# Quantum-like states on complex synchronized networks

Gregory D. Scholes
Department of Chemistry
Princeton University
Princeton, New Jersey, 08540 U.S.A.
<EMAIL_ADDRESS>

###### Abstract

Recent work has exposed the idea that interesting quantum-like probability
laws, including interference effects, can be manifest in classical systems.
Here we propose a model for quantum-like (QL) states and QL bits. We suggest a
way that huge, complex systems can host robust states that can process
information in a QL fashion. Axioms that such states should satisfy are
proposed. Specifically, it is shown that building blocks suited for QL states
are networks, possibly very complex, that we defined based on $k$-regular
random graphs. These networks can dynamically encode a lot of information that
is distilled into the emergent states we can use for QL like processing.
Although the emergent states are classical, they have properties analogous to
quantum states. Concrete examples of how QL functions are possible are given.
The possibility of a ‘QL advantage’ for computing-type operations and the
potential relevance for new kinds of function in the brain are discussed and
left as open questions.

##  1 Introduction

Quantum states characterize the microscopic world, where they convey the dual
wave-particle basis for quantum theory. Of special interest is the way
probability laws for quantum states allow for interference effects and other,
more exotic, correlations evidenced in the effect and outcome of measurements.
We thus find fascinating non-classical phenomena that can be exploited in
quantum information science [1]. Here, a goal is to leverage quantum
correlations at the molecular scale for function at the human scale. A well
known challenge, however, is that quantum states, at least the most
interesting ones, tend to be fragile—easily destroyed by environmental noise
and fluctuations (the process of decoherence[2]). Researchers therefore focus
on stabilizing simple states using techniques such as very low temperature.
Here we explore a way to produce robust states, comprising highly complex
systems, that exhibit "quantum like" properties.

Let’s start by defining what we mean, in the present work, by a quantum-like
(QL) state. An important requirement is that these states allow amplitude
level interferences that are explained by a measurement formulation like that
required by the quantum theory. This is a fairly broad criterion that can be
fulfilled by either: (a) states in vector spaces, where we think of amplitude
(vector) additions followed by measurement of probability as the amplitude
squared; (b) or we can work only with probabilities, but formulate a vector-
like model for measurement sequences (this is the essence of the Växjö model).
In the present work we would like to take the QL state framework further, and
show how eigenstates of classical systems can be combined (through
interactions between the systems serving as QL-bits) to produce states in a
state space that represents the tensor product of vector spaces or each QL-
bit. Since we are dealing with inner product spaces here, the vector spaces
are Hilbert spaces.

Why would complex quantum-like, systems be interesting? There are two major
motivations for this work. First, by exhibiting a concrete example of robust
quantum-like states we can suggest ways to design circuits with quantum like
functions. By considering such circuits, we can conceive of a experiments
designed to seek quantum-like correlations in a range of different systems.
Moreover, this platform could suggest a way to perform QL computing using
simplified hardware that is a step towards true quantum computing. Second, we
establish a hypothesis for what quantum-like states could ‘look like’ in
settings as complex as biology. The key point we will establish is the
feasibility of these states.

The possibility that some kinds of quantum effects provide function in
biological systems has intrigued researchers for over a century. Yet it is
easy to argue against such effects because the kinds of quantum states that we
are familiar with are fragile, and would rapidly decohere in a noisy
environment. This point is well recognized, yet, as enunciated by Penrose[3],
calls for discovery of new kinds of quantum systems that are not only robust
in a using environment, but "retain their manifest quantum nature at a much
larger scale". Even if such states are not present, or functional, in living
systems, elucidating the principles that enable them would open up new ways of
engaging quantum (like) correlations for technology.

In quantum states, classical correlations are prominent[4]. For example, in
the entangled state
$|\Psi_{-}\rangle=\frac{1}{\sqrt{2}}\Bigl{[}|0\rangle_{A}|1\rangle_{B}-|1\rangle_{A}|0\rangle_{B}\Bigr{]}$
it is obvious that if a measurement of qubit A indicates it is in state
$|0\rangle_{A}$, the qubit B (measured in the same basis) must be in the state
$|1\rangle_{B}$. In addition, we note that interference phenomena—a key
contributor to quantum function—is not confined to quantum systems. Thus there
is reason to believe that classical systems possessing the right kinds of
correlations and with the potential to exhibit interference effects might
replicate many (obviously not all) kinds of quantum phenomena. It is thus
proposed that it may not always be necessary to produce truly quantum devices
that exploit quantum correlations. It might often be sufficient to develop a
quantum-like mimic constructed from classical components. Not only does this
open new avenues for engineering systems that can capture some of the
properties of quantum systems, but it provides a framework for thinking about
quantum-like function in nature. The essential advantage would be that more
sophisticated probability laws can be leveraged.

Here we combine ideas from graph theory and operator theory to demonstrate how
complex networks can serve as a basis for generating complex. quantum-like
states and producing functions using those states. We formulate axioms that
guide the construction of the states and demonstrate examples of such states.
Importantly, we describe how these states are resilient in noisy environments.
In the discussion, we speculate about functions unique to these states and how
they provide a basis for testable hypotheses for exotic phenomena, including
how quantum-like processes might augment classical processes in neural
networks like the brain. Some of this might sound for-fetched, but keep in
mind that the central finding of this work is to define and analyze the
quantum-like states themselves. From this foundation, perhaps we can begin
asking questions that could not previously be well posed.

One inspiration for the work was to define physical realizations of QL states
that can show properties described by QL probabilities, as proposed by
Khrennikov in the framework of a contextual probabilistic model[5], termed the
Växjö model. The focus of the Växjö model is on non-Kolmogorov
probabilities—that is, models where probability can be thought of as a kind of
vector quantity. This allows probability amplitude to interfere, like in
quantum theory. However, there is no requirement of an underlying microscopic
quantum state, so the quantum-like model can be used to analyze problems as
far afield as financial markets, psychology, or even the mind. Our goal in
this paper is to propose a concrete model for QL states that are complex
enough to accommodate the idea of ‘context’ in measurement theory.

The main focus of the paper is to formulate examples of QL-states in a vector
space, closely analogous to quantum states, that can be realized in classical
setups. To do that we notice that emergent states of very complex systems are
incredibly robust and isolated in the spectrum, so they can potentially serve
as building blocks for QL states[6]. An emergent phenomenon is manifest when
numerous small interactions add coherently to produce a stunning collective
effect that appears above some threshold of the number of coherently
interacting systems. In quantum systems, an emergent state is signalled by a
single eigenvalue that splits away from the many other eigenvalues in the
spectrum, opening up an energy gap[7]. The questions that we solve in the
present work is how to design networks that produce stable superposition
states of two basis states, and how to combine those networks to produce a
larger state space that can be mapped onto a state space comprising linear
combinations of product states.

Emergent states are characteristic of synchronized complex networks of phase
oscillators. There are many examples of synchronization in a wide variety of
systems[8, 9, 10, 11, 12, 13, 14, 15, 16]. A few illustrative examples of
synchronization include power networks[17], bacterial networks interacting via
signalling[18], human networks of violin players[19], and mechanisms
underpinning memory processes in the brain[20]. We will discuss
synchronization in neural networks in more detail later in the paper. All
these kinds of systems attain a stable synchronized steady state by feedback
loops.

Synchronization is usually analyzed with the Kuramoto Model[9]; a system of
coupled differential equations that predict how initial conditions and
feedback acting in a set of phase oscillators allow synchronization to emerge.
Here we assume that the initial conditions (phase offsets) allow the system to
be synchronizable. We have studied the connections between network
synchronization and homomorphic quantum eigenstates in other work[21]. This
approach allows us to focus on the eigenstates rather than dynamics. In
particular, we are interested in an ordered eigenstate whose eigenvalue is
well separated from the eigenvalues of the many other random states of a
system. We construct the basis for these emergent states in terms of networks
(or graphs).

Figure 1:  Example of the way three QL-bits couple to produce any state in the
tensor product space. The QL-bits are comprised of graphs, labeled here for
later reference. Connections (edges) between the QL-bits are shown
schematically. The emergent state of the network produces a state that can be
mapped to the a state in the tensor product of Hilbert spaces of each QL-bit,
as indicated. In reality, there should be many edges between each pair of
subgraphs.

The main outcome of this paper is to combine these ideas—QL states and
emergent states of networks—to propose a network structure that serves as a
building block for the full tensor product representation of QL states. That
is, we show how to mimic linear combinations of product states. A schematic of
the construction that will be developed in the paper is shown in Fig 1. We
then discuss how those states might be exploited for function in ways that
have not been considered previously for typical network structures.

Detailed reviews of quantum information science can be found in various books
and reviews [1, 22, 23, 24, 25, 26]. In recent review-type papers[27, 4] we
describe the idea of entanglement and qubits and summarize the conceptual
ideas behind quantum information science for a broader readership.

##  2 Graphs and Emergent States

A network can be represented as a graph. A graph $G(n,m)$, that we will often
write simply as $G$, comprises $n$ vertices and a set of $m$ edges that
connect pairs of vertices. The size of a graph or subgraph, that is, the
number of vertices is written $|G|$. In the present work, the edges are
assigned semi-randomly—that is, there is an overarching ‘rule’ that guides how
the edges are placed randomly. In the present work, we only consider that
‘rule’ to be that we construct $k$-regular random graphs (we explain these
graphs below). In some cases, we also delete some fraction of the edges
randomly after building the graph[6]. Rather than assigning a specific number
of edges, sometimes we assign edges between vertices with probability $p$, so
the graph is formally $G(n,p)$. Here we discuss only graphs with no loops and
no multiple edges. For background see [28, 29, 30].

The adjacency matrix of a graph $A$ is the $n\times n$ matrix containing
entries $a_{ij}=a_{ji}=1$, with $i,j\in\\{1,2,\dots n\\}$, when an undirected
edge joins vertices $i$ and $j$. The adjacency matrix of a directed graph
contains entries $a_{ij}=1$ when an edge connects from vertex $i$ to vertex
$j$. We consider also signed graphs, where the edge entry in the adjacency
matrix is designated from $\\{0,1,-1\\}$.

Undirected graphs can be thought of as representing coupling between entities
located at the vertices—a kind of coupling map of the system. For instance,
below we will discuss networks of interacting oscillating dipoles, so the
interaction is the dipole-dipole coupling. That coupling can be negative,
indicating that the coupled oscillators are locked out-of-phase, then we use
the signed graph. Directed graphs indicate a flow, timing, or causal coupling
from one vertex to another. That is often found in physical systems[31],
including transportation networks, aspects of social networks, downstream
influence (common in biology), and communication.

The spectrum of a graph $G$ is defined as the spectrum (i.e. eigenvalues) of
its adjacency matrix $A$. Here we use the spectrum to detect and characterize
emergent states by their eigenvalues. The spectrum can tell us a lot about the
graph, and vice versa, see for example [32]. The graph can also be used as a
template to construct an ensemble of random matrices, whose spectrum is
analyzed[21].

In earlier work we noticed that certain graph types have a spectrum where the
lowest (or highest, depending on the signs of the edges) eigenvalue (or few
eigenvalues) are sufficiently separated from the rest of the spectrum so that
those states are resilient to decoherence[21]. Indeed, it is known that there
are families of graphs with exceptional properties of this kind—expander
graphs[33, 34, 35, 36, 37, 38]. In the present work, expander graphs of a
particular type will underlie our prototypical networks. Expander graphs
involve high connectivity throughout the graph, and are therefore optimal
structures for communication or random walks.

######  Definition 2.1.

(Expander graph) Let $G$ be the $n$-vertex graph with vertex set $V$ and take
$0<\epsilon\in\mathbb{R}$. $G$ is an $\epsilon$-expander if for every vertex
subset $Y$ of $V$ with $|Y|\leq\tfrac{1}{2}|V|$ we have

$|\partial Y|\geq\epsilon|Y|$ (1)

where $\partial Y$ is the boundary of $Y$, which means the set of edges in $V$
that have one endpoint in $Y$ and one endpoint in $V\setminus Y$.

######  Definition 2.2.

(Isoperimetric constant) The isoperimetric constant of $G$, $h(G)$, indicates
the minimum $\epsilon$ for the graph:

$h(G)=\min\Big{\\{}\frac{|\partial Y|}{|Y|}\Big{\\}}$ (2)

with $|Y|\leq\tfrac{1}{2}|V|$ .

The specific expander graphs we study here are the $k$-regular random graphs.

######  Definition 2.3.

($k$-regular graph) A graph $G$ is $k$-regular if every vertex has degree
(valency) $k$. That is, every vertex connects to $k$ edges.

######  Definition 2.4.

(Expander family, def 1.74 of [34]) Let $k$ be a positive integer. Let
$(G_{n})$ be a sequence of $k$-regular graphs such that
$|G_{n}|\rightarrow\infty$ as $n\rightarrow\infty$. $(G_{n})$ is an expander
family if the sequence $(h(G_{n}))$ is bounded away from zero.

Let the eigenvalues of $G$ be
$\lambda_{0}\geq\lambda_{1}\geq\dots\geq\lambda_{n-1}$. When $G$ is a
$k$-regular graph, then $\lambda_{0}=k$. This fact is easily demonstrated by
noting that the eigenvector associated with $\lambda_{0}$ is
$(1,1,1,\dots)\sqrt{n}$. The second-largest eigenvalue $\lambda_{1}$ is
bounded by the Alon-Boppana theorem[38, 39]. It can then be shown that the
smaller $\lambda_{1}$ is, the larger $h(G)$ is,

######  Definition 2.1.

(Prop 1.84 of [34]) Let $G$ be a $k$-regular graph, then

$\frac{k-\lambda_{1}}{2}\leq h(G)\leq\sqrt{2k(k-\lambda_{1})}$ (3)

Conversely, the larger $h(G)$ the larger the spectral gap
$\lambda_{0}-\lambda_{1}$ (recall that $\lambda_{0}=k$). This is the reason
that expander graphs are good candidates for emergent states.

The isoperimetric constant is not so easy to calculate. It is closely related
to the property of topological groups called weak containment, specifically a
quantity called Kazhdan’s property (T)[40], which can sometimes be more easily
bounded. However, for the present purposes, we simply point out that the
expansion property of a family of graphs is best estimated from graph spectra.
This is demonstrated for $k$-regular graphs by the Alon-Boppana bound[38, 39].

In recent work we have shown that $k$-regular random graphs retain their
characteristic spectral properties even when a substantial fraction of edges
have been removed at random[6]. We explain those numerical results on a more
formal level, with proofs, in another paper. We found that the ensemble of
graphs has the spectrum of an ensemble of $d$-regular random graphs, where
$d\leq k$ is the average valency of vertices for the ensemble of graphs.
Therefore, these graphs can be quite disordered, but still display a prominent
eigenvalue that is well-separated from those of the remaining ‘random’ states.
This particular point, and the reasons that the emergent state is
spontaneously stable, will be developed in another paper.

In other work, it has been shown that complex networks where nodes are coupled
with the structure of $k$-regular random graphs are highly synchronizable[41].
In that work the authors prove that for the case of regular graphs, spectral
expansion implies global synchrony. That is, a network of phase oscillators
coupled in the fashion of a $k$-regular graph will synchronize regardless of
their initial phases. In the present work we do not necessarily need perfectly
synchronizable networks. In other words, it may be advantageous to suffer some
initial phase arrangements that inhibit synchronization with the benefit of
less stringent construction requirements for the graph. After all, these
networks will be much more robust to disorder and implementable if they can
sustain a significant variance in structure. Our main requirement is that the
graphs are on average $d$-regular.

Some examples of $k$-regular random graphs on $n=40$ vertices, together with
their spectra, are shown in Fig 2. Here $k=20$, so the number of edges in the
graph without edge disorder is $kn/2=400$. We plot edge-disordered graphs,
with various numbers of edges removed. The deleted edges are indicated by
green lines, whereas the retained edges are light blue. We compare cases where
20, 100 and 300 edges have been randomly removed. The emergent state—evident
as the isolated, largest eigenvalue—is qualitatively stable (that is,
separated from the rest of the spectrum) until more than half the edges have
been deleted. The eigenvector of the emergent state is $(1,1,1,...)/\sqrt{n}$.

Figure 2:  Examples of $k$-regular random graphs on $n=40$ vertices with
$k=20$, together with their spectra. See text. (a) The number of edges is
$m=380$ (note that the maximum possible number of edges is 400). Here the
emergent state is clearly evident as the isolated state with greatest
eigenvalue. (b) The number of edges is $m=300$. (c) The number of edges is
$m=100$. Here the average valency (degree) of the vertices is too small for
the emergent state to separate clearly from the spectrum of random states.

It is useful to compare these graphs and spectra to those of signed graphs.
Let’s say the largest eigenvalue of the unsigned graph is $d$, then, if the
signs of all vertices are reversed, the smallest eigenvalue of the signed
graph is $-d$. It is associated with the eigenvector $(1,1,1,...)/\sqrt{n}$.
We show in Fig 3 examples where some fraction of the edges are randomly given
negative sign, according to probability $p_{n}$. The positive edges are blue,
while the negative edges are red. We compare case of $p_{n}=0.2$ to
$p_{n}=0.8$. It is evident that signed graphs also preserve the emergent
state, but only when the number of edges with one sign is sufficiently greater
than those of the other sign.

Figure 3:  Examples of $k$-regular random graphs with $n=40$, $k=20$, and
$m=300$ with signed edges. (a) The probability that en edge is negative is
$p_{n}=0.2$. Here, the positive edges dominate, so the emergent state has the
greatest eigenvalue in the spectrum. (b) The probability that en edge is
negative is $p_{n}=0.8$. Now, negative edges dominate, so the emergent state
has the least (most negative) eigenvalue in the spectrum.

##  3 Quantum-like states on networks

Here we define properties that QL states should possess by giving some brief
definitions of the state space we use for quantum systems and then proposing
some axioms that QL states should satisfy. After that, we build concrete
examples of systems that exhibit such states.

###  3.1 Background and Axioms

Our aim is to demonstrate stable states of a complex network that are defined
in some basis, $\\{|a_{1}\rangle,|a_{2}\rangle\\}$, such that we can produce
arbitrary states by arbitrary convex combinations of the basis states,

$|A\rangle=\alpha_{1}|a_{1}\rangle+\alpha_{2}|a_{2}\rangle,$ (4)

where $\alpha_{1},\alpha_{2}\in\mathbb{C}\>\textrm{or}\>\mathbb{R}$ and by
normalization $\alpha^{2}_{1}+\alpha^{2}_{2}=1$. These vectors will comprise a
Hilbert space $\mathcal{H}_{A}$.

It might be worth defining Hilbert space here, and for more detail see[42]. A
Hilbert space $\mathcal{H}$ is a (complete) vector space over a field
$\mathbb{K}$ (which can include $\mathbb{C}\>\textrm{or}\>\mathbb{R}$, for
example) with an inner product $\langle x,y\rangle$ which generalizes the idea
of a dot product of vectors in Cartesian space. For any vectors $x,y,z$ in
$\mathcal{H}$ and scalars $\alpha\in\mathbb{K}$ we have

$\displaystyle\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$
$\displaystyle\langle\alpha x,y\rangle=\alpha\langle x,y\rangle$
$\displaystyle\langle x,y\rangle=\overline{\langle y,x\rangle}$
$\displaystyle\langle x,x\rangle=\alpha\geq 0$

where the overbar means complex conjugate (or its equivalent for the relevant
field). Note also that the ‘vectors’ can be functions satisfying the
properties listed above. The norm is defined in terms of the inner product

$\lvert x\rvert=\langle x,x\rangle^{\frac{1}{2}}.$

We will define in the following sections QL states, constructed using networks
that have emergent states satisfying these properties. Here our vector spaces
are defined explicitly over $\mathbb{R}$. If we want to define them over
$\mathbb{C}$ we should use biased graphs[43, 44, 45], with edges taking values
from $\\{0,1,-1,i,-i\\}$.

We propose that those QL states and/or their underlying graphs should satisfy
the following axioms:

  1. 1.

The graphs will exhibit an emergent state that is distinguished in the
eigenvalue spectrum.

  2. 2.

Each graph will be robust and stable to disorder in its precise construction
as well as respect to disorder in the frequency of the oscillators represented
by its nodes (vertices).

  3. 3.

The basis QL bit is a two-state system, but one of the two states is not ‘off’
because the QL bit is defined by turning ‘on’ a network of oscillators.
Therefore we need the QL bit to comprise a basis of two distinguishable ‘on’
states. What we mean by ‘on’ and ‘off’ distinguishes QL states from quantum
states, where we are happy to have a two level system comprising the vacuum,
or ground, state and another level. In the case of QL states the network needs
to be activated (i.e. ‘on’) to function as a network and to be able to produce
superpositions by interacting with other networks of oscillating dipoles.

  4. 4.

Unitary operations on graphs and subgraphs should be defined.

  5. 5.

We need a construction that couples graphs or subgraphs in such a way that
allows production of a state space homomorphic to superpositions of product
states.

###  3.2 The quantum-like bit

Khrennikov and co-workers initiated a program for identifying how classical
systems could produce probabilistic outcomes like those peculiar to quantum
mechanical systems. Progress has largely concerned how probability laws are
modified, and how probability is measured, as we review in a later section. In
the course of many examples they refer to suitable QL states. In one paper[46]
a specific example is proposed for a single neuron. Here we take a different
approach by exploring the possibility that QL states can be composed from
interacting networks. A main motivation is that these states can be
surprisingly robust to decoherence—whereas typical quantum states are likely
to decohere quickly in disordered environments, which is the main objection to
the idea that quantum phenomena play any functional role in biological
systems. It is worth reiterating too, these states are not strictly quantum
mechanical states; they are QL states.

Building blocks proposed for QL states are networks, possibly very complex,
that we define using graphs. Networks have been extensively studied[47] and
can have the properties of Axioms 1 and 2, as discussed earlier in this paper.
Networks are known to provide insights into social networks, neural networks,
biochemical and biological networks, communications systems, and so on. They
can be functionally sophisticated and encode a lot of information. Networks
can evolve through feedback to shift their balance and condense consensus. In
short, these building blocks are rich and powerful structures, although in the
present work we solely consider their emergent eigenstates.

The emergent states of networks are classical. They arise from nonlinear
interactions among the vertices so that synchronization—the key characteristic
of the emergent state—arises from feedback[9, 10]. This produces a stable
eigenstate of the network of oscillators, which for $n$ oscillators is the
state $(1,1,1,\dots)/\sqrt{n}$. Our task is to develop a way to build from
these classical states, new states that have properties analogous key
characteristics of quantum states. Specifically, we first need to define a QL
bit $|A\rangle$ that ‘lives in’ a Hilbert space $\mathcal{H}_{A}$.

It is proposed that QL states can be built from QL bits defined by
partitioning an edge-disordered $k$-regular random graph $G$ into blocks,
giving the subgraphs $G_{a_{1}}$ and $G_{a_{2}}$, and the tensor sum structure
for the graph’s adjacency matrix, $A=A_{a_{1}}\oplus A_{a_{2}}$. We need to
ensure that there is a lower probability of edges interconnecting the
subgraphs than within each subgraph (for reasons discussed below). Thus we
should delete some fraction of the interconnecting edges (those spanning
$G_{a_{1}}$ and $G_{a_{2}}$). In the demonstrations below, we start from
unconnected subgraphs (no edges between $G_{a_{1}}$ and $G_{a_{2}}$), then
randomly add edges with some probability.

The edges within subgraphs, or those edges connecting subgraphs, can be
positive or negative—that is, these are signed graphs. The physical
interpretation of signed graphs is discussed below. The QL bit thus defined is
a bipartite graph with respect to the vertex sets of the constituent pair of
subgraphs $G_{a_{1}}$ and $G_{a_{2}}$. It has intrinsic $k$-regular random
structure, ensuring that the emergent states are robust.

Some examples of QL bits are shown in Fig 4a, b. Here each subgraph is on 30
vertices, $k=20$, and the probability of an edge between any two vertices
within a subgraph is approximately 0.23. The two subgraphs, $G_{a_{1}}$ and
$G_{a_{2}}$ are shown coloured blue and green respectively. Coupling edges,
that connect the subgraphs, are drawn in red. The left hand drawing shows the
graph and all vertices, whereas the middle drawing shows the separated
subgraphs, including all their edges, and the coupling edges are drawn
schematically. In Fig 4a, the probability of adding a coupling edge between
any two vertices, one in $G_{a_{1}}$ and the other in $G_{a_{2}}$, is 0.01.
That probability is 0.1 in Fig 4b. In these examples, the edges within the
subgraphs are all positive, whereas the coupling edges are negative. Note also
that the value of the coupling edge (the coupling strength) is set to 1
because we are plotting the spectra of the adjacency matrices. This
configuration makes the largest emergent eigenstate, shown in the spectra,
$\frac{1}{\sqrt{2}}(a_{1}-a_{2})$ and the second largest
$\frac{1}{\sqrt{2}}(a_{1}+a_{2})$, as indicated. Here, $a_{i}$ means the
emergent eigenstate of subgraph $G_{a_{i}}$. The order of these eigenstates
can be reversed by changing the signs of the coupling edges.

The emergent superposition state can be tuned to a mixed state—that is, where
it could be $a_{1}$ or $a_{2}$ with classical probability—by reducing the
number of coupling edges (or their coupling strength) so that the subgraphs
are not synchronized and the eigenvalues of $\frac{1}{\sqrt{2}}(a_{1}-a_{2})$
and $\frac{1}{\sqrt{2}}(a_{1}+a_{2})$ converge. This is seen by comparing the
graph spectra in Fig 4a and Fig 4b. Conversely, when the number of coupling
edges is increased, the graph $G$ becomes overall $k$-regular; it is no longer
described as a pair of coupled $k$-regular random subgraphs. Then what was in
the other examples the second largest eigenvalue converges into the density of
random states, Fig 4c. This is why we want to balance the density of edges
within each of $G_{a_{1}}$ and $G_{a_{2}}$ relative to those coupling edges
that connect the subgraphs.

To summarize, the keys to the proposal of these QL bits are:

  1. 1.

By coupling a pair of coupled $k$-regular random subgraphs $G_{a_{1}}$ and
$G_{a_{2}}$, we can produce convex combinations of the eigenstates of the
subgraphs. These states therefore serve as basis states that can be generated
as superposition states.

  2. 2.

The subgraphs are $k$-regular random graphs, which ensures that the basis
states and their superpositions are very robust for networks of arbitrary
complexity and are identified clearly by their emergent eigenvalue.

  3. 3.

The $k$-regular random subgraph construction also ensures that, regardless of
the number of nodes (vertices) in each subgraph, the emergent eigenvalues are
found at the same frequencies, that is, $d$, where $d$ is the average degree
of vertices in the subgraph. $d$ is less than $k$ when the graphs are
disordered and have randomly missing edges[6].

Figure 4: Examples of networks that produce QL states, as described in the
text. The left diagrams show the network. The middle pictures shows the
subgraphs $G_{a_{1}}$ and $G_{a_{2}}$ separated in space, and a schematic
indications of the coupling edges between them. On the right the spectra of
the graphs are shown. The eigenstates are written as linear combinations of
the eigenstates of the subgraphs, that is, $a_{1}$ is the emergent eigenstate
of $G_{a_{1}}$, and so on. In the graphs, the red edges are negative. Other
edges are taken to be positive in these examples.

How do we think about what these graphs represent physically? Our working
model is that each vertex is an oscillator. The concept of an oscillator is
very broad, but the the idea is that the vertex can flip periodically between
two states or phases. It might be the phase of a neuron firing, it might
represent the ‘for or against’ an opinion given some information specific to a
social sub-network, it might represent the phase of an electrical pulse in a
circuit, or perhaps the concentration—above or below a threshold—of a
biochemical marker. The coupling between vertices maps out interactions among
these states. In this paper we describe the networks within each QL bit as
perfectly correlated (all the edges within each subgraph have the same sign).
That is not necessary for these constructions, as suggested by Fig 3. We
simply need consensus within the network.

The QL bit can be synchronized to a superposition state of the constituent
subgraphs, or it can display either subgraph probabilistically, thus
satisfying Axiom 3. The physical concept of the graph is that it represents a
network of oscillating dipoles, one positioned at each vertex. Edges indicate
the dipole-dipole coupling between vertices. Negative edges denote coupling
between oppositely-phased dipoles.

Unitary operations, including QL gates, can be imagined for these networks as
graph homomorphisms. In particular, we simply need operations that flip
subgraph or coupling edge phases. We will explain this, with examples, in a
later section. The QL bits thus satisfy Axiom 4.

Lastly, these states are presented as spectra and associated states of graphs,
that is, of the graphs’ adjacency matrices. Therefore, the off-diagonal terms
are all set to unity. In a realistic system, these off-diagonal terms have
energy values reflecting the coupling and the diagonal entries can have
frequency disorder[21]. Here we do not need to examine various parameters
because we are attempting to establish ‘existence’ of the prescribed QL
states—are they feasible with the right kind of network? The key to the
argument in favour is that the eigenvalue gap between the emergent state and
the next state is fixed by the mean valency of the disordered $k$-regular
graph, $d$. For large networks, then, we simply need to ensure that $d$ is
‘large enough’ so that the QL-bit superposition state is emergent, that is,
distinct in the spectrum.

###  3.3 Quantum-like states

Two or more QL bits can be suitably coupled so that QL states are constructed.
By imagining the QL bit as an oscillating dipole (or transition density[48,
49]), then we can propose how ‘wiring diagrams’ can lock the relative phases
of subgraphs of each QL bit in lock-step across the multi-QL bit network. Then
the coupling between two QL bits labelled A and B, with basis states
$a_{1},a_{2}$ and $b_{1},b_{2}$ respectively, can be thought of, conceptually,
as a dipole-dipole coupling[49]. The coupling Hamiltonian is given by:

$H=\hbar\nu_{A}|A\rangle\langle A|+\hbar\nu_{B}|B\rangle\langle B|+\sigma
J(|A\rangle\langle B|+|B\rangle\langle A|)$ (5)

where $\nu_{A}$ is the oscillation frequency of QL bit A, $\nu_{B}$ is the
oscillation frequency of QL bit B, and $J$ is the coupling. We have
$\sigma=\pm 1$ depending on whether the QL bit pairs are coupled so that the
oscillations $a_{1}\rightarrow a_{2}$ and $b_{1}\rightarrow b_{2}$ are in-
phase or anti-phased, respectively. In our calculations of the the graph
adjacency matrices, we have $\nu_{A}=\nu_{B}=0$ and $J=1$. The basis states
are emergent states of the subgraphs, for example by $a_{1}$ we mean the
emergent state of subgraph $G_{a_{1}}$.

If we think of each QL bit conceptually as an oscillator in a superposition of
states of its subgraphs $|x_{1}\rangle$ and $|x_{2}\rangle$, where
$x\in\\{a,b,c,\dots\\}$, then we can envision how suitable connections among
QL bits can produce a variety of synchronized states reminiscent of the set of
states in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}\otimes\dots$. To
do this, we introduce signed edge sets that lock subgraphs of the QL bits in
or out of phase with each other, so that the entire graph is locked relative
to a global phase.

What we need to establish is a representation of the basis of the relevant
tensor product of Hilbert spaces as a pattern of phases (relative phases) of
the eigenstates of network subgraphs in the emergent eigenstate of the entire
network of coupled QL-bits. The product basis for quantum states has inbuilt
correlations that we can represent as linear combinations in a free vector
space that will serve as the QL representation, physically laying out the
phase correlations across the network. This is a key technical challenge that
we will address in future work and highlight some open questions. While the
method proposed here enables us to produce superpositions of states within any
excitation subspace, generating more complex states requires further work. The
idea we use is that a set of appropriate linear combinations of QL-bit states
(naturally produced by joining QL-bits by edges) allows us to construct a QL
state.

Here we only consider the Bell states in order to explain the concept, Fig 5.
Fig 5a shows how the superposition states are produced by coupling the QL
bits. Any of these four states can be selected as the emergent state by
suitable choice of signs of the couplings within each QL bit and between the
QL bits. The case of positive-signed edges between the QL bits is shown here.
The phases of the blocks of coefficients in the resulting eigenstates are
indicated, with ordering $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$. In Fig 5b, c we
show schematics of how to construct networks that emulate two of the Bell
states, $|\Psi_{-}\rangle$ and $|\Psi_{+}\rangle$. The pictures of the
networks shows the idea of how the coupling between subgraphs controls their
relative phases. The analogy of coupled dipoles shows the same concept and the
phase map sketches indicate how the subgraph phases are phase-locked to each
other. Calculated spectra are shown in the other panels, Fig 5d-g. First, for
reference, in Fig 5d we show the spectrum of $G_{a_{1}}\oplus G_{a_{2}}\oplus
G_{b_{1}}\oplus G_{b_{2}}$, where none of the subgraphs are coupled. This is
the perfectly mixed state. In Fig 5e, the spectrum of the network where the
subgraphs within each QL-bit are coupled (with random edges between subgraphs
within each QL-bit added with probability $p=0.2$), but the QL-bits are not
coupled to each other. So, here we have QL-bits but no entangled states. The
adjacency matrix is therefore represented as two blocks, $G_{A}\oplus G_{B}$.

In Fig f, g we show spectra where QL-bits are coupled, removing the block form
of the adjacency matrix, to produce an emergent state corresponding to the
Bell state $|\Psi_{-}\rangle$. The inter-QL-bit coupling edges are added with
$p=0.05$ and with signs as indicated in Fig 5b. In Fig 5g we show the spectrum
when all coupling edges are added with $p=0.05$, that is, both the intra-QL-
bit and inter-QL-bit edges. For each calculation, the composition of the
emergent eigenstate is indicated.

Figure 5: (a) The superposition states are produced by coupling the QL bits.
Here constructions of the four Bell states are shown. (b, c) Diagrams that
show how the QL-bit is analogous to a pair of coupled oscillating dipoles. To
change the state of the QL-bit from out-of-phase oscillations to in-phase
oscillations, the signs of the edges need to be reversed, as indicated. Red
edges are negative, blue edges are positive. The corresponding phase map
sketches are also shown for reference. (d) The spectrum of $G_{a_{1}}\oplus
G_{a_{2}}\oplus G_{b_{1}}\oplus G_{b_{2}}$, that is none of the subgraphs are
coupled. (e) The spectrum of the direct sum of QL-bit graphs, that is, the
subgraphs within each QL-bit are coupled (with random edge added with
probability $p=0.2$), but the QL-bits are not coupled to each other. (f) The
QL-bits are coupled to produce an emergent state corresponding to the Bell
state $|\Psi_{-}\rangle$. The coupling edges are added with $p=0.05$ and with
signs as indicated in part (b). (g) The QL-bits are coupled to produce an
emergent state corresponding to the Bell state $|\Psi_{-}\rangle$, like in
part (e). The difference in this calculation _all_ coupling edges are added
with $p=0.05$, see text.

We can characterize the resulting states according to the overall ‘excitation
number’ of the $|x_{2}\rangle$ states, that is, just like the excitation
subspaces of
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}\otimes\dots$.
Synchronization locks the QL bit array into a particular global state (which
can also be a mixed state, caused by insufficient synchronization). Just like
in the coupled dipole analogy, when $N$ QL bits are coupled, we can specify
$N$ states that map to superpositions of product states comprising one
$|x_{2}\rangle$ state and $N-1$ of the $|x_{1}\rangle$ states—the single
excitation subspace. We have ${N\choose 2}$ double excitation states, and so
on. A frequency difference between $|x_{2}\rangle$ and $|x_{1}\rangle$ is
allowed, but not essential, depending on how the states are detected.

The QL bits are networks that can be spatially separated from each other (like
shown in Fig 5), or they can be completely entwined. They can be different
sizes (different numbers of nodes) and can be slightly detuned from one
another (thus changing how the states mix). There is a rich variety of ways
that the system can be composed. The important rule is that the expander
subgraphs are approximately $d$-regular, so that the eigenvalues of each QL
bit are close to resonance with the others.

##  4 Context and the Växjö model

A model for QL measurements has been developed to apply to a wide range of
classical systems, the Växjö model[5]. These developments complement our
definition of QL states, which give concrete examples of how to engineer or
identify complex systems that yield quantum-like probability laws.

The Växjö model does not require specification of systems and their states.
Instead, it focuses on measurement and probability laws. Specifically, how
probability laws might be formulated so that interference-like phenomena are
accounted for. By appending our observables with extra structure that relates
to conditions of the measurement and other information potentially gathered
through prior measurements—context—Khrennikov shows how probabilistic laws can
incorporate interference effects. He defines the probability for obtaining
$a=\alpha$ by observing $a$ under context $\mathcal{C}$ as:

$p^{a}_{\mathcal{C}}(\alpha)\equiv\mathbb{P}(a=\alpha|\mathcal{C}),$ (6)

where each contextual probability satisfies $p^{a}_{\mathcal{C}}(\alpha)\geq
0$ and

$p^{a}_{\mathcal{C}}(\alpha_{1})+\dots+p^{a}_{\mathcal{C}}(\alpha_{n})+\dots=1$
(7)

for any context $\mathcal{C}$.

######  Definition 4.1.

(Khrennikov) Contextual expectation $\mathbb{E}[a|\mathcal{C}]$ of an
observable $a\in\mathcal{O}$ with respect to context $\mathcal{C}$ is given by

$\bar{a}_{\mathcal{C}}=\mathbb{E}[a|\mathcal{C}]=\alpha_{1}p^{a}_{\mathcal{C}}(\alpha_{1})+\dots+\alpha_{n}p^{a}_{\mathcal{C}}(\alpha_{n})+\dots.$
(8)

######  Definition 4.2.

(Växjö model) A contextual probability model is a contextual probability space
$\mathcal{P}=(\mathcal{C},\mathcal{O},\pi)$ such that $\mathcal{C}$ contains a
special subfamily of contexts
$\\{\mathcal{C}^{a}_{\alpha}\\}_{a\in\mathcal{O},\alpha\in X_{a}}$ which are
interpreted as $[a=\alpha]$-selection contexts. Context
$\mathcal{C}^{a}_{\alpha}$ corresponds to the selection with respect to the
result $a=\alpha$. Moreover, it is assumed that $\mathcal{C}^{a}_{\alpha}$
satisfies the condition

$\mathbb{P}(a=\alpha|\mathcal{C}^{a}_{\alpha})=1.$ (9)

It is assumed that, for each observable $a\in\mathcal{O}$ and its value
$\alpha$, the selection context $\mathcal{C}^{a}_{\alpha}$ is uniquely
determined in the class of contexts $\mathcal{C}$ of the model.

We also need to define what Khrennikov terms ‘transition probabilities’ using
a conditional probability construction:

$p_{\beta|\alpha}\equiv\mathbb{P}(b=\beta|a=\alpha)=\mathbb{P}(b=\beta|\mathcal{C}^{a}_{\alpha}),$
(10)

because QL probability laws will be evidenced by measurements on two
observables, $a$ and $b$. The reader is referred to [5] for more details.

The formula for total probability in the usual, Kolmogorov, model (i.e.
without the structure of contexts) is

$p^{b}(\beta)=\sum_{\alpha}p^{a}(\alpha)p_{\beta|\alpha}.$ (11)

But, as Khrennikov argues, this formula is not guaranteed to hold in the Växjö
model, where probability distributions in the set of data for
$a,b,\mathcal{C}$ may not be described by a single probability space. For
instance, the context for observation $a$ might vary depending on whether we
previously learned something about $b$, through observation (or vice versa).
That concept is exploited by the ‘question order effect’, discussed below.

The deviation of the contextual total probability from the Kolmogorov law
serves as a kind of interference term, $\delta(\beta|a,\mathcal{C})$:

$\mathbb{P}(b=\beta|\mathcal{C})\equiv
p^{b}_{\mathcal{C}}(\beta)=\sum_{\alpha}p^{a}_{\mathcal{C}}(\alpha)p_{\beta|\alpha}+\delta(\beta|a,\mathcal{C}).$
(12)

The interference term is the key to connection with quantum systems, where
interference of probability amplitudes is a defining feature. The interference
term identified here is the classical counterpart to conditional expectation
in quantum theory[50].

The QL states of classical synchronized systems proposed in the previous
section can represent states encoded with context. That context essentially
means that the QL bit networks or collective QL states respond and change as
information is gathered by other QL bits—that is, we make measurements. This
could be accomplished in various ways by connecting QL bits and/or adding
additional ‘context’ networks.

##  5 Examples of quantum-like function

Here we provide two simple, but very different, examples of how QL bits might
serve as a the basis for information processing. The first example integrates
the QL bits into Khrennikov’s concept of the question order effect, that
exploits interference effects that come from context, reflecting the order
that information is gathered. It indicates how to think about measurement of
QL states. The second example is a textbook protocol that exploits
entanglement. It provides a reference point for assessing QL state functions.

It is important to note that quantum states and QL-states do not necessarily
enable functions that are impossible to achieve by classical processes, but
the different mechanisms that they allow to achieve the function could have
advantages. The examples below illustrate different mechanisms of quantum
functions.

###  5.1 The Question Order Effect

Khrennikov and co-workers have studied how QL measurement theory can be useful
for explaining psychological measurements[51, 52, 53]. That work provides an
example of how the action of measurement matters when we are dealing with QL
states. A prototypical example is the ‘question order effect’. The question
order effect refers to the statistical observation that the answers to
questions, for example in a survey, can depend on the order in which the
questions are presented. Here we describe the QL measurement model for the
question order effect qualitatively, using Figure 6. We refer the reader to
the paper by Ozawa and Khrennikov for the details[52].

Figure 6: Diagram that shows the way networks interact for the steps needed to
understand the question order effect described in the text.

The system comprises a pair of ‘entangled’ QL states. These projections $A$
and $B$ on the Hilbert space of the system, $\mathcal{H}$, correspond to
questions labelled $A$ and $B$ respectively. For each question we can think
about eigenvalues corresponding to answers of yes or no. To measure the
quantum state of our system, for example to measure the answer to question A,
we need to interact the system with a measurement probe. This probe also
comprises an entangled pair of QL states with projections on the Hilbert space
$\mathcal{K}$. During measurement of $A$, for example, the composite system
evolves according to a unitary operator $U_{A}$ acting on
$\mathcal{H}\otimes\mathcal{K}$. The interaction encodes the probe with the
eigenvalue of $A$ that can be read out using a meter observable. The
interaction concomitantly changes the state of the system, so that a
subsequent measurement on $B$ can give a different answer than if this
measurement were carried out before the measurement on $B$. This interplay
between system and measurement can model the question order effect.

The example shows how an appropriate ‘read out’ of states is important for the
kinds of sophisticated functions enabled by networks, in particular when a
sequence of evolving answers or iterations may be needed. A good example is
interactions among the states to reach ‘consensus’.

###  5.2 Superdense Coding for Sensing or Processing

Superdense coding is a strategy whereby a quantum resource (entanglement) can
mediate communication of two classical bits of information, potentially over a
long distance, In an illustrative example, we send two classical bits (00, 10,
01, or 11) using a single qubit[1].

Two QL bits can be ‘entangled’, that is, synchronized, to produce a reference
Bell-like state. Now apply a quantum gate to QL-bit A in isolation. In
practice, this means we switch phases appropriately of the subgraphs in the QL
network A. This can be understood intuitively using the analogy where each QL
bit comprises a pair of coupled oscillating dipoles. In practice, we simply
need to flip the signs of the edges within subgraphs and/or of the coupling
edges. The details for each quantum gate are shown in Fig 7. Here we use the
usual notation[4] for the states and name QL-bit A Alice and QL-bit B Bob. We
can associate $|0\rangle_{A}$ with $|a_{1}\rangle$ and $|1\rangle_{A}$ with
$|a_{2}\rangle$, etc.

Bell-like states correspond to the four unique ways the two pairs of
oscillating dipoles are locked in sync, Fig 7. Relative to the fixed phases of
QL bit B, then, we can rotate the phases in QL bit A. The information sent to
QL bit B amounts to a single QL bit. However, a readout of the total Bell-like
state of the A-B QL-bit pair translates into two classical bits of
information. Two qubits are needed to encode two classical bits. Therefore we
have the same total information supplied by four Bell states as four classical
pair states. However, by making use of entanglement, QL bit A can communicate
a pair of classical bits by sending only a single qubit. Notice that Alice’s
QL-bit has four possible states, but pairs of them differ only by an overall
phase, so they do not convey different information in isolation. It is only
with the context of the QL state that the four QL-bit states translate to four
different interpretations. The key is that context is provided by referencing
phase mutually among the QL-bits.

Figure 7:  A table that shows how Alice rotates the state of her QL-bit by a
series of quantum gates and how that changes the QL state observed by Bob. The
operations within the networks that produce the QL states are indicate
schematically as a‘ phase map’.

The total Bell-like state of the A-B QL-bit pair could be detected by its
spectrum. It is not obvious that the scheme would be useful for network
communication when the entangled network states are close together, because we
cannot exploit ‘spooky action at a distance’. However, the protocol might
allow QL-bit A to serve as a quantum sensor, with read-out facilitated by QL-
bit B. The proof of principle may also inspire ways that QL circuits could be
constructed for more sophisticated processing operations.

A main conclusion from this example is that it is feasible to implement
various quantum-like gates on the network states. For instance, Hadamard gate
that ‘rotates’ $|a_{1}\rangle$ to the superposition
$(|a_{1}\rangle+|a_{2}\rangle)/\sqrt{2}$ could be achieved by activating the
coupling edges between subgraphs and activating the network corresponding to
$G_{a_{2}}$. Then the state with the frequency of the emergent state—recall
that is fixed by the valency of the $k$-regular graphs—is becomes the required
superposition. These gates are quantum _like_ because the states and gates are
composed by dynamic structures of the network, not by matrix operations.
Nevertheless, then set of ways the network can be reconfigured offers a
representation for the set of matrix operations that, in turn, represent maps
on states in the relevant Hilbert space.

##  6 Discussion

In this study we have asked: Is it worth looking for quantum effects in very
complex, ‘messy’ systems? The standard answer is that quantum states are too
fragile to exist in such systems, including biological settings. However,
given the many relevant new ideas that have emerged over the past
decades—including, for instance, more sophisticated understandings of
networks, synchronization phenomena, and advances in solving quantum dynamics
for large and complicated systems, we argue that the time is right to revisit
this question. Furthermore, perhaps we do not need true ‘quantum states’, but
instead sufficiently ‘quantum like’ (QL) states will suffice. That inspires a
shift from the common idea that we need to make chemical systems pristine in
order to host quantum states in the form of qubits, to the idea that there are
new opportunities to discover by identifying more abstract, but surprisingly
robust states. These states can even be realized in electronic circuits. The
main objective of this work is to propose concrete examples of very large and
complex networks that host states resilient to disorder and displaying QL
properties.

By considering suitably constructed graphs with expander properties, it turns
out that we can find robust QL states that can inhabit huge, complex systems
and that satisfy the axioms proposed in Sec. 3. A requirement for states of
any large and complex quantum system, whose structure is, in a sense,
‘organic’, is that they must especially stable with respect to various kinds
of disorder. This is where the concept of emergent or collective phenomena is
important because it allows the state of interest to be distinguished from the
many other states in the spectrum. This emergence property thus protects the
state. To abstract a suitable model we need to account for an underlying
structure, that could be encoded by some kind of rule (here the rule is that
the graph is $k$-regular). Then we allow disorder in that structure or on the
structure so that there is a large margin for flexibility in the precise
structure. Frequency disorder at sites in a network is an example of disorder
_on_ the structure[21]. Disorder caused by randomly deleting edges from the
network is an example of disorder _in_ the structure[6]. We explicitly
included this latter kind of disorder in the examples exhibited in the present
work.

We identified $k$-regular graphs as examples of networks that can serve as the
basis for QL states. However, any expander graph would serve this purpose. The
best possible expanders are Ramanujan graphs[54, 55, 56], but ideal expansion
properties do not appear to be crucial because in many interesting or useful
‘real world’ scenarios, it is desirable for the graphs to be disordered—so
that precision construction is not needed. The expander graphs also have
intrinsically interesting properties that are desirable for networks involved
in communication and information transfer and processing[36, 57, 58].
Expanders are not necessary for the constructions, just sufficient. Other
graphs might be of interest, including chiral graphs that could host states
analogous to electron spins or photon polarization.

By positing a QL bit structure based on coupled expander graphs and combining
that with the concept of QL states, here an ‘existence proposal’ for huge
quantum states was described. The work opens up many questions for future
work. Below we discuss two open questions: (i) Is there a ‘quantum-like’
advantage for function? (ii) Could QL states and function be used by neural
networks in the brain?

###  6.1 Is there a QL advantage?

Synchronized networks of QL states could serve as a hardware platform for
information processing. There are three big questions: (i) what is the QL
advantage for certain functions? (ii) What platforms could be designed to
produce and test QL states? (iii) What experiments could test the QL functions
of QL states?

The main point argued in the present work is that we defined construction of a
QL bit $|A\rangle$ so that its emergent states ‘live in’ a Hilbert space
$\mathcal{H}_{A}$, just like quantum states. A set of axioms is proposed that
guide the properties the states need to have. We exhibited examples of these
states and the way they are measured. It was also shown how QL gates can be
implemented by controlling phases—specifically, signs of edges within each QL
bit. Based on this framework, one can in principle adapt circuit designs from
quantum processing systems. This is a key direction for future work that will
provide answers to how QL states can have a QL advantage for function.

Compelling demonstration of QL state function requires a hardware platform for
producing those states. The obvious approach is to design circuits of coupled
electronic oscillators[59]. These kinds of systems could be miniaturized, and
may be well suited for tasks like pattern recognition[60]. QL states might
also be implemented in the framework of spiking neural networks, thereby
taking advantage of neuromorphic hardware systems[61].

Later work can adapt the concepts for more complex networks in optical,
biological, chemical, and other systems. Biological systems are certainly an
intriguing target because they comprise a huge number and diversity of
networks. Manytimes these networks are so dense and complicated that they are
referred to as ‘hairballs’[62]. The QL-states proposed here provide a platform
for re-thinking quantum biology because it is much more likely to find QL-
states that quantum states in such complex systems.

###  6.2 Quantum neural networks

It is well-known that neurons are connected to form the complex networks of
the brain. It is now generally accepted that the ways these networks process
signals and information to produce responses and functions do not simply
involve signal transmission. Rather, information can be encoded in the timing
and spatial coherence of signal oscillations[20, 63, 64, 65, 66, 67, 68, 69].
We are not attempting to review that field here, but simply to sketch a small
part of what is known to give examples of how networks function in what is
arguably nature’s most sophisticated machine—the brain.

One key discovery is that information processing seems to be facilitated by
oscillations of neuron activity. There are characteristic frequencies for
these oscillations, grouped into bands denoted delta (<4 Hz), theta (4–8 Hz),
alpha (8–12 Hz), and beta (12-30 Hz), in the low frequency range, to the high
frequencies in the gamma band (30–80 Hz) and above.

One example is shown in Fig 8. Here gamma band oscillations are studied in the
visual cortex of a monkey as it was presented with an image[70]. In the
experiment, neuronal activity was detected using an electrocorticographic grid
that monitored the left brain hemisphere of a macaque monkey. A number of
prior studies had shown that neuronal gamma-band synchronization in visual
cortex is associated with various kinds of functions. This study suggested
that 50 to 80 Hz gamma-band activity plays a key role in the process of
viewing. Fig 8 shows the main result of the paper, a unipolar local field
potential trace taken as the monkey looks at two oranges.

Figure 8: Gamma band oscillations observed in the visual cortex of a monkey as
it was presented with an image. The monkey’s focus is tracked by the line on
the image. Reproduced with permission from [70].

A second example serves to illustrate that brain function involves the
interaction of widely distributed cortical regions. Long-range synchronization
has been implicated in enabling the coordination of these regions across the
brain and to communicate between brain regions[66]. For example, how do we map
sound to meaning[65]? In [71] the authors study the perception process. They
used an analysis of EEG recordings in humans that allowed correlation of brain
oscillations to be mapped. The experiment involved the subject watching a
screen where two bars come together, then move apart. There is an audio cue
when the bars meet. The measured power response during the stimulus is
collected in maps, depicting the theta (4–8 Hz), alpha (8–16 Hz), beta (16–32
Hz), low gamma (32–64 Hz), and high gamma (64–128 Hz) band activity across the
brain. The main finding of the work was enhanced long-range beta-synchrony
during stimulus processing, which was accompanied by local suppression beta-
band activity.

There is an immense body of work exploring brain function. A key question is
how different sensory inputs are linked to appropriate processing circuits to
draw conclusions. It has been proposed that ‘binding’ of these units is
facilitated by phase synchronization[63]. It has also been proposed that the
oscillation frequencies allow these disparate computations to associate[67].
Further ideas hypothesize that a complex interplay of oscillations is
important for perception[69].

In sum, on one hand the brain functions using a complex network of networks
that become synchronized during function. Here is where there is compelling
evidence that coherent oscillations on complex networks are functionally
operative in the brain, so a connection with the ideas hypothesized in present
paper—that QL function is possible—is feasible. However, it remains an
enticing idea that requires further work. In particular, the way the dynamic
networks interplay in the brain, and even how coherence is exploited are
questions that have not yet been elucidated in detail. The present work at
least provides a framework for devising testable hypotheses for additional,
likely surprising, mechanisms that might be at play in the sophisticated
networks of the brain.

##  7 Conclusion

We proposed a quantum-like (QL) bit structure based on coupled expander
graphs. Combining that with the concept of QL states, we suggest a way that
huge, complex systems can host states that can process information in a QL
fashion.

Specifically, it was suggested that building blocks suited for QL states are
networks, possibly very complex, that we defined using graphs. These networks
can be functionally sophisticated and encode a lot of information that is
distilled into the emergent states we can exploit for QL like processing. The
emergent states of networks are classical. They arise from feedback. We
developed a way to build from these classical states, new states that have
properties analogous to quantum states. Specifically, we based this
construction on a QL bit that ‘lives in’ a Hilbert space $\mathcal{H}_{A}$. We
demonstrate a candidate QL bit by partitioning an edge-disordered $k$-regular
random graph $G_{A}$ into two blocks, giving the subgraphs $G_{a_{1}}$ and
$G_{a_{2}}$, then connect those blocks to yield superposition states of the
emergent states of each subgraph. The intrinsic $k$-regular random structure
ensures that the emergent states are robust.

The work opens up many questions for future work. We explicitly discussed two
major open questions: (i) Is there a ‘quantum-like’ advantage for function?
Perhaps this could be leveraged in hardware for QL-computing, which would be
greatly simplified from that needed for true quantum computing, but may offer
an attractive compromise between classical and quantum architectures. (ii)
Could QL states and function be used by neural networks in the brain? We
conclude that quantum-like states can exist in arbitrarily complex systems and
that it is worthwhile investigating further examples and working out where
they can have a functional advantage. The primary advance of the paper was to
overcome the main objection for various proposals of quantum states in
biology, the brain, and other complex systems—that is, that quantum states are
too fragile to serve a functional role. This was done by proposing, first that
we should think about ‘quantum like’ states[5], not quantum states. Second,
that we can devise a blueprint for suitable states, satisfying the axioms
proposed above, by exploiting expander properties of graphs so that complex
networks yield a single key state in their spectrum.

## Acknowledgments

This material is based upon work supported by the National Science Foundation
under Grant No. 2211326 and the Gordon and Betty Moore Foundation through
Grant GBMF7114. Graziano Amati and Andrei Khrennikov are thanked for helpful
suggestions for improving the paper.

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