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{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{19.91684pt}{-51.21504pt}\pgfsys@lineto{19.91684pt}{-42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{90.89847pt}{-64.75343pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{Classical channel}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{82.51282pt}{-56.90552pt}\pgfsys@lineto{20.54015pt}{-23.10226pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.8779}{0.47885}{-0.47885}{-0.8779}{20.54015pt}{-23.10228pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{89.14844pt}{-34.35623pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{Quantum channel}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{82.51282pt}{-25.6073pt}\pgfsys@lineto{66.07646pt}{-17.38933pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.89442}{0.4472}{-0.4472}{-0.89442}{66.07646pt}{-17.38934pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}$ (12)

In a LOCC protocol, one channel is considered “classical” while the other is
considered “quantum”; however, this language masks the fact that both channels
are physical. As pointed out in [104], all media supporting classical
communication are physical, and interactions with these media are always local
measurements or preparations. Hence the two channels in a LOCC protocol are
physically equivalent – both are TQFTs implemented by $B$ – although their
conventional semantics are different.

Diagram (12) can, clearly also represent externally-mediated communication
between any two functional components of a system, e.g., macromolecular
pathways within a cell or functional networks within a brain. We show in [98]
that whenever $Q_{1}$ and $Q_{2}$ are deployed by distinct – technically,
separable or mutually decoherent – “observers” or “systems,” they fail to
commute, i.e., the commutator $[Q_{1},Q_{2}]=Q_{1}Q_{2}-Q_{2}Q_{1}\geq h/2$,
where again $h$ is Planck’s constant. As shown in [57], Theorem 3.4 using the
CCCD representation, non-commutativity of QRFs induces quantum contextuality,
i.e., dependence of measurement results on “non-local hidden variables” that
characterize the measurement context [105, 106, 107]. In the current context,
such hidden variables characterize the action of $H_{B}$ on $\mathscr{B}$,
affecting what $A$ will observe next in every cycle of $A$-$B$ interaction.

As shown in [63], such context dependence can, in principle, be captured
classically if sufficient measurements of the context can be implemented. Such
measurements would, however, have to access all of $B$. The existence of an MB
prevents such access; in the current setting, $A$ has access to $B$ only via
$\mathscr{B}$. The finite energetic cost of measurement, and consequent
requirement for a thermodynamic sector $F$, prevents measurement even of all
of $\mathscr{B}$ by any finite physical system. Hence, we can expect physical
systems, including all biological systems, to employ only local context-
dependent control to switch between mutually non-commuting (sets of) QRFs. How
context switches implemented by QRF switches induce evolution, development and
learning was introduced in [22]. Some specific of context switching will be
discussed §5.

##  3 Tensor network representation of control flow

###  3.1 Tensor networks and holographic duality

Entanglement and quantum error correction, two concepts developed in quantum
information theory, have been proved to have a fundamental role in unveiling
quantum gravity [108]. At the origin of this consideration there has been the
discovery by Bekenstein and Hawking [109, 110, 111, 112] that the second law
of thermodynamics can be preserved in the gravitational field of a black hole,
if this latter has an entropy proportional to the area of its horizon, by the
inverse of the Newton gravitational constant G. This entropy is maximal, as
implied by the second law itself, providing an upper bound for possible
configurations of matter within a region of the same size [113, 114].

Nonetheless, the scaling of the local degrees of freedom counted by the
entropy does not increase as the volume, hinging toward the formulation of the
holographic conjecture [115], suggesting a division between the information
that can only be retrieved on the boundary world, and a merely apparent bulk
world. AdS/CFT realized the holographic conjecture, postulating a duality
between gravity in asymptotically AdS space and quantum field theory on the
spatial infinity of the AdS space [116]. Giving literal meaning to the
duality, Ryu and Takayanagi (RT) proposed that entanglement of a boundary
region fulfils the same law as for the black hole entropy, replacing the area
of the black hole horizon with an extremal surface area that bounds the bulk
region under scrutiny.

While on the boundaries the theory can be individuated assigning a specific
conformal field theory (CFT), in the bulk the geometry can be associated to
specific entanglement structures of the quantum systems. This is for instance
what happens to the ground states of a CFT associated to an AdS space: the RT
area surface increases less fast than the volume of the boundary. When the
boundary is at equilibrium, in a thermal state of finite temperature, the bulk
geometry corresponds to that of a black hole, its horizon being parallel to
the boundary and its size increasing with the temperature. The RT surface is
then confined between the boundary and the back hole horizon, approaching the
boundary at higher temperature and increasing its entropy. These
considerations suggest the existence of a subtle link interconnecting the
structure of space-time and quantum entanglement, and hence that a theory of
quantum gravity must be fundamentally holographic, where its states satisfy
the RT formula for some bulk geometry.

The existence of an exact correspondence between bulk gravity and quantum
theory at the boundary may hinge toward possible inconsistencies with
locality. This has been discussed in the literature, in terms of local
reconstruction theory [117, 118, 119]: variables in the bulk (e.g. bulk spins)
can be controlled instantaneously from the boundary, but requiring
simultaneous access to a large portion of the boundary: locality and upper
speed of light do not hold exactly in this theory. Nonetheless, local
observers confined in small regions at the boundary still fulfil locality and
the existence of an upper limit of the speed of information exchange, in a way
that is reminiscent of quantum error correction code (QECC) in quantum
information theory: information is stored redundantly, in such a way that when
part of it is corrupted, a reconstruction of information is still possible.
Locality in the bulk is therefore a QECC property of the encoding map that
realizes the duality between bulk and boundary. On the other hand, these
properties are strictly connected to RT, which provides the necessary resource
of entanglement for QEEC to emerge.

The RT formula and QECC are properties fulfilled by different classes of
models, among which TNs [120]. These have been first introduced in condensed
matter physics as variational wave-functions of strongly correlated systems
[121, 122]. TNs are many-body wavefunctions that can be derived composing few-
body quantum states, which are indeed tensors. A prototype of TN is e.g. the
Einstein-Podolsky-Rosen (EPR) entangled pairs of qubits: in an entangled
basis, measured qbits are in some entangled pure state and can be composed
with remaining ones with increasing complexity: complicated quantum
entanglement can be derived by only entangling a few qubits [123].

Particularly relevant for its implications on the reconstruction of the space-
time structure is the multi-scale entanglement renormalization ansatz (MERA)
[124]. TNs can be naturally related to holography duality by considering that
their entanglement entropy can be controlled by their graph geometry. Some
versions of TNs that are characterized by RT entanglement entropy and QEEC
have been constructed resorting to stabilizer codes [125, 126] and random
tensors with large bond dimension [127]. TNs with random tensors at each node
can be regarded as random states restricted by the topology of the network.
Exactly as random states are almost maximally entangled, random TNs show
through the RT formula an almost maximal entanglement, providing a large
family of states with interesting properties to explore holographic duality.
Furthermore, for random TNs the RT formula holds in generic spaces with not
necessarily hyperbolic geometry, hinging toward an extension of holographic
duality beyond AdS, to more general configurations in quantum gravity.
Nonetheless, at least in three dimensions, random tensor networks have been
related to the gravitational action, by means of the Regge calculus [128].

On the other hand, since geometry emerges as a specification of the
entanglement structure, one may consider that the Einstein equations should be
connected as well to the dynamics of entanglement. For small perturbations
around the ground state of a CFT in boundary, linearized Einstein equations
have been derived from the RT formula [129, 130]. Indeed, the conformal
symmetry enables a relations between the energy momentum and the entanglement
entropy, and consequently the area of the extremal surface c an be connected
to the energy-momentum distribution at the boundary — this is equivalent to
the linearized Einstein equations.

The dynamics on the boundary, on the other side, shows a chaotic behaviour,
with scrambling of the single-particle operators, which evolve into multi-
particle operators [131]. Maximal chaotic behaviour recovered in the ladder
operators commutator growth, is encoded in the out-of-time-ordered correlation
(OTOC) functions, characterized by exponential growth in time and temperature.
A model endowed with this properties is e.g. the Sachdev-Ye-Kitaev model,
developed to describe certain systems in condensed matter physics, such as
Gapless spin-fluid [132, 133, 134]. On the other hand, operator scrambling is
also related to QEEC: the chaotic dynamics at the boundary instantiates QECC
preserving quantum information, efficiently hidden (and protected) behind the
horizon. Nevertheless this has led to may questions, concerning the
information behind the horizon being eventually accessible from the boundary
though non-local measurements, the fate of the local degrees of freedom
hitting the singularity, the relation among the causal structure of the bulk
and a smooth geometry across the horizon.

###  3.2 General results

We can move to prove a general result:

######  Theorem 2.

A system $A$ exhibits non-trivial control flow if, and only if, its control
flow can be represented by a TN.

and examine some of its corollaries. We begin by defining:

######  Definition 1.

Control flow is trivial if a system deploys only one QRF.

As any collection of mutually-commuting QRFs can be represented as a single
QRF [57, 59], any system that deploys only mutually-commuting QRFs exhibits
trivial control flow.

Systems that deploy only a single QRF “do the same thing” regardless of
context, and so do not qualify as “interesting” in the sense used here. As
noted above, no finite physical system can measure the entire state of its
boundary with a single QRF, so no such system can simultaneously measure and
act on its entire context. Any system $A$ that deploys multiple QRFs in
sequence cannot, as noted above, avoid contextuality due to unobservable
effects, mediated by the action of $H_{B}$, of the action of $Q_{i}$ on the
state measured by $Q_{j}$. Every action taken by an “interesting” system, in
other words, at least transiently increases the VFE at its boundary.

Consider, then, a system $A$ that deploys multiple, distinct QRFs
$Q_{1},Q_{2},\dots Q_{n}$, where $n\ll N=dim(H_{AB})$. Classical control flow
in $A$ can then be represented by a matrix $\mathbf{CF}=[P_{ij}]$, where
$P_{ij}$ is the probability of the control transition $Q_{i}\rightarrow
Q_{j}$. As noted earlier, any such transition has an energetic cost, which
must be paid with free energy sources from $F$.

The matrix $\mathbf{CF}$ is a 2-tensor. Theorem 2 states that this tensor can
be decomposed into a TN. We prove it as follows:

###### Proof (Thm. 2).

Suppose first that control flow in a system $A$ can be represented by a TN. A
TN is, by definition, a factorization of a tensor operator into a network of
tensor operators. This network can be either hierarchical or flat; if it is
hierarchical, each layer can be considered a flat TN. Hence no generality is
lost in considering just the case of a flat TN, which is an operator
contraction $T=\dots T_{ij}T_{jk}T_{kl}\dots$, where summation on shared
indices is left implicit. In general, $T_{jk}\neq T^{T}_{jk}=T_{kj}$, hence
these expressions do not commute. They therefore represent non-trivial control
flow. Conversely, any non-trivial control flow can be written, at any fixed
scale or level of abstraction, as a linear sequence of (in general
probabilistic) operators. The fixed order of operators in the sequence can be
encoded formally by adding “spatial” indices as needed to allow contraction
over shared indices. Hence any non-trivial control flow at a fixed scale can
be written as a flat TN. This construction can be repeated at each larger
scale to produce a hierarchical TN over a collection of “lowest-scale” TNs. ∎

We can now examine two corollaries of this result:

######  Corollary 1.

Decoherent reference sectors exist on $\mathscr{B}$ if and only if control
flow can be implemented by a TN.

###### Proof.

Decoherence sectors require independently-deployable, non-commuting QRFs. This
requires a control structure that factors, by Theorem 2 a TN. Conversely, a TN
factors the control structure, making QRFs independently deployable, which
renders their sectors decoherent. ∎

Equivalently, the GM factors if, and only if, control flow can be implemented
by a TN.

######  Corollary 2.

The TN of any system compliant with the FEP is a decomposition of the
Identity.

###### Proof.

The FEP applies to systems with a NESS, and drives such systems to return to
(the vicinity of) the NESS after any perturbation. Hence at a sufficiently
large scale, the TN of any such system is a cycle, i.e., a decomposition of
the Identity. ∎

Many standard TN models, e.g., MERAs, assume boundary conditions
asymptotically far, in numbers of lowest-scale operators, from the region of
the network that is of interest. Identifying such asymptotic boundary
conditions yields a cyclic system.

Theorem 2, together with its corollaries, provides a natural, formal means of
classifying systems by their control architectures. At a high level, two
characteristics distinguish systems with different architectures:

  * •

Hierarchical depth indicates the number of “virtual machine” layers [135] the
architecture supports. The interfaces between these layers implement coarse-
graining, removing from the higher-level representation all dimensions, and
hence all information, which is contracted out of the lower-level operators.

  * •

Number and location of contractions that yield unitary operators, and hence
build in entanglement between lower-level operators. The natural limit is a
MERA, in which every pair of lower-level operators is entangled at every
hierarchical level [48].

The control-flow architecture, in turn, specifies the structure of the
“layout” of distinguishable sectors on $\mathscr{B}$ and hence of detectable
features/objects in the environment. Locality on $\mathscr{B}$ requires a
hierarchical TN; detectable entanglement requires a MERA-like TN. Locality is
required for detectable features/objects to appear to have components with
nested decomposition. Any QRF for geometric space, and hence for spacetime,
must be hierarchical, and must be a MERA if entanglement in space is to be
detected. A MERA is required, in particular, if the use of coherence between
spatially-separated systems as a computational or communication resource is
detectable.

To illustrate the classification of systems by hierarchical level, consider
the ten-step cyclic TN shown in Diagram (13):

$\begin{gathered}\leavevmode\hbox to122.75pt{\vbox
to32.5pt{\pgfpicture\makeatletter\hbox{\hskip 18.6969pt\lower-8.9359pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@curveto{8.5359pt}{4.71431pt}{4.71431pt}{8.5359pt}{0.0pt}{8.5359pt}\pgfsys@curveto{-4.71431pt}{8.5359pt}{-8.5359pt}{4.71431pt}{-8.5359pt}{0.0pt}\pgfsys@curveto{-8.5359pt}{-4.71431pt}{-4.71431pt}{-8.5359pt}{0.0pt}{-8.5359pt}\pgfsys@curveto{4.71431pt}{-8.5359pt}{8.5359pt}{-4.71431pt}{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{4.71431pt}{33.16707pt}{8.5359pt}{28.45276pt}{8.5359pt}\pgfsys@curveto{23.73845pt}{8.5359pt}{19.91685pt}{4.71431pt}{19.91685pt}{0.0pt}\pgfsys@curveto{19.91685pt}{-4.71431pt}{23.73845pt}{-8.5359pt}{28.45276pt}{-8.5359pt}\pgfsys@curveto{33.16707pt}{-8.5359pt}{36.98866pt}{-4.71431pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.40936pt}{-3.41666pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{82.10481pt}{-3.41666pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$J$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-8.5359pt}{0.0pt}\pgfsys@curveto{-17.8969pt}{1.65054pt}{-9.50531pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{16.64359pt}{22.76228pt}{26.03555pt}{22.76228pt}{42.67914pt}{22.76228pt}\pgfsys@curveto{59.32272pt}{22.76228pt}{68.71469pt}{22.76228pt}{85.35828pt}{22.76228pt}\pgfsys@curveto{94.86359pt}{22.76228pt}{103.25517pt}{1.65054pt}{93.89418pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}$ (13)

and its extension to a hierarchy as shown in Diagram (14):

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to112.57pt{\pgfpicture\makeatletter\hbox{\hskip
37.38866pt\lower-34.94322pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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where red, blue, and green colors indicate distinct hierarchical “layers” of
tensor contractions. We have trained artificial neural networks (ANNs) to
execute these TNs as the sequences of state transitions shown in Table 1. The
first sequence (Dataset 1) is a ten-step cycle shown Diagram (13); the second
sequence (Dataset 2) layers the coarse-grained state transitions of Diagram
(14) onto this ten-step cycle. In Dataset 2, a two-bit tag is used to
differentiate the “low-level” from the coarse-grained “high-level” cycles. An
example state state transition from a randomly-generated initial state is
shown in Fig. 2; the red-on-green bit pattern effectively moves “up” one step
on each state-transition cycle.

> Table 1: Datasets used in ANN simulations. Dataset 1 specifies a ten-step
> cycle A $\rightarrow$ B $\rightarrow~{}\dots~{}\rightarrow J\rightarrow A$.
> Dataset 2 specifies this same cycle, with three coarse-grained cycles
> layered on top. The tags (0,0), (0,1), (1,0), and (1,1) distinguish the data
> for the low- and high-level cycles.

Figure 2: Example state transition from Dataset 1.

We trained two ANNs, one to execute each of the control cycles shown in Table
1. The networks are each composed of three layers, as illustrated in Fig. 3,
with network sizes of [10, 50, 10] and [10, 200, 10], respectively, for the
input, hidden, and output layers. The units in the hidden layer use the
rectified linear unit (ReLU) nonlinear activation function and the neurons in
the output layer use the hyperbolic tangent activation function. The network
is connected in a feedforward way where a neuron in one layer connects to
every neuron in the next layer. Since the ANN serves as a switch state
controller, we use a training scheme, similar to one-class classification
[136], where the training data are the only data that the network learns to
produce. In so doing, the network learns to overfit the training data, and any
input outside of the designated state-encoding is discarded. The network is,
therefore, not expected to deviate from the learned pattern. The network
learns both control regimes wth 100% accuracy after training with 3,000
randomly-generated 10-bit inputs.

Figure 3: Feed-forward network archtecture used to learn the control cycles
specified in Table 1. Each node is connected to every node of the next layer,
as shown here for the first and last nodes only. The labels ‘T’ and ‘T+1’
indicate time steps in the executed control flow.

In the more realistic case of noisy input data, where binary states can be
flipped, the Bidirectional Associative Memory (BAM), a minimal two-layer
nonlinear feedback network [137], is a viable alternative to a shallow feed-
forward ANN. The architecure is shown in Fig. 4. This BAM network learns to
associate between the two initial and final states in Table 1, with similar
performance to that of the feed-forward network.

Figure 4: Architecture of the Bidirectional Associative Memory (BAM) network
employed here. As in Fig. 3, only the connections of the first and last nodes
are shown explicitly.

##  4 Implementing control flow with TQNNs

Tensor Networks can be naturally associated to the matrix elements of physical
scalar products among topological quantum neural networks (TQNNs). Physical
scalar products encode indeed the dynamics of TQFTs, since they fulfill their
constraints of imposing flatness of the curvature and gauge invariance. Thus,
the matrix elements associated to scalar products can be seen as evolution
matrix elements for the spin-network states that span the Hilbert spaces of
TQNNs.

###  4.1 Tensor networks as classifiers for TQNNs

A notable example is provided by BF theories [138], a class of TQFTs
particularly well studied in the literature of mathematical physics that
enables expressing effective theories of particle physics, gravity and
condensed matter, and provides as well a general framework for implementations
of models of quantum information and quantum computation, machine learning
(ML) and neuroscience. These are defined on the principal bundle $M$ of a
connection $A$ for some internal gauge group $G$, with algebra $\mathfrak{g}$,
according to the action on a _d_ -dimensional manifold $\mathcal{M}_{d}$

$\mathcal{S}=\int_{\mathcal{M}_{d}}{\rm Tr}[B\wedge F]\,,$ (15)

where $B$ is an ad$(\mathfrak{g})$-valued _d-2_ -form, $F$ denotes the field-
strength of $A$, which is a _2_ -form, and the trace Tr is over the internal
indices of $\mathfrak{g}$, ensuring gauge invariance of the density Lagrangian
$\mathcal{L}={\rm Tr}[B\wedge F]$ of the BF theory.

Variation with respect to the conjugated variables, the connection $A$ and the
$B$ frame-field, closing a canonical symplectic structure, provide the
equations of motion of the theory [138]:

$F=0\,,\qquad d_{A}B=0\,,$ (16)

respectively the curvature constraint, imposing the flatness of the
connection, and the Gauß constraint, imposing invariance under gauge
transformations, having denoted with $d_{A}$ the covariant derivative with
respect to the connection $A$.

At the quantum level, the states of the kinematical Hilbert space of the
theory, fulfilling by construction the Gauß constraint, can be represented in
terms of cylindrical functionals $Cyl$, supported on graphs $\Gamma$ that are
unions of segments $\gamma_{i}$, the end points of which meet in nodes $n$,
and with holonomies – elements of the group $G$ – $H_{\gamma_{i}}[A]$ of the
connection $A$ assigned to $\gamma_{i}$ and intertwiner operators – invariant
tensor product of representations – $v_{n}$ assigned to the nodes $n$.

For $G=$ SU(2), spin-networks $|\Gamma,j_{\gamma},\iota_{n}\rangle$, supported
on $\Gamma$ and labelled by the spin $j_{\gamma}$ of the irreducible
representations of the group elements assigned to $\gamma$ and by the quantum
intertwiner numbers $\iota_{n}$ associated to $v_{n}$, represent a basis of
the kinematical Hilbert space of the theory. In terms of functionals of $Cyl$,
one can provide the holonomy representation, which is related to the “spin and
intertwiner” representation of $|\Gamma,j_{\gamma},\iota_{n}\rangle$ by means
of the Peter-Weyl transform. This allows us to decompose the spin-network
cylindric functional as [139]:

$\Psi_{j_{\gamma_{ij}},\iota_{n_{i}}}(h_{\gamma_{ij}})=\left(\bigotimes_{n}\iota_{n}\right)\cdot\left(\bigotimes_{\gamma_{ij}}D^{(j_{\gamma_{ij}})}(h_{\gamma_{ij}})\right)\,,$
(17)

with $D^{(j)}$ are Wigner matrices providing representation matrices of the
SU$(2)$ group elements.

The functorial evolution among spin-networks is ensured by the projector
operator [140], which implements the curvature constraint in the physical
scalar product among states, i.e.

$\langle{\rm in}|P\,|{\rm out}\rangle\,,\qquad{\rm with}\qquad
P=\int\mathcal{D}[N]\exp(\imath\int{\rm Tr}[NF])\,.$ (18)

We may then regard $|{\rm in}\rangle$ as elements of the Hilbert space, and
without loss of generality pick up those ones resulting from composing
tensorially in $Cyl$ _k_ -representations of holonomies. We may further denote
them as $|j_{1}\dots j_{k}\rangle$, with some ordering prescription to
associate the topological structure of $\Gamma$ to the sequence of spin
labels. Physically evolving states $P|{\rm in}\rangle$ are distinguished from
the former ones by labelling them as $|\widetilde{j_{1}\cdots j_{k}}\rangle$.
Similarly, we introduce $|{\rm out}\rangle$ as the tensor product of (_n-
k_)-representations of holonomies, and denote these states as $|i_{1}\dots
i_{n-k}\rangle$. Then the matrix elements of $\langle{\rm in}|P\,|{\rm
out}\rangle$ naturally give rise [98] to an _n_ -tensor, i.e.

$\langle i_{1}\dots i_{n-k}|\widetilde{j_{1}\cdots j_{k}}\rangle=T_{i_{1}\dots
i_{n-k}j_{1}\dots j_{k}}\,.$ (19)

###  4.2 Geometric RG flow for TQNNs and TNs

The mathematical structures of TQNNs we summarized in Sec. 4.1 are picturing
systems “at equilibrium”, for which TQFTs characterize a topological stability
that percolates into the related transition amplitudes. Nonetheless, it is
worth considering as well how stochastic noise might interfere with the
topological order ensured by TQFTs, and study the role of “out-of-equilibrium”
physics in the analysis of the evolution of the systems under scrutiny.

Out-of-equilibrium dynamics is instantiated considering a heat-flow evolution
of the fundamental fields of the theory, with respect to a thermal time
$\tau$. Typical Langevin equations, complemented with stochastic noise,
provide through their convergence toward the equations of motion of the theory
the relaxation toward equilibrium of the field configurations representing
specific systems [141]. In general, given some fields $\phi_{\sigma}$, with a
classical equation of motion derived, according to the variational principle
${\delta\mathcal{S}}/{\delta\phi_{\sigma}}$, from an action $\mathcal{S}$ over
a Euclidean manifold $\mathcal{M}$, the associated Langevin equations read:

$\frac{\partial}{\partial\tau}\phi_{\sigma}=-\frac{\delta\mathcal{S}}{\delta\phi_{\sigma}}+\eta_{\sigma}\,,$
(20)

with $\eta_{\sigma}$ a stochastic noise term. The theory at equilibrium is
characterized by the symmetries of the equations of motion
${\delta\mathcal{S}}/{\delta\phi_{\sigma}}=0$ that are broken in the transient
phase [142]; these symmetries are consistent with – and in the case of BF
theories, actually generated by – the theories at equilibrium.

A prototype of geometric heat-flow was introduced by Hamilton, and then used
by Perelman to prove the Poincaré conjecture, which goes under the name of
Ricci flow. Here the gravitational field $g_{\mu\nu}$ is the basic
configurational space field, while the drift terms are the Einstein equations
of motion in the vacuum, which indeed are expressed by requiring that the
components of the Ricci tensor vanish, i.e. $R_{\mu\nu}=0$. The Ricci flow
then reads

$\imath\frac{\partial}{\partial\tau}g_{\mu\nu}=-2R_{\mu\nu}\,,$ (21)

having considered now a Lorentzian manifold $\mathcal{M}$. The Ricci flow
equations can be further complemented introducing the Ricci target
$R^{T}_{\mu\nu}=\kappa^{2}(T_{\mu\nu}-1/2g_{\mu\nu}T)$, expressed in terms of
the Newton constant $G=\kappa^{2}/(8\pi)$ and the energy-momentum tensor of
matter $T_{\mu\nu}$, so as to obtain at equilibrium the Einstein equations:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa^{2}T_{\mu\nu}\,,\qquad{\rm
or\,\,equivalently}\qquad R_{\mu\nu}=R^{T}_{\mu\nu}\,.$ (22)

The stochastic version of the Ricci flow, with heat equation turning into a
Langevin equation, has been introduced and deepened in [142] for a generic
gravitational system in the presence of matter fields, describing an action
$\mathcal{S}$ for gravity and matter. Moving then from:

$\imath\frac{\partial}{\partial\tau}g_{\mu\nu}=-\frac{1}{\kappa^{2}}\frac{\delta\mathcal{S}}{\delta
g^{\mu\nu}}+\eta\,g_{\mu\nu}\,,$ (23)

in which a multiplicative noise $\eta_{\mu\nu}=\eta\,g_{\mu\nu}$ appears, the
Hamiltonian analysis of the stochastic Ricci flow (SRF) in the Adomian
decomposition method (ADM) variables has been derived [142].

An essential by-product of the discussion, from the Ricci flow perspective, is
that the equilibration trajectories corresponds to those of a renormalization
group (RG) flow. The thermal time $\tau$ plays the role of scale parameter
that individuates a dimension in the bulk, which is out-of-equilibrium. The
boundaries are recovered asymptotically in $\tau$, in the infra-red regime,
and are by definition at equilibrium and thus symmetric.

For a particular class of TQFTs, the BF theories we have introduced in Sec.
4.1 for implementing TQNNs and TNs, the geometric RG flow acquire a specific
expression as the TQFT equivalent of the gravitational Ricci flow [143].

###  4.3 TNs as a generalization of the main model architectures in ML

The use of TNs is an emerging topic in the ML community. The integration
between the two appears quite immediate. A TN structure can be viewed as an ML
model in which the parameters are properly adjusted to learn the
classification of a data set. Yet, as Ref. [144] mentions, machine learning
can aid, in turn, in determining a factorization of a TN approximating a data
set. Moreover, TNs are also used to compress the layers of ANN architectures,
besides a variety of other uses. Tensor networks are becoming more and more
popular to the extent that they are a powerful tool for representing and
manipulating high-dimensional data, as in the case of image and video
classification tasks in which the data is represented as a high-dimensional
tensor. High efficiency, flexibility, and easy to use are making them a
dominant choice for many AI applications. Furthermore, besides being used to
represent data, TNs can be used to process data by exploiting a number of
operators. This feature makes them an effective technique for processing data
in ML applications.

As it is well known, TNs are particularly well suited for representing quantum
many-body states in which the dimension of the Hilbert space is exponentially
large in the number of particles. The corresponding ML approach consists in:

  * •

Lifting data to exponentially higher spaces;

  * •

Applying any linear classifier $f(x)=W^{\star}\Phi(X)$ to a non-linear space;

  * •

Compressing the weights by using TNs.

The output of the model is a separation of classes that would not be linearly
separable in a linear space. In particular, the decision function is the
overlap of the weight tensor $W$ with the feature map tensor $\Phi$ as in Fig.
5. The weight tensor $W$ can be approximated by the decomposition in Fig. 6.

Figure 5: Representation of the decision function (see [145]). Figure 6:
Matrix product decomposition (again see [145]).

Regularization and optimization are built as a constructive product of low-
order tensors while weight compression is performed by using the Matrix
Product States (MPS) decomposition. If we look at Deep Neural Networks as a
piecewise composition of linear discriminators (logistic regression
functions), then the TN framework appears as a generalization of the main
model architectures found in the ML literature, e.g. Support Vector Machines,
Kernel models, and Deep Neural Networks.

The literature concerning the use of tensor theory in traditional ML is
becoming large. A short review starts with a seminal paper by Stoudenmire and
Schwab [146], which demonstrated how algorithms for optimizing TNs can be
adapted to supervised learning tasks by using MPS (tensor trains) to
parametrize non-linear kernel learning models. Novikov, Trofimov, and
Oseledets [147] have shown how an exponentially large tensor of parameters can
be represented in a factorized format called Tensor Train (TT), with the
consequence of obtaining a regularization of the model. van Glasser, Pancotti,
and Cirac [148] explored the connection between TNs and probabilistic
graphical models by introducing the concept of a “generalized tensor network
architecture” for ML. Ref. [149] then designed a generative model, i.e. a
traditional machine learning model that learns joint probability distributions
from data and generates samples according to it, by using MPS. Ref. [150] made
use of autoregressive MPSs for building an unsupervised learning model that
goes beyond proof-of-concept by showing performance comparable to standard
traditional models. Finally, Ref. [151] analyzes the contribution of
polynomials of different degrees to the supervised learning performance of
different architectures.

##  5 Implications for biological control systems

Scale-free biology requires a smooth transition from quantum-like to
classical-like behavior. Typical representations of metabolic, signal-
transduction, and gene-regulatory pathways are entirely classical, even though
many of their steps involve electron-transfer or other mechanisms that are
acknowledged to require a quantum-theoretic description [87, 152]. As noted
earlier, free-energy budget considerations suggest that both prokaryotic and
eukaryotic cells employ quantum coherence as a computational resource [92].
Emerging empirical evidence for longer-range entanglement in mammalian brains
suggests that large-scale networks may also be using quantum coherence as a
resource [93]. Control flow models must, therefore, support the possibility of
quantum computation in biological systems. Hierarchical TNs that include
unitary components, e.g., MERA-type models, provide this capability.

In prokaryotes, the primary tasks of control flow are adapting metabolism to
available resources via metabolite-driven gene regulation [153] and initiating
DNA replication and cell division when conditions are favorable. We can,
therefore, expect shallow hierarchies of effectively classical control
transitions in these organisms. Eukaryotes, however, are characterized by both
intracellular compartmentalization and morphological degrees of freedom at the
whole-cell scale. We have shown previously that the FEP will induce
“neuromorphic” morphologies – i.e. morphologies that segregate inputs from
outputs and enable a fan-in/fan-out computational architecture – in any
systems with morphological degrees of freedom [154]. Such systems can be
expected to have deep control hierarchies at the cellular level, with
hierarchical structure correlating with morphological structure in
morphologically-complex cells such as neurons [155], and in multicellular
assemblages at all scales. As well as managing metabolism and replication,
such systems must implement active exploration of the environment,
communication with other systems, and – crucially for cognition – the writing
and reading of stigmergic memories. Thus we can expect such systems to
implement QRFs for spacetime and for specific kinds of objects, e.g.,
conspecifics and suitable substrates for recording stigmergic memories. Such
QRFs rely on symmetries, and hence on redundancy of encoded (or encodable)
information; they depend, in other words, on the availability of error-
correcting codes [25, 156]. The implementation of spacetime as a quantum
error-correcting code by TNs has been extensively studied by physicists; see
[157] for review and [98] for a detailed analysis using the present formalism.
The use of spacetime as an error-correcting code by organisms – e.g., the
implementation of translational and rotational invariance of objects by dorsal
visual processing in mammals [158, 159] – is well-understood
phenomenologically, but the details of neural implementation remain to be
elucidated.

Both the context-sensitivity of, and the occurence of context effects due to
non-commutativity of QRFs in, control networks can be expected to increase
with their complexity and hierarchical depth. “Bowtie” networks with high fan-
in/fan-out to/from multi-use proteins or second messengers such as Ca2+ are
increasingly recognized as ubiquitous in high eukaryotic cells [160]. Such
networks have the general for of the CCCD depicted in Diagram (3). Frequently,
such networks evolve via compression of information (e.g. toward share second
messengers, as in [Ca2+]-based interactions [161, 162]) as an efficiency-
increasing mechanism. Bowties introduce semantic ambiguities that must be
resolved by context. Each incoming signal has its own governing semantics, but
the relevant context can depend on boundary conditions which can be
exceedingly difficult (if not impossible) to predetermine (see e.g., [163,
164] for general discussions of the history and semantic depth of this
problem). As pointed out in [22], a context change $x\mapsto y$ is
semantically problematic if for a fixed set $\\{o_{i}\\}$ of observations, the
conditional probability distributions $\rm{P}(o_{i}|x)$ and $\rm{P}(o_{i}|y)$
are well defined, but the joint distribution $\rm{P}(o_{i}|x\vee y)$ is not
[106]. This occurs whenever the QRFs for $x$ and $y$ do not commute [57, Th
7.1]. As suggested by Diagram (3), this context-switching problem affects deep
learning using VAEs [165]; see e.g., the application to antimicrobial peptides
in [166]. In general, the structure of Diagram (3) can serve as a convenient
benchmark for distinguishing signal transduction networks that incorporate co-
deployable versus non-co-deployable QRFs [57].

“Quantum” context effects due to non-commutativity have, interestingly, been
reported even at the scale of human language use. The “Snow Queen” experiment
[167] challenged subjects with distinct, mutually-inconsistent meanings of
terms such as ‘kind’, ‘evil’, or ‘beautiful’ in different contexts, and
detected statistically-significant context effects using the CbD formalism
[62, 63]. Such effects cannot be explained by linguistic ambiguity,
misreading, etc. Such language-driven contextuality is taken up in the setting
of psycholinguistics and distributional semantics in [168], which combines CbD
and the sheaf theoretic [60, 61] methods to systematically study semantic
ambiguity as creating meaning/sense discrepancies in statements like “It was
about time”, “She had time on her hands to win the heat”, “West led with a
queen”, etc.

While the notion of “languages” has thus far been applied to cells, tissues,
and even non-vertebrates in a mostly metaphorical way, we can speculate that
linguistic approaches to understanding the interplay between context
dependence and semantic ambiguity may be useful to biology in general. Immune
cells (e.g., T cells) are, for example, “programmed” or “trained” by their
progenitor cells to respond to local cellular signals and ambient conditions
in particular ways. Unexpected context changes may induce dysfunctional (at
the organism scale) responses, including chronic disorders [169]; these can be
considered consequenes of discrepancies between the “actual’ semantics of
incoming signals and the semantics expected by the immune systems’ “language.”
This suggestion of possible “linguistic” contextuality seems in consonance
with the hypothesis of [170] that the immune system is a cognitive (living)
system implementing its exclusive system of language-grammar, which may be
prone to analogous disorders of communication as those discussed in [168].
Similar context effects have been observed in microbiological systems [171];
here discrepancies in experimentally derived classical probabilities arising
from lactose-glucose interference signaling in E. Coli can only be explained
in terms of non-classical probabilities. We note that the expression ‘quantum-
like’ [172] is often used for such effects; however, their formal structure is
exactly that given by quantum theory.

We expect that further research into quantum biology will unfold significant
perspectives on human/mammalian physiology and cognitive capabilities along
the lines suggested in the present article. For example, allostatic
maintenance, as briefly alluded to in §2.1, can be seen as a process
regulating a body’s physiological conditions relative to costs and benefits
while dynamically allocating resources for the purpose of overall adaptability
of an organism within its internal environment. Implementing the allostatic
and anticipatory mechanisms are the visceromotor cortical regions generating
autonomic, hormonal, and immunological predictions leading to interoceptive
inference [68, 69, 70, 82, 173, 174, 176, 177]. This process of inference in
humans and mammals putatively utilizes predictive coding for the processes of
homeostasis-allostasis through a hierarchy of cellular to organ-level systems,
in turn connecting interoception to the processes of extercoception and
proprioception [82, 173, 177, 178, 179]. The basic principles follow from how
allostasis provides protection against potential surprise by utilizing a
framework somewhat beyond the error signaling necessary for homeostatic
maintenance (it is essentially through minimizing the free energy of internal
state trajectories towards combatting surprise, as discussed in §2.1). The net
effect of the process is consonant with the Good Regulator theorem of [175],
showing how regulation of a given system requires an internal model of that
system. A further perspective is to emphasize the predictive nature of an
integrated, complex, allostatic-interoceptive cortical system capable of
supporting a spectrum of psychological phenomena including memory and emotions
[177] (cf. [82]). Accordingly, cognitive conditions such as depression and
autism have been described as abnormalities of allostatic-interoceptive
inference, so impairing predictive coding mechanisms due to aberrant
assimilation and mistuning of prediction errors (putatively a connectivity
issue), conceivably leading to a root cause of many known cognitive conditions
[82, 173, 178].

We anticipate that this fully general, context sensitive model of control flow
will be important for understanding morphogenesis, which is not simply a feed-
forward emergent system, but rather a highly context-sensitive error-
minimizing process [23]. Specifically, the collective intelligence of cells
during embryonic development, organ regeneration, and metamorphosis can create
and repair specific complex structures despite a wide range of perturbations
[180]. Changes in the genome, the number of cells, or the starting
configuration can often be overcome: bisected embryos result in normal twins,
amputated salamander limbs re-grow back to normal, and planarian fragments
result in perfect little worms [181]. The competency of cellular collectives
to reach the correct target morphology despite even drastic interventions
requires an understanding of how they navigate, via context-sensitive control
flow, problem spaces including anatomical morphospace [182], physiological,
and transcriptional spaces [156, 183]. Understanding the navigation policies
used by unconventional collective intelligences can help not only understand
creative problem-solving on rapid timescales (such as the ability to regulate
genes to accommodate an entirely novel stressor [184] without evolutionary
adaptation), but may also have implications for predicting and managing the
goals and behavioral repertoires of synthetic beings [185].

##  6 Conclusion

We have shown here how the problem of defining control flow arises in active
inference systems, and provided three formal representations of the problem.
We have proved that control flow in such systems can always be represented by
a tensor network, provided illustrative examples, and shown how the general
formalism of topological quantum neural networks can be used to implement a
general model of control flow. These results provide a general formalism with
which to characterize context dependence in active inference systems at any
scale, from that of macromolecular pathways to that of multi-organism
communities. They suggest that the concept of communication by language is not
just metaphorical when applied to biological systems in general, but rather an
appropriate and productive description of interactional dynamics.

We view these results as a further step toward fully integrating the formal
models, concepts, and languages of physics, biology, and cognitive science.
This integration is not reductive. It rather allows us to classify systems
using natural measures of organizational and computational complexity, and to
understand how interactions between simpler systems can implement the more
complex behavior of the larger systems that they compose.

## Acknowledgements

K.F. is supported by funding for the Wellcome Centre for Human Neuroimaging
(Ref: 205103/Z/16/Z), a Canada-UK Artificial Intelligence Initiative (Ref:
ES/T01279X/1) and the European Union’s Horizon 2020 Framework Programme for
Research and Innovation under the Specific Grant Agreement No. 945539 (Human
Brain Project SGA3).M.L. gratefully acknowledges funding from the Guy
Foundation and the John Templeton Foundation, Grant 62230. A.M. wishes to
acknowledge support by the Shanghai Municipality, through the grant No.
KBH1512299, by Fudan University, through the grant No. JJH1512105, the Natural
Science Foundation of China, through the grant No. 11875113, and by the
Department of Physics at Fudan University, through the grant No.
IDH1512092/001.

## Conflict of interest

The authors declare no competing, financial, or commercial interests in this
research.

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