Quantum Neural Networks - Computational Field Theory and Dynamics

In order to produce a general computational framework on which to discuss QuANNs, we need to consider a directed graph (digraph) structure for the $n$ neuron network extended with a Hilbert space structure and a set of conditional unitary operators, one operator for each neuron, with the unitary computation conditional on the input neural connections, formally this digraph can be defined as follows:

The computational dynamics of QuANNs can be addressed as a system of spinors on a network (Gonçalves, 2017). In the spinor model, when each neuron can have just two base computational patterns of activity, firing or nonfiring, corresponding to two energy levels, the Hilbert space for the full network is given by $n$ tensor product copies of the Hilbert space $\mathbb{H}_{2}$, which is spanned by the standard qubit basis $B_{2}=\left\{\left|0\right\rangle,\left|1\right\rangle\right\}$, where the ket vector $\left|0\right\rangle$ represents a nonfiring neural activity and $\left|1\right\rangle$ a firing neural activity, we are using the standard Dirac’s notation where the “ket” vector $\left|w\right\rangle$ represents a column vector and the “bra” vector $\left\langle w\right|$ is its conjugate transpose.

The Hilbert space for the network, in the case of a binary firing pattern is defined as $\mathbb{H}_{Net}=\mathbb{H}_{2}^{\otimes n}$ and spanned by the firing pattern basis (Gonçalves, 2017):

$$B_{2}^{\otimes n}=\left\{\left|s_{0},s_{1},...,s_{n-1}\right\rangle:s_{k}=0,1;k=0,1,...,n-1\right\}$$

(2)

A generalization of this model, for a finite number of logical states encoded in quantum a neural firing activity, is obtained by expanding the single neuron basis to $B_{l}=\left\{\left|0\right\rangle,\left|1\right\rangle,...,\left|l-1\right\rangle\right\}$, spanning the single neuron $l$-dimensional Hilbert space $\mathbb{H}_{l}$, which would lead to the neural network’s Hilbert space $\mathbb{H}_{Net}=\mathbb{H}_{l}^{\otimes n}$, spanned by the generalized firing pattern basis:

$$B_{l}^{\otimes n}=\left\{\left|s_{0},s_{1},...,s_{n-1}\right\rangle:s_{k}=0,1,...,l;k=0,1,...,n-1\right\},$$

(3)

when $l=2$, this last basis reduces to the standard two-level firing pattern basis. In this section, we work with the generalized formalism, since it contains the two-level as a special case. In section 3, the example is worked for $l=2$.

Now, the classical information states encoded in a neural firing dynamics, formalized as basis vectors in $B_{l}^{\otimes n}$, are not the only possible information states, we can also have a superposition of different neural firing patterns, which can be formalized as a normalized ket vector $\left|\psi\right\rangle$ on the network’s Hilbert space, expanded in the firing pattern basis as follows:

$$\left|\psi\right\rangle=\sum_{s_{0},s_{1},...,s_{n-1}}\psi(s_{0},s_{1},...,s_{n-1})\left|s_{0},s_{1},...,s_{n-1}\right\rangle$$

(4)

The squared modulus of each quantum amplitude $\psi(s_{0},s_{1},...,s_{n-1})$ provides for a statistical weight associated with each neural firing pattern, so that, for an ensemble of identical independent neural networks, the probability of the network exhibiting a specific firing pattern $s_{0},s_{1},...,s_{n-1}$ is given by $|\psi(s_{0},s_{1},...,s_{n-1})|^{2}$.

Different interpretations of quantum mechanics interpret this statistical measure differently, for instance, in an Everettian interpretation (Everett, 1957; 1973), each alternative firing pattern with a non-zero amplitude corresponds, in this case, to a projected dimension of systemic activity, with a projective intensity coinciding with the squared norm of the projection which, in turn, corresponds to the squared modulus of the amplitude $|\psi(s_{0},s_{1},...,s_{n-1})|^{2}$, so that, considering an ensemble of identical independent neural networks all described by the same multidimensional projective pattern (the same ket vector), the ensemble density operator description leads to a statistical measure (as a relative frequency) exactly coincident with $|\psi(s_{0},s_{1},...,s_{n-1})|^{2}$, which means that, choosing at random one network in the ensemble, the probability associated with that network’s exhibiting a given projected firing pattern coincides with the squared modulus of the quantum amplitudes, this point was addressed in Gonçalves (2015b).

Other interpretations of the probability link are possible (Cramer, 2016; Gonçalves, 2019a,b), we will not, however, assume here a specific interpretation quantum mechanics, with the formalism holding for different interpretations. The relevant point is the link to the probabilistic description since further on we will need to work with quantum averages for quantum fields on the network that rely on the above correspondence between the squared modulus of the amplitudes and statistical weights in an ensemble of identical independent networks.

Now, quantum computations on the network are formalized by way of the operations of the unitary gates in $\Theta\left(\mathbb{H}_{Net}\right)$ on the normalized ket vectors $\left|\psi\right\rangle$, considering this point, in order to introduce the concept of a unitary neural map, let $p:\left\{0,1,...,n-1\right\}\mapsto\left\{0,1,...,n-1\right\}$ represent a permutation of the neuron indices, then, we can formalize the concept of a unitary neural map as follows:

In this way, a unitary neural map is a product of the unitary gates in $\Theta\left(\mathbb{H}_{Net}\right)$ in an order corresponding to a neuron activation sequence that matches the permutation $p$. Given a unitary neural map, we can define the sequence of iterations of the map as:

$$\left|\psi(t)\right\rangle=F\left|\psi(t-1)\right\rangle,$$

(6)

which expands the unitary maps, worked in quantum chaos theory (Stöckmann, 2000; Braun, 2001), to the quantum neural computational setting.

In this case, $t$ represents the iteration step, and the QuANN’s dynamics is addressed in terms of a unitary quantum map which matches the quantum computing circuit described by equation (5). Given equation (6), and letting $\left|\psi(0)\right\rangle$ be the input for the network, we get the output at iteration t as:

$$\left|\psi(t)\right\rangle=F^{t}\left|\psi(0)\right\rangle$$

(7)

The above iteration scheme allows us to deal with QuANNs as quantum networked dynamical systems. Now, in order to better address the network’s dynamics we need to introduce a quantum neural computing field theory.

QuANNs, when addressed as quantum networked dynamical systems, lead to a bridge between quantum field theory and quantum computing, indeed, to address QuANNs as quantum networked dynamical systems implies the need to develop a quantum field theory on the network. Working with the formalism introduced in the previous subsection, we can develop a formalism for quantum fields on the network by introducing a general quantum neural field operator $\alpha(k)$, as a field operator on the network, formally:

Given the above definition, it follows that the operators commute, and, for any normalized ket vector on the network of the form:

$$\left|\psi\right\rangle=\left|\phi\right\rangle\otimes\left|\alpha_{s}\right\rangle\otimes\left|\varphi\right\rangle$$

(9)

where $\left|\phi\right\rangle\in\mathbb{H}_{2}^{\otimes(k-1)}$ and $\left|\varphi\right\rangle\in\mathbb{H}_{2}^{\otimes(n-k)}$, the following eigenvalue equation holds:

$$\alpha(k)\left|\psi\right\rangle=\alpha_{s}\left|\phi\right\rangle\otimes\left|\alpha_{s}\right\rangle\otimes\left|\varphi\right\rangle$$

(10)

for $s=0,1,...,l$.

Now, under the action of the quantum neural map, the field dynamics can be adddressed in the Heisenberg picture as follows:

$$\alpha(k,t)=F^{\dagger}\alpha(k,t-1)F$$

(11)

Therefore, in the Heisenberg picture, the field operators undergo the unitary evolution while the vectors stay at their initial configuration, so the unitary map’s iteration rule applies to the field operator rather than to the vector.

Assuming $\alpha(k,0)=\alpha(k)$, recursive application of the map leads to the following link:

$$\alpha(k,t)=(F^{\dagger})^{t}\alpha(k)(F)^{t}$$

(12)

Now, given the initial ket vector $\left|\psi(0)\right\rangle$, the quantum average of the field at iteration step $t$ and at neuron $k$, in the Heisenberg picture, is given by:

$$\bar{\alpha}(k,t)=\left\langle\psi(0)\left|\alpha(k,t)\right|\psi(0)\right\rangle$$

(13)

applying equation (12) we can transition from the Heisenberg to the Schrödinger picture since:

$$\bar{\alpha}(k,t)=\left\langle\psi(0)\left|\alpha(k,t)\right|\psi(0)\right\rangle=\left\langle\psi(0)\left|(F^{\dagger})^{t}\alpha(k)(F)^{t}\right|\psi(0)\right\rangle,$$

(14)

which leads to the equivalent result for the quantum mean field value at neuron $k$ obtained from the quantum averages calculated in the Schrödinger picture:

$$\bar{\alpha}(k,t)=\left\langle\alpha(k)\right\rangle_{t}=\left\langle\psi(t)\left|\alpha(k)\right|\psi(t)\right\rangle,$$

(15)

so that both pictures lead to equivalent results.

Taking the sequence of real-valued quantum averages $\left\langle\alpha(k)\right\rangle_{t}$, for an $n$ neuron network we can embed the sequence in $\mathbb{R}^{n}$, so that we get a sequence of points $\left\langle\alpha\right\rangle_{t}=\left(\left\langle\alpha(0)\right\rangle_{t},\left\langle\alpha(1)\right\rangle_{t},...,\left\langle\alpha(n-1)\right\rangle_{t}\right)$, which leads to a trajectory in $\mathbb{R}^{n}$.

Taking advantage of the Euclidean space metric topology of $\mathbb{R}^{n}$, for any sample path $\left\{\left\langle\alpha\right\rangle_{t}:t=t_{0},t_{0}+1,...,t_{0}+T-1\right\}$, we can calculate the distance matrix $\mathbf{S}$ that stores the distances for each pair of points, with entries defined as:

$$\mathbf{S}_{t,t^{\prime}}=\left\|\left\langle\alpha\right\rangle_{t}-\left\langle\alpha\right\rangle_{t^{\prime}}\right\|,$$

(16)

where $\left\|.\right\|$ stands for the Euclidean metric defined on $\mathbb{R}^{n}$.

The distance matrix is symmetric of rank $T$, with the main diagonal entries all equal to zero and it contains the information about recurrences in a sample trajectory.

Taking advantage of the Euclidean metric topology of $\mathbb{R}^{n}$, the pattern of recurrences can be extracted from the distance matrix using a closed $\delta$-neighborhood structure, which leads to the binary recurrence matrix $\mathbf{R}_{\delta}$ with entries:

$$\mathbf{R}_{t,t^{\prime}}^{\delta}=\begin{cases}0,&\left\|\left\langle\alpha\right\rangle_{t}-\left\langle\alpha\right\rangle_{t^{\prime}}\right\|>\delta\\ 1,&\left\|\left\langle\alpha\right\rangle_{t}-\left\langle\alpha\right\rangle_{t^{\prime}}\right\|\leq\delta\end{cases}$$

(17)

The matrix $\mathbf{R}_{\delta}$ is, thus, binary and symmetric with an entry containing the value 1 when two points in the sample trajectory are not apart from each other more than $\delta$, in a closed Euclidean neighborhood, which is the definition of a recurrence event, of course, given this property the diagonal of the matrix, which corresponds to the cases where $t=t^{\prime}$, is comprised only of 1s.

We are using a closed neighborhood because it allows us to identify fully periodic dynamics since, when the dynamics is fully periodic, if the radius is set equal to zero, all diagonal lines parallel to the main diagonal corresponding to the period in question will have a value of 1 in each matrix entry, otherwise the value will be 0. When the dynamics is not periodic there is a cutoff radius below which we do not get any recurrences. In the nonperiodic case, diagonal lines with a value of 1 in each entry, at a given radius, correspond to a periodic or quasiperiodic skeleton that the dynamics revisits, these are 100% recurrence lines, that is, lines where the percentage of points (recurrence matrix entries) that are recurrence points is equal to 100%. These lines are particularly important in identifying periodic, quasiperiodic dynamics and, even, chaotic dynamics.

For the analysis of sequences of quantum averages extracted from the iterations of the quantum neural map we calculate the following three recurrence measures (Gonçalves, 2017, 2020):

• •

  The recurrence probability: this is the number of diagonals below the main diagonal with recurrence points, divided by the total number of diagonals below the main diagonal in the recurrence matrix, since the recurrence matrix is symmetric only the diagonals below the main diagonal are counted, in this case, this metric provides for the probability of finding a line with recurrence, in a random selection of diagonal lines below the main diagonal.

• •

  The recurrence strength: this is the sum of the number of points that fall within a distance no greater than the radius in each diagonal, below the main diagonal, divided by the total number of diagonals below the main diagonal with recurrence, this measure evaluates how strong on average the recurrence is, if all lines with recurrence had 100% recurrence, for the radius chosen, then this number would be equal to 1, the lower this statistic is, that is, the closer to zero it is, the more interrupted the diagonals there are, which occurs for stochastic systems and also for dererministic chaotic dynamics.

• •

  The conditional 100% recurrence probability: this is the probability that a diagonal line with recurrence has 100% recurrence, for the radius chosen.

These recurrence statistics can be used alongside the visual analysis of a recurrence plot that plots the recurrence matrix (black and white plot) or the distance matrix (colored recurrence plot), in the black and white plot, which we will use in the present work, a point is painted in black if it is a recurrence point and white if not, this plot is a key element in addressing recurrence properties of both low and high-dimensional dynamical systems (Gonçalves, 2017, 2020; Eckmann et al., 1987; Gao and Cai, 2000), and have been employed frequently in the analysis of neural network models as well as in studies on brainwave dynamics (Thomasson et al., 2002; Acharya et al., 2011; Aladag et al., 2010; Lopes et al., 2020), in Gonçalves (2017, 2020) it was also applied to QRNN simulations.

In the present work, we will apply it to the study of the behavior of the quantum neural activity field with a two-level neural firing pattern, which can be built from the fermionic raising and lowering operators on the Hilbert space $\mathbb{H}_{2}$, these raising and lowering operators are defined as:

$$a=\left|0\right\rangle\left\langle 1\right|=\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right),\;a^{\dagger}=\left|1\right\rangle\left\langle 0\right|=\left(\begin{array}[]{cc}0&0\\ 1&0\end{array}\right)$$

(18)

For the anticommutator $\left\{A,B\right\}=AB+BA$, these operators obey the following relations:

$$\left\{a,a^{\dagger}\right\}=1_{2}=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right)$$

(19)

$$\left\{a,a\right\}=\left\{a^{\dagger},a^{\dagger}\right\}=\left(\begin{array}[]{cc}0&0\\ 0&0\end{array}\right)$$

(20)

$$a^{\dagger}a\left|0\right\rangle=0\left|0\right\rangle$$

(21)

$$a^{\dagger}a\left|0\right\rangle=1\left|1\right\rangle$$

(22)

From the above equations, we can introduce the special case of a quantum neural field operator which is the neural activity field operator $N$ on the network defined as:

$$N(k)=1_{2}^{\otimes(k-1)}\otimes a^{\dagger}a\otimes 1_{2}^{(n-k)},$$

(23)

which has the eigenvalue spectrum obeying the following equation:

$$N(k)\left|...,s_{k},...\right\rangle=s_{k}\left|...,s_{k},...\right\rangle,$$

(24)

thus, considering the neural firing basis, at each neuron, the field operator yields a value of 0 when the corresponding neuron is not firing (not active) and of 1 when it is firing (active).

For the sequence $\left|\psi(t)\right\rangle$ with the expansion:

$$\left|\psi(t)\right\rangle=\sum_{s_{0},s_{1},...,s_{n-1}}\psi_{t}(s_{0},s_{1},...,s_{n-1})\left|s_{0},s_{1},...,s_{n-1}\right\rangle,$$

(25)

the quantum averages for the neural activity field at each neuron coincide with the squared modulus of the quantum amplitudes, which coincide, in turn, with the statistical measure for the neuron to be active (firing):

	$\displaystyle\left\langle N(k)\right\rangle_{t}=\left\langle\psi(t)\left|N(k)\right|\psi(t)\right\rangle=$		(26)
	$\displaystyle=0\sum_{s_{0},...,s_{k-1},s_{k+1},...,s_{n-1}}\left|\psi(...,s_{k}=0,...)\right|^{2}+$
	$\displaystyle+1\sum_{s_{0},...,s_{k-1},s_{k+1},...,s_{n-1}}\left|\psi(...,s_{k}=1,...)\right|^{2}$
	$\displaystyle=\sum_{s_{0},...,s_{k-1},s_{k+1},...,s_{n-1}}\left|\psi(...,s_{k}=1,...)\right|^{2}$

Using the $\mathbb{R}^{n}$ embedding $\left\langle N\right\rangle_{t}=\left(\left\langle N(0)\right\rangle_{t},\left\langle N(1)\right\rangle_{t},...,\left\langle N(n-1)\right\rangle_{t}\right)$, $\left\langle N\right\rangle_{t}$ corresponds to the configuration of the mean neural activity field at each neuron. As a final point regarding the quantum neural activity field, it is relevant to stress the relation between the neural activity field operator and the local (neuron level) neural firing energy Hamiltonian operators, formally these Hamiltonians can be defined as:

$$H_{k}=\omega\hbar N(k)$$

(27)

where $\omega=2\pi f$ with $f$ corresponding to a neural firing frequency in Hertz. Given the above equations, the energy eigenvalue spectrum for the neuron is given by:

$$H_{k}\left|...,s_{k},...\right\rangle=\omega\hbar s_{k}\left|...,s_{k},...\right\rangle,$$

(28)

so that the energy is zero when the neuron is nonfiring and $\omega\hbar$ when the neuron is firing.

These operators commute and the total neural firing energy for the network is given by:

$$H=\sum_{k=0}^{n-1}H_{k}=\sum_{k=0}^{n-1}\omega\hbar N(k)$$

(29)

with the eigenvalue spectrum:

$$H\left|s_{0},s_{1},...,s_{n-1}\right\rangle=\omega\hbar\sum_{k=0}^{n-1}s_{k}\left|s_{0},s_{1},...,s_{n-1}\right\rangle$$

(30)

Having addressed the field theory, we now address the issue of the local (neuron-level) von Neumann entropy dynamics.

Formally, in a QuANN, due to the networked nature of the quantum computation, the quantum dynamics at the local neuron level leads to entanglement between the neurons’ quantum dynamics, which means that, locally, the neuron operates as an open quantum system. In general, with entanglement, we cannot describe the neuron in terms of a vector, but rather by a local density operator, tracing out the rest of the network’s degrees of freedom, which lead to the local (neuron-level) densities:

$$\rho_{k}(t)=Tr_{k}\left(\left|\psi(t)\right\rangle\left\langle\psi(t)\right|\right)$$

(31)

The quantum information dynamics of the network, at the local neuron level, can be addressed by calculating the von Neumann entropy with a binary basis, indeed considering the general formula:

$$S(\rho)=-Tr\left(\rho\log_{2}\rho\right),$$

(32)

which is equal to $0$ for a pure density, that is, a density given by a projector $\left|\varphi\right\rangle\left\langle\varphi\right|$, with $\left|\varphi\right\rangle$ being a normalized vector, we can study the local neuron-level entropy dynamics employing similar recurrence analysis techniques as those introduced in the previous subsection, indeed, calculating the local von Neumann entropies for each neuron in binary basis we get the entropy sequences:

$$S_{k}(t)=S(\rho_{k}(t))=-Tr\left(\rho_{k}(t)\log_{2}\rho_{k}(t)\right)$$

(33)

that we can embed in $\mathbb{R}^{n}$, $\left(S_{0}(t),S_{1}(t),...,S_{n-1}(t)\right)$, and to which we can apply the recurrence analysis techniques in order to analyze the main dynamics for the local entropies.

When the input for the network is given by an initial pure density (a projector) the entropy for the full network is zero and remains zero under the evolution of the quantum neural map, since the map is unitary and therefore does not change the global entropy, the local neuron-level entropies, however, are not necessarily zero, due to the entanglement dynamics associated with quantum networked computation.

Since the neuron-level densities are usually not equal to a projector, there is ususally some level of entropy fluctuations at the local neuron-level, indeed, the neuron-level networked dynamics tends to a far-from-equilibrium dynamics that does not stabilize in a fixed maximum entropy level, that is, while, in certain iterations, the local neuron level’s entropy can be led to close to the maximum entropy this is not always the case, and there can be entropy reductions to close to zero entropy, followed by entropy increases. In the case of class 4 dynamics, for instance, we can also get class 4 dynamics at the level of the entropy fluctuations themselves, with the entropy fluctuating in a fluctuation band that is not maximal.

Having introduced the main concepts and framework, we now address the example of the most elementary family of quantum recurrent neural networks (QRNNs) the QRNNs comprised of two neurons characterized by two-level neural firing activity (nonfiring and firing with a fixed energy level).

The most elementary QRNN is a network comprised of two neurons characterized by a two-level neural firing activity, which, following the previous section’s formalism, is defined by:

$$G_{QRNN}=\left(\{n_{0},n_{1}\},\{(n_{0},n_{1}),(n_{1},n_{0})\},\mathbb{H}_{2}^{\otimes 2},\Theta\left(\mathbb{H}_{2}^{\otimes 2}\right)\right)$$

(34)

The neural firing pattern basis for this network is given by:

$$B_{2}^{\otimes 2}=\left\{\left|0,0\right\rangle,\left|0,1\right\rangle,\left|1,0\right\rangle,\left|1,1\right\rangle\right\}$$

(35)

The set of operators $\Theta\left(H_{Net}\right)=\left\{U_{0},U_{1}\right\}$ is, in turn, defined such that each operator is a conditional unitary operator that follows the neural connections, namely:

$$U_{0}=U_{0,0}\otimes\left|0\right\rangle\left\langle 0\right|+U_{0,1}\otimes\left|1\right\rangle\left\langle 1\right|$$

(36)

$$U_{1}=\left|0\right\rangle\left\langle 0\right|\otimes U_{1,0}+\left|1\right\rangle\left\langle 1\right|\otimes U_{1,1},$$

(37)

where $U_{r,s}$, for $r,s=0,1$, are elements of the unitary group U(2).

From the above equations, it follows that, under the connection $(n_{0},n_{1})$, the unitary gate $U_{1,1}$ is applied at the second neuron when the first neuron is firing, while the unitary gate $U_{1,0}$ is applied at the second neuron when the first neuron is nonfiring. In the reverse direction, a similar conditional computation is performed, so that when the second neuron is firing the computation at the first neuron is given by the operator $U_{0,1}$ while, when the second neuron is nonfiring, the computation at the first neuron is given by the operator $U_{0,0}$.

Now, for a specific operator set $\Theta\left(\mathbb{H}_{Net}\right)$, there are two possible activation orders for a unitary neural map $U_{1}U_{0}$ or $U_{0}U_{1}$, the second alternative activates first the connection $(n_{0},n_{1})$ and then the feedback (recurrent) connection $(n_{1},n_{0})$, the first alternative activates first the connection $(n_{1},n_{0})$ and then the feedback (recurrent) connection $(n_{0},n_{1})$. In what follows, we will be working with the activation sequence $U_{0}U_{1}$.

The neural map that we will be analyzing has the following structure:

$$F=U_{0}U_{1}$$

(38)

$$U_{1}=\left|0\right\rangle\left\langle 0\right|\otimes I+\left|1\right\rangle\left\langle 1\right|\otimes U_{r}$$

(39)

$$U_{0}=I\otimes\left|0\right\rangle\left\langle 0\right|+U_{r}\otimes\left|1\right\rangle\left\langle 1\right|$$

(40)

$$U_{r}=\cos\left(\frac{r\pi}{2}\right)\left(\left|0\right\rangle\left\langle 0\right|+\left|1\right\rangle\left\langle 1\right|\right)+\sin\left(\frac{r\pi}{2}\right)\left(-\left|0\right\rangle\left\langle 1\right|+\left|1\right\rangle\left\langle 0\right|\right)$$

(41)

In matrix representation, the two operators $U_{1}$ and $U_{0}$ are given by:

$$U_{1}=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\left(\frac{r\pi}{2}\right)&-\sin\left(\frac{r\pi}{2}\right)\\ 0&0&\sin\left(\frac{r\pi}{2}\right)&\cos\left(\frac{r\pi}{2}\right)\end{array}\right)$$

(42)

$$U_{0}=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&\cos\left(\frac{r\pi}{2}\right)&0&-\sin\left(\frac{r\pi}{2}\right)\\ 0&0&1&0\\ 0&\sin\left(\frac{r\pi}{2}\right)&0&\cos\left(\frac{r\pi}{2}\right)\end{array}\right)$$

(43)

which leads to the following matrix structure of the neural map $F$

$$F=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&\cos\left(\frac{r\pi}{2}\right)&-\sin^{2}\left(\frac{r\pi}{2}\right)&-\sin\left(\frac{r\pi}{2}\right)\cos\left(\frac{r\pi}{2}\right)\\ 0&0&\cos\left(\frac{r\pi}{2}\right)&-\sin\left(\frac{r\pi}{2}\right)\\ 0&\sin\left(\frac{r\pi}{2}\right)&\sin\left(\frac{r\pi}{2}\right)\cos\left(\frac{r\pi}{2}\right)&\cos^{2}\left(\frac{r\pi}{2}\right)\end{array}\right)$$

(44)

Considering the Schrödinger picture iteration:

$$\left|\psi(t)\right\rangle=F\left|\psi(t-1)\right\rangle=U_{0}U_{1}\left|\psi(t-1)\right\rangle$$

(45)

and assuming the two expansions:

$$\left|\psi(t-1)\right\rangle=\sum_{s_{0},s_{1}}\psi_{t-1}(s_{0},s_{1})\left|s_{0},s_{1}\right\rangle$$

(46)

$$\left|\psi(t)\right\rangle=\sum_{s_{0},s_{1}}\psi_{t}(s_{0},s_{1})\left|s_{0},s_{1}\right\rangle$$

(47)

the amplitudes at $t-1$ and at $t$ are linked by:

$$\psi_{t}(0,0)=\psi_{t-1}(0,0)$$

(48)

	$\displaystyle\psi_{t}(0,1)=\cos\left(\frac{r\pi}{2}\right)\psi_{t-1}(0,1)-$		(49)
	$\displaystyle-\sin^{2}\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,0)-\sin\left(\frac{r\pi}{2}\right)\cos\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,1)$

$$\psi_{t}(1,0)=\cos\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,0)-\sin\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,1)$$

(50)

	$\displaystyle\psi_{t}(1,1)=\sin\left(\frac{r\pi}{2}\right)\psi_{t-1}(0,1)+$		(51)
	$\displaystyle+\cos\left(\frac{r\pi}{2}\right)\sin\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,0)+\cos^{2}\left(\frac{r\pi}{2}\right)\psi_{t-1}(1,1)$

For $r=0$, $\psi_{t}(s_{0},s_{1})=\psi_{t-1}(s_{0},s_{1})$, so that we get a class 1 dynamics, that is, a fixed point, since $\left|\psi(t)\right\rangle=\left|\psi(t-1)\right\rangle$, so that the following holds:

$$\left|\psi(t)\right\rangle=\left|\psi(0)\right\rangle$$

(52)

for every normalized initial ket vector. With respect to the neural activity field, introduced in the previous section, we get the quantum averages:

$$\left\langle N(0)\right\rangle_{t}=\left|\psi_{0}(1,0)\right|^{2}+\left|\psi_{0}(1,1)\right|^{2}$$

(53)

$$\left\langle N(1)\right\rangle_{t}=\left|\psi_{0}(0,1)\right|^{2}+\left|\psi_{0}(1,1)\right|^{2}$$

(54)

which leads to a class 1 dynamics when the embedding in $\mathbb{R}^{2}$ is performed, since the dynamical point for the mean field is a fixed point:

$$\left\langle N\right\rangle_{t}=\left(\left|\psi_{0}(1,0)\right|^{2}+\left|\psi_{0}(1,1)\right|^{2},\left|\psi_{0}(0,1)\right|^{2}+\left|\psi_{0}(1,1)\right|^{2}\right)$$

(55)

On the other hand, when $r=1$, the dynamics is class 2. The reason for this requires a closer look at the iteration steps, in this case from $\left|\psi(t-1)\right\rangle$ to $\left|\psi(t)\right\rangle$ we get the transition sequence:

	$\displaystyle\left|\psi(t)\right\rangle=\psi_{t-1}(0,0)\left|0,0\right\rangle-\psi_{t-1}(1,0)\left|0,1\right\rangle-$		(56)
	$\displaystyle-\psi_{t-1}(1,1)\left|1,0\right\rangle+\psi_{t-1}(0,1)\left|1,1\right\rangle$

	$\displaystyle\left|\psi(t+1)\right\rangle=\psi_{t-1}(0,0)\left|0,0\right\rangle+\psi_{t-1}(1,1)\left|0,1\right\rangle-$		(57)
	$\displaystyle-\psi_{t-1}(0,1)\left|1,0\right\rangle-\psi_{t-1}(1,0)\left|1,1\right\rangle$

$$\left|\psi(t+2)\right\rangle=\left|\psi(t-1)\right\rangle$$

(58)

Therefore, we have a 3-cycle, that is, after three iterations, the ket vector returns to the configuration in which it was three iterations before, which leads to the embedded sequence:

	$\displaystyle\left\langle N\right\rangle_{t}=$		(59)
	$\displaystyle=\left(\left|-\psi_{t-1}(1,1)\right|^{2}+\left|\psi_{t-1}(0,1)\right|^{2},\left|-\psi_{t-1}(1,0)\right|^{2}+\left|\psi_{t-1}(0,1)\right|^{2}\right)$

	$\displaystyle\left\langle N\right\rangle_{t+1}=$		(60)
	$\displaystyle=\left(\left|-\psi_{t-1}(0,1)\right|^{2}+\left|-\psi_{t-1}(1,0)\right|^{2},\left|\psi_{t-1}(1,1)\right|^{2}+\left|-\psi_{t-1}(1,0)\right|^{2}\right)$

$$\left\langle N\right\rangle_{t+2}=\left\langle N\right\rangle_{t-1}$$

(61)

While only class 1 and 2 dynamics are possible for these two parameters, more complex dynamics arise when $0<r<1$, for different initial conditions, as we now show.

In figure 1, we show the simulated sequences for the mean neural activity field values at each neuron, $\left\langle N(0)\right\rangle_{t}$ and $\left\langle N(1)\right\rangle_{t}$, when $r=0.0005$, for $\left|\psi(0)\right\rangle=\left|+\right\rangle\otimes\left|+\right\rangle$, with $\left|+\right\rangle=\left(\left|0\right\rangle+\left|1\right\rangle\right)/\sqrt{2}$.

In this case, we get an emergent pattern which follows a sinusoidal curve for both neurons, with a high level of synchronization in the sinusoidal pattern between the two neurons.

It is important to stress that the plot in figure 1 is a actually a scatterplot, the appearance of a continuous periodic curve is due to the close proximity of the dots, which means that the sinusoidal pattern holds as an emergent pattern that appears in the dots’ sequence for each iteration.

The sequence of dots is actually quasiperiodic, since for a recurrence radius of 0 we get a recurrence probability equal to 0, however, the quasiperiodicity is following an emergent continuous periodic curve, that is, we get an emergent brainwave-like pattern that determines the dynamics at the level of the mean field at each neuron which, while being quasiperiodic, follows, in fact, an emergent continuous periodic shape. Since the two emergent periodic curves are synchronized, we get a pattern that is like an emergent synchronized neural wave driving the network’s dynamics.

While $r=0$ leads to a fixed point dynamics, increasing the parameter to $r>0$, for low values of this parameter we get an emergent “brainwave” for different initial conditions.

In figure 2, we show the corresponding sequences of mean neural activity field values at each neuron for the initial conditions $\left|\psi(0)\right\rangle=\left|0,1\right\rangle$, $\left|\psi(0)\right\rangle=\left|1,0\right\rangle$ and $\left|\psi(0)\right\rangle=\left|1,1\right\rangle$, and $r=0.0005$, in all three cases we get an emergent pattern that is like a continuous wave, with a high correlation between the mean field dynamics, however, for the first two cases, the sample correlation is negative, which corresponds to a neural inhibitory dynamics, while for the last case it is positive, corresponding to a neural reinforcing dynamics. In the inhibitory cases this leads to an emergent negatively correlated excitation-relaxation cycle between the two neurons with a plateau in the excited phase and a smooth but faster relaxation phase.

The dynamics, shown in figures 1 and 2 are also characterized by emergent continuous periodic dynamics driving the neurons’ entanglement dynamics with a signature at the von Neumann entropy levels, which is different from Everett’s quantum automaton dynamics where the entanglement leads to a fixed branching entanglement pattern (Everett, 1957, 1973), the difference is that Everett’s quantum automata theory is linked to a formalization of a laboratory-based theory of quantum measurement, and the measurement is a feedforward single interaction between the observer and the observed system plus apparatus, in this sense at the level of the observer’s description there is an entanglement-related local decoherence (Tegmark, 2000; Joos et al., 2003).

In QuANNs there is, however, no fixed/stable decoherence identifiable as a von Neumann entropy rise to a fixed local maximally mixed density, instead, due to the networked interaction, the entropy fluctuates, so we do not have the same type of framework as is assumed in Everett (1957, 1973) regarding quantum cognition, which, as stated, is an expansion from a laboratory-based framework focused on a specific type of interaction which is a quantum measurement, instead, for QuANNs, we need to consider the issue of complex entropy dynamics associated with interacting quantum systems in order to address quantum networked processing at the local neuron level, where each neuron’s computational dynamics operates far-from-equilibrium.

In order to better understand this point, let us consider the local neuron description. In the case of the above network, the local neuron-level entropy, like the mean neural activity field, is also driven by an emergent periodic continuous dynamics and exhibits a complex dynamical relation with respect to the mean field values as we show in figure 3, for $\left|\psi(0)\right\rangle=\left|+\right\rangle\otimes\left|+\right\rangle$. In this case, the initial entropy for each neuron was zero, since the input vector for the network was separated into a tensor product of two ket vectors. As the iterations proceed, we find that even though the entropy sequence is discrete, like the mean neural activity field values, it follows a continuous smooth curve which is periodic in pattern, the entropy fluctuations range from near zero entropy which corresponds to a pure density, to near 1, which corresponds to a maximum entropy level associated with a depolarized mixed density, the mean entropy being around 0.63 bits, as shown in table 1.

There is also a relation between the mean neural activity field at each neuron and the respective entropy, in this case, we get an eye-like structure, such that when the mean field value at the neuron is near 0.5, the entropy is either near 0 or near 1. For higher values of the mean neural activity field, the dispersion in entropy fluctuations diminish converging on a high but non-maximal entropy value for the maximum mean neural activity field value, which also corresponds to maximum mean energy at the neuron level.

Throughout the network’s iterations each neuron is operating as an open quantum computing system with fluctuations in entropy that can range from a value close to zero to a value close to the maximum entropy level, which illustrates the point that we do not get the basic fixed decoherence pattern that is addressed in the context of quantum measurement theory (Everett, 1957, 1973; Tegmark, 2000; Joos et al., 2003). This is also the case for the entropy dynamics associated with the initial conditions of figure 2.

Emergent smooth periodic curves driving the mean neural activity field and entropy values are not the only patterns that are present in this network’s dynamics. When the network is initialized for $\left|\psi(0)\right\rangle=\left|+\right\rangle\otimes\left|+\right\rangle$, as $r$ is increased, in the region of periodic emergent brainwave-like patterns, the wavelength of the resulting neural waves decreases, so that for $r$ very near 0, the wavelength is longer, but, as $r$ is increased, the wavelength decreases, as well as the sample correlation. As shown in Gonçalves (2020), these correlations eventually transition from from positive to negative values, with two main shapes characterizing the dynamics of the mean neural activity field at neuron $n_{0}$ versus at neuron $n_{1}$, one with a triketa-like shape¹¹1It can be obtained from intersections of three ellipsoids. and the other with the shape of a trifolium, with the transition from the triketa to the trifolium being progressive as $r$ is increased, exhibiting complex quasiperiodic dynamics. Two examples of these attractors are shown in figure 4.

The attractor on the left corresponds to a value of $r$ where the two neurons show a close to zero correlation²²2In this case, the sample correlation for figure 4’s simulation is 1.85638176e-05., while, for the attractor on the right, the correlation is negative³³3In this case, the sample correlation for figure 4’s simulation is -0.49961075.. While positive correlation indicates a dominance of an excitatory dynamics between the two neurons, and a negative correlation indicates the presence of an inhibitory relation, in the close to zero and negative correlation region, unlike the positive correlation region, the dynamics is characterized by complex quasiperiodic structures, mainly the triketa which is the dominant geometrical structure, with the trifolium only emerging for $r$ greater than 0.99. It is important to stress that even though we get the same triketa shape, different values of the parameter $r$ lead to different complex quasiperiodic patterns with different recurrence structures as systematized in Gonçalves (2020).

The nonlinear relation between the sequences of mean field values at each neuron emerges as $r$ increases, and the triketa becomes the dominant emergent dynamical geometry both for near zero correlation and negative correlation. In this sense, the linear correlation measure becomes misleading as measure of the relation between the mean neural activity field at each neuron.

This point becomes particularly relevant when considering the near zero correlation case, since, while there is no dominant excitatory or inhibitory dynamics, there is still a nonlinear relation between the neurons which is characterized by a dynamics that exhibits, in the recurrence structure, both signatures of dynamics with multiple periodicities characterized by long resilient diagonals with 100% recurrence as well as broken diagonals and isolated dots that ususally appear in stochastic or chaotic systems.

The dynamics is actually not chaotic nor periodic, instead, it is closer, in regards to the recurrence structure, to the edge of chaos, such types of dynamics have been identified in other QuANN models including recurrent networks interacting with an environment (Gonçalves, 2017), as reviewed in the introduction. The recurrence plots for the near zero correlation case are shown in figure 5, illustrating this point.

As can be seen in figure 5, we no longer have the sequence of mean field values following an emergent continuous periodic pattern, instead, we get a dispersion in the form of a cloud of points, with the complex pattern only being visible for higher number of iterations. The edge of chaos recurrence signatures show up upon an analysis of the recurrence quantification measures.

In table 2, we show the results from a recurrence analysis on the ordered pairs $\left(\left\langle N(0)\right\rangle_{t},\left\langle N(1)\right\rangle_{t}\right)$, for 20,000 iterations after 10,000 initial iterations being dropped for transients, with $r=0.550129597$.

The table shows how the recurrence statistics change with the increasing radius, for radius 0, the recurrence probability is zero since there are no diagonals, below the main diagonal, with recurrence, which is indicative of an nonperiodic dynamics, the recurrence probability and the mean recurrence strengths rise with the radius reaching a 94.1097% recurrence probability for the radius 0.1, the probability of finding a diagonal line with 100% recurrence conditional on the diagonal having recurrence points, however, rises initially with the radius but then starts dropping, this indicates that the new recurrence points that appear with the rise in radius are predominantly isolated and clustered dots more characteristic of a noisy recurrence structure, in this case, there is a resilient quasiperiodic skeleton of lines with 100% recurrence, but the remaining recurrence points do not tend to produce a 100% recurrence, which leads to a mix of an emergent stochastic-like recurrence pattern intermixed with a few long diagonals, characterizing a complex quasiperiodic dynamics, this is characteristic of edge of chaos signatures.

To evaluate the periodicities involved, we can calculate the distances between the 100% recurrence lines, these distances provide for an evaluation of the quasiperiodic skeleton, in this case, for a radius of 0.1, we find the present of three cycles, a 5 iterations cycle, a 21 iterations cycle and a 26 iterations cycle, the fact that there are different cycles present is characteristic of quasiperiodic dynamics, in this case, there are two dominant cycles, for a radius of 0.1, the first is the 21 iterations cycle, which occurs 836 times, followed by the 5 iterations cycle which occurs 352 times, by contrast, the 26 iterations cycle only appears 25 times, as shown in table 3.

Now, if we consider the von Neumann entropy dynamics, for $r=0.550129597$, we need to consider the sequences of entropy values associated with neurons $n_{0}$ and $n_{1}$, respectively, $S_{0}(t)$ and $S_{1}(t)$, calculated from the respective neuron-level reduced densities, as shown in figure 6 we also get a an nonperiodic dynamics, with multiple diagonals but also a high number of interrupted diagonals and isolated clusters, the power spectrum has multiple spikes at the high frequency level for both neurons entropy sequences.

There is, in this case, a predominant cycle which corresponds to a 47 iterations cycle that occurs 248 times with respect to the 100% recurrence lines at a radius of 0.1, the second cycle, in importance, is a 68 iterations cycle, that occurs 88 times and a 115 iterations cycle, that occurs 20 times, as shown in table 4.

Given the above results, we do not have, again, the stabilization in a fixed entropy regime, the entropy fluctuations exhibit a complex dynamics, associated with changes in entanglement levels and quantum amplitudes.

As shown in table 5, the lowest entropy value is close to zero, while the highest entropy value is around 0.819 bits, no neuron is ever led to the maximum entropy level associated with a depolarized mixed density, and the mean entropy is around 0.498, so each neuron is operating as a far-from-equilibrium open quantum system.

For $r=0.999$, the entropy also has a complex pattern, as shown in figure 7.

While, in figure 7, the entropy series values follow an emergent smooth complex periodic curve, the process is very nontrivial, indeed, working with the power spectrum, we find the presence of a power law decay in the spectrum obtained for the von Neumann entropy sequences $S_{0}(t)$ and $S_{1}(t)$ with an additional rise for a significant peak at the high frequency scale, indicating the presence of a strong periodicity at that higher frequency. The recurrence plot shows elements of a periodic skeleton but also local neighborhood fluctuations in the sequence.

As shown in table 6, the interval of values for the entropy fluctuations is similar to the previous one, the lowest entropy value is close to zero, the highest entropy value is around 0.843 bits, the mean entropy is, however, lower that in the previous case, around 0.261 bits, again, no neuron is ever led to the maximum entropy level and each neuron is computing far from the maximum entropy level.