plot the results of the iterations of the above quantum map as a scatterplot for the two projectors’ quantum averages with respect to the iteration index, which shows the cycle through the above values, also in Figure 1 (left) we show the corresponding recurrence plot which plots the recurrence matrix for a radius of zero, the plot visibly shows the periodic orbit obtained for the quantum averages with the long fully filled diagonals surfacing for this radius.

Figure 1.

Recurrence plot (left) for a radius of 0 from a 20 iterations’ simulation of the quantum Walsh-Hadamard map for a single qubit, scatterplots (right) for the sequence of quantum averages of the two computational basis projectors.

The result is a regular grid-like pattern for the recurrence plot. Since the matrix is symmetric the full recurrence structure can be researched using either the diagonals above the main diagonal or below, for which different metrics can be calculated.

The first metric that we use is to count the number of diagonals above the main diagonal with recurrence points, this number corresponds to the total number of diagonal lines above the main diagonal with recurrence points.

The second counting is the number of diagonals above the main diagonal with 100% recurrence, that is, a diagonal that is fully filled with recurrence points, finally, the last counting is the number of diagonals above the main diagonal.

From these three metrics, one can calculate the recurrence probability, that is, the probability that a randomly chosen diagonal above the main diagonal contains recurrence points, this metric can be calculated by dividing the number of lines above the main diagonal with recurrence points by the total number of diagonals above the main diagonal.

The second metric that can be calculated is the probability that a randomly chosen diagonal above the main diagonal with recurrence points is a 100% recurrence diagonal, this conditional probability can be calculated by dividing the number of diagonals above the main diagonal with 100% recurrence by the number of diagonals above the main diagonal with recurrence.

A third metric that can be calculated is the average recurrence strength. In this case, one starts by dividing the number of recurrence points in a diagonal with recurrence by the length of the diagonal, which provides for the proportion of the diagonal that is filled with recurrence points, this proportion is 0 for a diagonal with no recurrence points and 1 for a 100% recurrence diagonal.

After calculating this proportion for each diagonal above the main diagonal, one can calculate the mean of this proportion which provides for the average recurrence strength.

Finally, we research the distribution of distances between diagonals with 100% recurrence, measured in terms of the number of iterations needed to reach the next 100% recurrence diagonal, this leads to the periodicity of a cycle associated with 100% recurrence diagonals and becomes a critical statistic for addressing possible cycles, or, in the case of nonperiodic dynamics, the possibility of quasiperiodic structures.

This distribution will play a major role in the analysis of the class 4 dynamics as we will see.

It turns out that these metrics are effective as we will see in distinguishing between the four dynamical classes.

If we calculate these metrics for Figure 1’s simulation, we obtain the results shown in Table 1.

Metrics
Number of lines with 100% recurrence	9
Number of lines with recurrence	9
Number of diagonals	19
Average recurrence strength	1
P[Recurrence]	0.4737
P[100%recurrence|recurrence]	1

Table 1.

Recurrence metrics for the recurrence plot shown in Figure 1.

The recurrence metrics reflect the periodic structure, in this case, there were 20 iterations. The distance matrix is, therefore, a square matrix of rank 20, which leads to 19 diagonals above the main diagonal, the cycle is period 2, which means that we get 9 diagonals all with full recurrence, the probability of a randomly chosen diagonal with recurrence being a 100% recurrence diagonal is, therefore, equal to 1, on the other hand, the probability of a diagonal being a diagonal with recurrence points is equal to 9/19 which is approximately 0.4737.

The average recurrence strength is equal to 1 since all diagonals with recurrence are 100% recurrence diagonals.

Now, since there are nine 100% recurrence diagonals and it takes two iterations to reach the next 100% recurrence diagonal, the distances between 100% recurrence diagonals are all equal to 2 and occur eight times, as expected, from a period 2 dynamics. The above dynamics is an example of a class 2 dynamics.

A more complex dynamics is obtained when dealing with QNNs, as an example, we now address a two-neuron quantum recurrent neural network.

2.3 Two-neuron recurrent neural network

Considering a two-neuron quantum recurrent neural network with two-level firing patterns, the general architecture is shown in Figure 2.

Figure 2.

Quantum recurrent neural network architecture.

In this case, the Hilbert space for the network is spanned by the computational basis $B_2=\left\{|00>,|01>,|10>,|11>\right\}$, such that $\mid \mathit{rs}>$ with $r,s\in \left\{0,1\right\}$ encodes a neural firing pattern, that is, if r is equal to 0 then the first neuron is not firing, while, if it is equal to 1, the first neuron is firing, the same holds for s, in relation to the second neuron.

The main projectors for this basis lead to the neural firing pattern projectors $P_{\mathit{rs}}=\mid \mathit{rs}><\mathit{rs}\mid$, which means that the quantum averages’ tuple P(t) can be embedded in four-dimensional Euclidean space. For each neuron, we can also calculate the local neural firing operators $N_0=P_1\otimes I$ and $N_1=I\otimes P_1$, where I is the rank 2 identity matrix, the first operator yields the value of 1 when the first neuron is firing and 0 otherwise, and the second operator yields the value of 1 when the second neuron is firing and 0 otherwise.

In the network that we will be addressing, the two operators $U_1$ and $U_0$ are conditional operators that conditionally transform the quantum register of the target neuron, conditional on the firing pattern of the input neuron, following the network’s links:

$$U_0=I\otimes \mid 0><0\mid +U_r\otimes \mid 1><1\mid$$E10

$$U_1=\mid 0><0\mid \otimes I+\mid 1><1\mid \otimes U_r$$E11

$$U_r=cos\left(\frac{\mathit{\pi r}}{2}\right)\left(|0><0|+|1><1|\right)+sin\left(\frac{\mathit{\pi r}}{2}\right)\left(-|0><1|+|1><0|\right)$$E12

Assuming that the first neural connection that is activated is from the first neuron to the second and that the second neural connection to be activated is from the second neuron to the first, running the network through a full computational cycle is given by the operator product:

$$F=U_0{U}_1$$E13

Repeated sequential full computational cycle operations for this network lead to the quantum map:

$$\mid \psi \left(t+1\right)>=F\mid \psi \left(t\right)>=U_0{U}_1\mid \psi \left(t\right)>$$E14

To illustrate the dynamics we will work here with the initial superposition state:

$$\mid \psi \left(0\right)>=\frac{\mid 00>+\mid 01>+\mid 10>+\mid 11>}{2}$$E15

The dynamics of this network is already nontrivial, it depends upon the value of r. For low values of r, the quantum averages for the projectors in which at least one neuron is active track down a sinusoidal curve.

In Figure 3, we show the corresponding sequence for the quantum averages associated with the projector set. It should be stressed that the figure is actually a scatterplot, for a discrete-time sequence.

Figure 3.

Simulation of the quantum recurrent neural network with r equal to 0.002, 10,000 iterations are shown after the first 1000 iterations were dropped for transients, the trajectory of each component of the projectors’ tuple P(t) is shown.

From the results shown in Figure 3, a few points can be noticed, first, the quantum average for the projector where the two neurons are nonfiring is constant as can be seen in the top graph, this is a class 1 dynamics.

The quantum averages for the remaining projectors follow the periodic sinusoidal curves which are class 2 dynamics, in this case, when the value rises for the projector where the two neurons are active, which corresponds to a reinforcing neural firing activity, it falls for the projectors where the two neurons are in reverse neural firing activity, which corresponds to inhibitory neural firing activity, on the other hand, the quantum averages, for the cases where the two neurons show reverse neural activity, follow a synchronized sinusoidal curve.

As shown in [12, 26], for the values of r that lead to class 2 dynamics, the dynamics for the averages of the local neural firing operators $N_0$ and $N_1$ are synchronized and also follow a sinusoidal curve.

As the parameter r is increased the period associated with the sinusoidal curves decreases as well as the correlation between the quantum averages for the two-neurons’ local neural firing operators, until there is a transition from positive to negative correlation. The point at which the dynamics transitions from positive to negative correlation is also a point after which the dynamics no longer follows the sinusoidal periodic curve transitioning to dynamical regimes characterized by class 4 dynamics [12, 26].

While the quantum average for the projector $P_{00}$ remains in its fixed point dynamics for all values of r, for the remaining projectors, two major attractor shapes emerge with the increase of the parameter r and the disappearance of the sinusoidal pattern.

The main attractor has a triketa-like shape, which can be obtained from intersections of three ellipsoids, and is largely characterized by class 4 dynamics, the other has the shape of a trifolium, the trifolium is characterized by class 2 dynamics resulting from a progressive change in the triketa attractor which loses its middle section leading to the trifolium as r is increased.

In Figure 4, we show both attractor shapes and corresponding quantum averages for the projectors shown in scatterplots, in the case of the triketa, the plotted dynamics is for a parameter with near-zero correlation between the sequence of quantum averages of $N_0$ and $N_1$.

Figure 4.

Simulation of the quantum recurrent neural network with r = 0.550129597 (top) and with r = 0.999 (bottom), 10,000 iterations are shown in each case after the first 1000 iterations were dropped for transients, the scatterplot trajectory of each component of the projectors’ tuple P(t) is shown (right) along with the attractor (left).

2.4 Experimental implementation of the two-neuron recurrent neural network

The experiments in the quantum device involve the standard application of quantum measurement for a quantum map, which is discussed in Section 2.1. In this case, for each value of t, in order to extract the sequence P(t), we need to apply the quantum map t times, and then measure the two quantum registers’ states at the end of t, this needs to be repeated multiple times (multiple rounds) in order to obtain the empirical frequency distribution from which P(t) can be estimated calculating the relative frequencies for the different computational basis values.

For each iteration index t, the t applications of the quantum map are automatically converted into a quantum circuit that implements the transformation $F^t$, this process is automated in IBM’s systems, and the Qiskit code for the Jupyter Notebook used for the experiments NSimul.ipynb is provided on Github at the QNeural project’s website [31, 32], the project which also contains the simulation basis for the theoretical model and that was used in [12, 22, 26].

Figure 5 shows 1000 iterations of the quantum map performed on IBM’s quantum device “ibm_q_belem,” for the triketa attractor.

Figure 5.

Simulation of the quantum recurrent neural network on the “ibm_q_belem” device, for r = 0.550129597, with 5000 rounds sent for each iteration and 1000 iterations kept after the first 1000 iterations were dropped for transients.

As can be seen in Figure 5, the resulting sequence of values for P(t) matches well the theoretical sequence, however, there are random fluctuations associated with both the quantum measurement and environmental noise, which, in this case, while low, can still lead to fluctuations in the relative frequencies for each measurement, this is evident in the cloud of points indicating a dispersion, we will return to this point further on.

In Figure 6 (left), we show the colored recurrence plot for the triketa attract0r’s theoretical simulation, also using 1000 iterations, and in Figure 6 (right), we show the corresponding colored recurrence plot for Figure 5’s simulation.

Figure 6.

Colored recurrence plots obtained for the tuples P(t), from 1000 iterations of the quantum recurrent neural network with r = 0.550129597, for the theoretical simulation (left) and the implementation on “ibm_q_belem” (right) shown in Figure 5. Lighter colors represent shorter distances, the color map used was Python’s matplotlib’s cm.inferno_r.

The two recurrence plots are very similar and both exhibit the presence of a quadratic grid disturbed by random fluctuations, the quadratic grid is evidence of the presence of periodicities.

The parameter for which the simulation is produced, as stated, is one in which the local neural firing operators’ sequence of quantum averages transition from positive to negative correlation, with their dynamics embedded in two-dimensional space corresponding to a two-dimensional projection of the above attractor [12, 26].

In Table 2, we provide the main recurrence metrics for both the theoretical simulation and the empirical results from the quantum device, using a radius of 0.1. In this case, for 1000 iterations, the differences between the theoretical simulation and the physical device’s implementation are not large, the values are close to each other.

Metrics	Theoretical	Empirical
Number of lines with 100% recurrence	156	135
Number of lines with recurrence	964	978
Number of diagonals	999	999
Average recurrence strength	0.9558	0.9289
P[Recurrence]	0.9650	0.9790
P[100%recurrence|recurrence]	0.1618	0.1380

Table 2.

Recurrence metrics for the recurrence matrices of Figure 6’s theoretical simulation and empirical simulation obtained for a Euclidean radius of 0.1.

The main differences are that the physical device simulation has a slightly higher number of lines with recurrence, but it has a lower number of lines with 100% recurrence, leading to a higher recurrence probability but a lower probability of a randomly chosen line with recurrence being a line with 100% recurrence, the average recurrence strength is also smaller for the physical device.

This is consistent with a higher number of interrupted diagonals in the physical device implementation which is, in turn, consistent with the presence of noise associated with the physical device and the quantum measurement itself.

In both cases, the probability of a randomly chosen diagonal to contain recurrence points is high, however, the probability of a randomly chosen diagonal with recurrence points to be a 100% recurrence line is low, which indicates that the majority of lines with recurrence are either broken lines or contain disconnected dots, the dominance, in this case is of interrupted diagonals (unstable cycles), since the average recurrence strength in both the theoretical simulation and the empirical implementation in the physical device have a higher than 90% average recurrence strength, which indicates that, on average, the diagonals with recurrence have more than 90% of their size occupied by recurrence points, for this radius. This indicates that there are strong recurrence signatures.

The existence of lines with 100% recurrence, and the grid-like pattern in the colored recurrence plot is evidence of regularities possibly associated with a resilient periodic or quasiperiodic structure that needs to be researched.

In this case, for the theoretical simulation, we find the presence of three 100% recurrence distances corresponding to three cycles, a 21 iterations cycle, a 26 iterations cycle and a 47 iterations cycle. As the total number of iterations is increased and the corresponding 100% recurrence lines are calculated, we find that only these three cycles appear associated with these lines.

The frequencies with which these cycles appear increase linearly with the total number of iterations as can be seen in Figure 7, which shows the absolute frequencies of the estimated distances between 100% recurrence lines, estimated for the recurrence matrices with 0.1 radius, with the total number of iterations increasing from 1000 to 10,000 in steps of 1000.

Figure 7.

Statistical distribution of the distances between the 100% recurrence lines for the theoretical simulation with r = 0.550129597 and the total number of iterations increasing from 1000 to 10,000 in steps of 1000.

This increase in frequency with the iterations and the multiple distances corresponding to different cycles in the recurrence matrix constitute evidence favorable to the presence of a resilient quasiperiodic structure.

The dominant cycle is 21, followed by the 26-period cycle and, finally, the 47-period cycle. These quasiperiodic signatures are resilient class 2 recurrence signatures that intermix with the class 3 recurrence signatures, in this way, we have evidence of a class 4 recurrence structure, that is, a dynamics that intermixes elements of class 2 and 3.

For the experimental simulation in the quantum device, these cycles also appear, however, other 100% recurrence lines also appear and the cycles are unstable, that is, they last for long sequences of iterations but they are eventually interrupted.

This is linked to two factors: random fluctuations associated with quantum measurement and environmental noise.

In this way, a smaller resilience of these quasiperiodic signatures is expected in the experimental implementation, however, there are a few features that become relevant.

Of the different cycles, there is one that appears often as dominant in the recurrence structure, this is the 47-period cycle, which by contrast is less frequent in the theoretical simulation, this is one of the major differences between the theoretical and the empirical simulation.

Other cycles also appear at some iterations. However, all these cycles are eventually interrupted, and even so, the basic quasiperiodic skeleton resurfaces.

These results are shown in Figure 8, which shows, for the empirical simulation in the physical device, the absolute frequencies of the estimated distances between 100% recurrence lines, estimated for the recurrence matrices with 0.1 radius, with the total number of iterations increasing from 1000 to 10,000 in steps of 1000.

Figure 8.

Statistical distribution of the distances between the 100% recurrence lines for the empirical simulation on “ibm_q_belem” with r = 0.550129597, and the total number of iterations increasing from 1000 to 10,000 in steps of 1000, using 5000 rounds for extracting the quantum averages.

Figure 8 contrasts with Figure 7 on several points, first, besides the 21-, 26- and 47-period cycles there emerges a 68-period cycle, a 115-period cycle and a 162-period cycle, and the 47-period cycle is dominant, instead of the 21 and 26 periods, as stated.

Secondly, these cycles, while present at a certain number of iterations, tend to disappear for a higher number of iterations and then resurface, which means that the 100% recurrence lines, in the physical device, while appearing in iterations with a long number of steps, with the 47-period cycle and the 68-period cycle standing out for a 7000 iterations simulation, disappear as the number of iterations is increased which means that the dynamics for the quantum averages is recurrently revisiting the neighborhood of unstable periodic orbits much in the same way as a classical chaotic dynamical system does.

The recurrence structure is driven closer to the topological signatures that also occur in classical chaotic dynamics near the onset of chaos, but already in the chaotic regime.

In the physical device, the dynamics is influenced by the noise and the random fluctuations associated with quantum measurement, leading to fluctuations around the theoretical quantum averages, this factor, and the intrinsically complex mixture of the class 2 and class 3 dynamics, effectively leads to this type of topological signatures that are close to the recurrence signatures of a chaotic dynamics near the onset of chaos.

Now, going beyond the above recurrent network comprised of only two neurons, there is a generalization of this network that intermixes feedforward and recurrent connections.

3. The ring network

The network that we will, now, be addressing is the quantum ring network shown in Figure 9. The network is simultaneously a multilayer deep neural network but can also be considered a variant of a quantum cellular automaton.

Figure 9.

Quantum ring network.

Defining the binary string s as $s=s_0{s}_1\dots s_{K-1}$, with $s_j\in \left\{0,1\right\}$, for $j=0,1,\dots ,K-1$, the general state of this network, expanded in the firing pattern basis, is given by the normalized state vector:

$$\mid \psi >=\sum _s{\psi }_s\mid s>$$E16

With the unitary operators defined as:

$$U_0=I\left({\otimes }_{j=1}^{K-2}I\right)\otimes P_0+U_r\left({\otimes }_{j=1}^{K-2}I\right)\otimes P_1$$E17

$$U_l=\left({\otimes }_{j=0}^{l-2}I\right)\otimes P_0\otimes I\left({\otimes }_{j=l+1}^{K-1}I\right)+\left({\otimes }_{j=0}^{l-2}I\right)\otimes P_1\otimes U_r\left({\otimes }_{j=l+1}^{K-1}I\right),0<l<K-1$$E18

$$U_{K-1}=\left({\otimes }_{j=0}^{K-3}I\right)\otimes P_0\otimes I+\left({\otimes }_{j=0}^{K-3}I\right)\otimes P_1\otimes U_r$$E19

The operator $U_r$ is given by Eq. (12) and the unitary map for the network is defined by:

$$F=U_{K-1}\dots U_1{U}_0$$E20

In the network’s simulations, we assume the initial state as:

$$\mid \psi \left(0\right)>=\frac{{\sum }_s\mid s>}{\sqrt{2^N}}$$E21

In this case, instead of the quantum averages for the projectors, we study the local neural firing operators defined as:

$$N_j=\left({\otimes }_{j=0}^{j-1}I\right)\otimes \mid 1><1\mid \left({\otimes }_{i=j+1}^{K-1}I\right)$$E22

Leading to the eigenvalue equation:

$$N_j\mid \dots s_j\dots >=s_j\mid \dots s_j\dots >$$E23

We will be studying, in the simulations, the dynamics of the tuple of quantum averages for these projectors, defined as:

$$N\left(t\right)=\left(⟨\psi \left(t\right)\left|N_0\right|\psi \left(t\right)⟩,⟨\psi \left(t\right)\left|N_1\right|\psi \left(t\right)⟩,\dots ,⟨\psi \left(t\right)\left|N_{K-1}\right|\psi \left(t\right)⟩\right)$$E24

The above tuple’s sequence can be embedded in K-dimensional Euclidean space. In Figure 10, we show the dynamics for r = 0.95, 5000 iterations and a three-neurons’ network, for which we embed the sequences of the quantum operators $N_j$’s averages in three-dimensional Euclidean space, we also show the iterations’ sequence.

Figure 10.

Embedded dynamics (left) and scatterplot for the local neuron operators’ quantum averages (right) for a simulation of the ring network with three neurons, r = 0.95, 5000 iterations are shown after 1000 iterations were dropped for transients.

Considering the iterations sequence, the dynamics for the local neural firing operators’ quantum averages seems to be random-like, however, there is an attractor structure visible in the three-dimensional embedding.

When the number of neurons is increased to four, as shown in Figure 11, the dynamics is random and the three-dimensional projection of the four-dimensional attractor is a random scatterplot. In Figure 12, we show the colored recurrence plots for each of the two simulations.

Figure 11.

Three-dimensional projection (left) and scatterplot for the local neuron operators’ quantum averages (right) for a simulation of the ring network with four neurons, r = 0.95, 5000 iterations are shown after 1000 iterations were dropped for transients.

Figure 12.

Colored recurrence plot for the three-neurons’ network’s simulation (left) and for the four-neurons’ network’s simulation (right).

Considering the recurrence structure, for the colored recurrence plots and the recurrence metrics for a radius of 0.1 (shown in Table 3), we find that the recurrence probability is high for both networks (near 95%), however, the average recurrence strength is low for both networks, which contrasts with the previous results for the two-neurons’ network, indicating the emergence of random-like topological signatures that tend to occur in chaotic dynamics.

Metrics	K = 3	K = 4
Number of lines with 100% recurrence	10	0
Number of lines with recurrence	4769	4747
Number of diagonals	4999	4999
Average recurrence strength	0.0357	0.0056
P[Recurrence]	0.9540	0.9496
P[100%recurrence|recurrence]	0.0021	0

Table 3.

Recurrence metrics for the recurrence matrices for the three- and four-neurons’ networks’ simulations obtained for a Euclidean radius of 0.1.

In this case, the four-neurons’ network is close to class 3, since it has high recurrence probability, low average recurrence strength and no line with 100% recurrence, while the three-neurons’ network has 10 lines with 100% recurrence, the three-dimensional projection also shows a random-like distribution of points. Since the sequence of quantum averages results from the application of the unitary quantum map, the sequence is actually deterministic, thus, the sequence is actually pseudorandom.

In the case of the three-neurons’ network, the dominant part of the recurrence structure is also random-like, indeed, the average recurrence strength is low, which means that, on average, the diagonals with recurrence are only 3.57% occupied with recurrence points and the probability of finding a diagonal with 100% recurrence from a random selection of diagonals with recurrence is just 0.21%, while 95.40% of the diagonals have recurrence points. This means the signatures are also of a predominant class 3 recurrence structure.

Also, for the three-neurons’ network, as the number of iterations is increased, unstable cycles emerge, with 100% recurrence lines appearing and then disappearing with the increase in the number of iterations, which are topological features of chaos. This might lead one to consider that the three-neurons’ network was a class 3 network with topological markers similar to those of a chaotic dynamics. However, it still has features of class 4 dynamics, with a resilient quasiperiodic skeleton appearing.

Indeed, considering a sequence of simulations from 2000 to 10,000 iterations of this network, in steps of 1000, and calculating the number of cycles, linked to the distances between 100% recurrence lines, for each case, we find that there are unstable cycles appearing as 100% recurrence lines and then disappearing with an increasing number of iterations, this includes a 79-period cycle and a 580-period cycle.

However, there are other cycles that appear as distances between 100% recurrence lines as the number of iterations is increased and that seem to rise with the total number of iterations, these cycles are 299, 388, 499, 617, 728 and 887, Figure 13 shows these cycles’ counts with the iterations increasing from 2000 to 10,000 in steps of 1000. Of notice, all of these resilient cycles tend to increase with the number of iterations in a ladder-like way, except for the 887 cycle which appears only one time and remains as a distance between two diagonals throughout the 10,000 iterations.

Figure 13.

Persistent cycles associated with distances between 100% recurrence lines for increasing number of iterations for the three-neurons’ network, with the number of iterations increasing from 2000 to 10,000 in steps of 1000 and r = 0.95.

Another important point is that the attractor for the three-neurons’ network, shown in Figure 10, is fractal with a box-counting dimension of 2.1498, using a four-decimal places approximation. Figure 14 shows the dimension estimation with a fitted line, the resulting R² is of 99.7101% and the p-value associated with the slope is 0.0. The box-counting dimension shows that the attractor in Figure 10 has a dimension that is between 2 and 3 dimensions.

Figure 14.

Estimation of box-counting dimension for the attractor in Figure 10.

The occurrence of fractal attractors in QNNs’ dynamics is not exclusive of the above three-neurons’ network, indeed, for the physical simulation of the previously studied two-neurons’ network, in the case of 10,000 iterations, the “ibm_q_experience,” which, as we saw produced topological markers similar to those of chaos, leads to a form of a sheet that mirrors the triketa shape but with some dispersion, this sheet also has a fractal structure.

In Figure 15, we show the resulting attractor along with the estimation of the box-counting dimension, in this case, the estimated dimension is 1.9225, the resulting R² is of 99.8504% and the p-value associated with the slope is 0.0.

Figure 15.

Simulation of the two-neurons’ quantum recurrent neural network on the “ibm_q_belem” device (left), for r = 0.550129597, with 5000 rounds sent for each iteration and 10,000 iterations kept after the first 1000 iterations were dropped for transients, box-counting dimension estimation (right).

In this case, unlike the three-neurons’ network, which had an estimated fractal dimension between 2 and 3, the estimated fractal dimension for the triketa-like attractor is between 1 and 2, which has the form of a sheet, due to the dispersion induced by noise. In [26] for a quantum stochastic process introduced for this network which included noise in the quantum map also led to a fractal dimension between 1 and 2 for the resulting attractor for the same parameter as the one of Figure 15, in that specific case, the estimated dimension was of 1.8927.

As an added note, while most of the points follow the attractor shape, environmental noise fluctuations also led to additional quantum errors that led to the occurrence of a few residual number of points outside the attractor, this can be observed as a straight line that occurred outside the triketa and that is visible in Figure 15 (left).

These points are not linked to initial transients. They occurred due to noise, corresponding to residual points that are registered in the topological signatures of the quantum averages as random fluctuations that did not lead to a deviation in the scaling of the attractor with the boxes falling close to a straight line in the log-log plot, showing evidence of the fractal scaling.

The fact that the dimension is between 1 and 2 is linked to the noise-like fluctuations which, as can be seen in Figure 15, leads actually to a sheet-like structure dispersion of points that follows the triketa, a sheet-like fractal structure as stated also occurred in [26] for the triketa attractor in the case of the stochastic quantum map.

In terms of quantum chaos research, these findings also reinforce other recent work in quantum chaos that also used sequences of averages and embedding methods having identified the emergence of attractors in open quantum systems with the application of topological data analysis methods [29].

The difference, in the case of QNNs, is that one does not start from a classically chaotic system, but rather from a quantum computing network, that is, one is dealing with networked computation.

The emergence of complex attractors in this last case results from the distributed computing dynamics itself. Furthermore, in the case of the unitary map, the dynamics is globally conservative but locally dissipative, that is, each neuron’s dynamics is described by a dissipative quantum map.

As studied in detail in [12], for the two-neurons’ recurrent network, the entanglement changes with time, leading to local neuron-level von Neumann entropy fluctuations, the study of these fluctuations reveals also complex patterns, that is, the complexity of the network’s dynamics extends to the von Neumann entropy dynamics itself. Also, for the global unitary operators, in the case of QNNs, the eigenstates are, in general, entangled states for the networks, which reflects the networked nature of the quantum system in question.

4. Conclusions

We reviewed the application of the concept of a unitary quantum map to a quantum recurrent neural network comprised of two neurons and a QNN with ring topology that includes a feedforward and a recurrent connection, finding, in both cases, complex dynamics with topological signatures typical of chaos and a quasiperiodic resilient skeleton, which characterizes the dynamics as a class 4 recurrence dynamics (edge of chaos).

The quasiperiodic skeleton was also observed in the simulation on IBM’s quantum device “ibm_q_experience,” but the quasiperiodic signatures are shown to be unstable, and revisited recurrently but occurring as broken long diagonals in the recurrence matrix corresponding to unstable cycles.

Class 3 (random dynamics) and class 2 (regular periodic or quasiperiodic dynamics) were also illustrated. Finally, for both the three-neurons’ cycle network and the two-neurons’ network’s simulation in the quantum device, fractal attractors were shown to be present.

The results reinforce and expand on the previous research into QNNs as complex dynamical systems, also showing how the application of quantum unitary maps and recurrence analysis methods can be effectively employed to study the dynamics of these networks, and how concepts from classical complexity research carry over to quantum complexity research, including the four classes of dynamics that occur for cellular automata, random Boolean networks and coupled map lattices, that can also be identified in the context of recurrence analysis with effectiveness in the characterization of the dynamics of these networks.

In terms of quantum computer science and technologies, these results have implications for the context of quantum distributed computation and the possible dynamical patterns that may occur.

In particular, in the case of class 4 dynamics, the intermixing of chaos-like topological signatures with interrupted diagonals and scattered dots in the recurrence structure, associated with unstable cycles, combined with resilient quasiperiodic signatures means that a form of dynamical memory may be possibly computationally exploited when dealing with quantum computing networks operating as quantum dynamical systems.

On the other hand, the emergence of random-like sequences of quantum averages with the increase in the number of neurons in the ring network demands further study.

From the standpoint of quantum chaos research, these networks may provide for additional models that may lead to complex attractors with fractal signatures and chaos-like recurrence signatures. In this case, the methods of quantum chaos research into unitary and dissipative quantum maps are already shown to be effective when dealing with the dynamics of QNNs.