# Complex Analysis of Intelligent Systems

M.W. AlMasri<EMAIL_ADDRESS>
###### Abstract.

Logic gates can be written in terms of complex differential operators where
the inputs and outputs are analytic functions with several variables. Using
the polar representation of complex numbers, we arrive at an immediate
connection between the oscillatory behavior of the system and logic gates. We
explain the universal programming language (UPL) used by physical objects to
process information. To assure the causality structure in UPL, we introduce
the concept of layers that characterizes the computations for each time scale.

##  1\. Introduction

Understanding how nature “thinks” is undoubtedly the ultimate aim of our
scientific research. Since the last decade, we have witnessed great progress
in our computational capabilities, paving the way for revolutions in the field
of artificial intelligence. Interestingly, despite the numerous physical
theories of computation, we still lack a decisive answer to the question: how
nature “thinks”? Understanding this vital issue would complete the current
perspective and enables us to develop further technologies at maximum capacity
in terms of energy efficiency and computation capabilities. It has been
suggested a few decades ago that our universe might be a giant computer[1].
This has inspired researchers to strongly believe that information is
physical[2].

Computers are physical systems: the laws of physics dictate what they can and
cannot do. Concretely, the speed with which a physical device can process
information is limited solely by its energy, and the amount of information
that it can process is limited by the number of degrees of freedom it
possesses. Therefore, there must be an intimate relationship between the
dynamical evolution of physical states and their behavior regarding processing
and storing information [3, 4, 5, 6, 7].

In order to have a true description of how nature thinks, we start with the
properties of elementary particles and their behaviors. The justification of
this ideology stems from the feeling that nature uses one universal
programming language for encoding and decoding information at different
scales. While it is possible to artificially construct other ways for coding
and encoding information, the way that nature chooses must be the fastest and
therefore the most efficient way for processing information in order not to
violate the causality principle. In other words, if we assume a physical
system $\mathcal{S}$ to process information at some time scale $t_{0}$ then it
must be equivalent or less efficient than the way the container $\mathcal{C}$
process information at the same time scale $t_{0}$. To a large extent, the
container is the universe that contains all the subsystems evolving on the
same time scale. We shall explain these in detail and introduce the concept of
layers in computation.
According to quantum mechanics, subatomic particles behave like waves
propagating in spacetime. The wave-particle duality was proposed by de Broglie
[8]. The de Broglie wavelength $\lambda$ associated with a particle is

(1) $\displaystyle\lambda=\frac{h}{p},$

where $h$ is Planck’s constant and $p$ is the particle’s momentum. In the
Davisson-Germer experiment, electrons scattered by the surface of a crystal of
nickel metal produced a diffraction pattern similar to waves111C. Davisson and
L. H. Germer, _Phys. Rev._ 30 (1927) 705. Interestingly, changing diffraction
patterns, as we shall see in the incoming sections, are intimately connected
to the operations of logic gates.

The correspondence between oscillatory behavior and logic gates computation is
assured mathematically by writing logic gates in terms of complex differential
operators acting on analytic functions. Oscillatory systems are omnipresent in
the universe. Some notable examples are the attractor cycle oscillators,
rotors, neural oscillation patterns in the brain, synchronized chaos, and
Josephson junctions. Other examples in biology and physics can be read from
[9, 10].

##  2\. Hilbert Space of Analytic Functions

Let $U$ be a non-empty open set in $\mathbb{C}^{d}$ and let $\mathcal{H}(U)$
denote the space of holomorphic functions on $U$. A complex-valued function
$f:U\rightarrow\mathbb{C}$ is said to be holomorphic if it is differentiable
in a neighborhood of each point in $U$. Let $\mathcal{H}L^{2}(U,\omega)$
denotes the space of $L^{2}$-holomorphic functions with respect to the weight
$\omega$, that is,

(2)
$\displaystyle\mathcal{H}L^{2}(U,\omega)=\\{F\in\mathcal{H}(U)|\int_{U}|F(z)|^{2}\;\omega(z)\;dz<\infty\\}.$

Here $dz$ denotes the $2d$-dimensional Lebesgue measure on
$\mathbb{C}^{d}=\mathbb{R}^{2d}$ not a line integral. Some examples of
holomorphic function spaces are the weighted Bergman and the Segal-Bargmann
[11, 12, 13, 14]. In this work, we will use the Segal-Bargmann spaces .

The Segal-Bargmann spaces (also known as Bargmann spaces)
$\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{t})$ are spaces of holomorphic functions
with Gaussian integration measure $\mu_{t}=(\pi)^{-d}e^{-|z|^{2}/t}$ and inner
product of the form [11]

(3) $\displaystyle\langle f|g\rangle=(\pi
t)^{-d}\int_{\mathbb{C}^{d}}\overline{f}(z)\;g(z)\;e^{-|z|^{2}/t}dz,$

Where $|z|^{2}=|z_{1}|^{2}+\dots+|z_{d}|^{2}$ and $dz$ is the $2d$-dimensional
Lebesgue measure on $\mathbb{C}^{d}$ and $t$ is a constant equals to $\hbar$
if the physical states under study are quantized.

The Segal-Bargmann transform is a unitary map
$B_{t}:L^{2}(\mathbb{R}^{d},dx)\rightarrow\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{t})$
defined as

(4) $\displaystyle B_{t}f(z)=(\pi
t)^{-d/4}\int_{\mathbb{R}^{d}}e^{\left(-z^{2}+2\sqrt{2}z\cdot
x-x^{2}\right)/2t}f(x)\;dx.$

The reproducing kernel is

(5) $\displaystyle
K(z,w)=\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!t^{n}}}\frac{\overline{w}^{n}}{\sqrt{n!t^{n}}}=\sum_{n=0}^{\infty}\frac{1}{n!}(\frac{z\overline{w}}{t})^{n}=e^{z\overline{w}/t}$

The kernel $K(z,w)$ is holomorphic in $z$ and anti-holomorphic in $w$. It
satisfies the following properties:

1- $K(z,w)=\overline{K(w,z)}$

2- For each fixed $z\in\mathbb{C}$, $K(z,w)$ is square-integrable $d\mu_{t}$.
For all $f(z)\in\mathcal{H}L^{2}(\mathbb{C},\mu_{t})$, we may write

(6) $\displaystyle f(z)=\int_{\mathbb{C}}K(z,w)\;f(w)\;\mu_{t}(w)\;dz.$

3- For all $z,w\in\mathbb{C}$ ,

(7) $\displaystyle\int_{\mathbb{C}}K(z,u)\;K(u,w)\;\mu_{t}(u)\;du=K(z,w)$

The monomials $\\{\frac{z^{n}}{\sqrt{n!t^{n}}}\\}$ form an orthonormal basis
in $\mathcal{H}L^{2}(\mathbb{C}^{n},\mu_{t})$. The orthonormality condition
with $t=d=1$ is

(8)
$\displaystyle\frac{1}{\pi}\int_{\mathbb{C}}e^{-|z|^{2}}\;\overline{z}^{n}z^{m}dz=n!\;\delta_{mn}.$

Using the properties of the Bargamnn spaces, one could calculate the numerical
values of the $n$th derivative of any complex functions at point $z_{0}$ by
integration over the Bargamnn space. Let $f(z)$ be an analytic function of
complex variable $z$. The Taylor series of $f$ at a complex number $z_{0}$ is
[15]

(9)
$f(z)=f(z_{0})+\frac{f^{\prime}(z_{0})}{1!}\left(z-z_{0}\right)+\frac{f^{\prime\prime}(z_{0})}{2!}(z-z_{0})^{2}+\dots=\sum_{n=0}^{\infty}\frac{f^{(n)}(z_{0})}{n!}\left(z-z_{0}\right)^{n}.$

When $z_{0}=0$, the Taylor series reduces to a Maclaurin series i.e.
$\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^{n}$. Let us compute the inner
product $\langle z^{n}|f(z)\rangle$ where
$f(z)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^{n}$ is a Maclaurin series, we
find

(10) $\displaystyle f^{(n)}(0)=\langle z^{n}|f(z)\rangle$

which is very interesting since it gives a closed analytical expression for
the $n$th derivative of any analytic complex function at $z=0$ by computing
the inner product of its Maclaurin series expansion with monomial $z^{n}$ in
$\mathcal{H}L^{2}(\mathbb{C},\mu_{1})$. Practically, this means computing
Gaussian integrals of the form
$\frac{1}{\pi}\int_{\mathbb{C}}\overline{z}^{n}f(z)e^{-|z|^{2}}dz$. For the
Taylor series, the formula 10 becomes

(11) $\displaystyle f^{(n)}(z_{0})=\langle(z-z_{0})^{n}|f(z)\rangle.$

The derivative of a given analytic function $f(z)$ is

(12) $\displaystyle f^{\prime}(z)=\lim_{h\rightarrow 0}\frac{f(z+h)-f(z)}{h},$

where $h$ is the step size. For infinitesimally small $h$, we may compute the
derivative approximately using a forward-difference expression,

(13) $\displaystyle f^{\prime}(z)\approx\frac{f(z+h)-f(z)}{h},$

Using 11, the forward-difference expression at point $z_{0}$ assumes the form

(14)
$\displaystyle\langle(z-z_{0})|f(z)\rangle\approx\frac{1}{h}\left(f(z_{0}+h)-f(z_{0})\right)$

Analogously, the second-order forward-difference expression is

(15)
$\displaystyle\langle(z-z_{0})^{2}|f(z)\rangle\approx\frac{1}{h^{2}}\left(f(z_{0}+2h)-2f(z_{0}+h)+f(z_{0})\right).$

Thus, we have written the finite differences in terms of integrals over the
space of holomorphic function spaces with Gaussian integration measure.
Consequently, one could use the numerical methods for integrals to compute
derivatives approximately instead of finite differences[16]. This is more
efficient since for higher-order derivatives we would have long expressions
using the finite-difference method. However, using our method we can compute
higher-order derivatives using one integral similar to 10.

Now let us look closely at 14, using the fundamental theorem of calculus we
find

(16) $\displaystyle
f^{\prime}(z_{0})=\langle(z-z_{0})|f(z)\rangle\approx\frac{1}{h}\int_{z_{0}}^{z_{0}+h}f^{\prime}(z)dz.$

Interestingly, the derivative of function $f$ at point $z_{0}$ is
approximately equal to the integral of $f^{\prime}(z)$ between $z_{0}+h$ and
$z_{0}$. This means that we can equate integrals over holomorphic function
with Gaussian integration measure with integrals over ordinary complex space.

It is clear that working in the Bargamnn spaces enhances the space of
solutions by using the inner product properties of these spaces. For example,
the generalized hypergeometric function is

(17)
${}_{p}F_{q}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}\frac{z^{n}}{n!},$

where $(x)_{n}=x(x+1)(x+2)\dots(x+n-1),\;\;n\geq 1$ is the Pochhammer symbol
with $(x)_{0}=1$. Using 10, we find

(18)
$\displaystyle\frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}=\langle
z^{n}|_{p}F_{q}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)\rangle.$

Thus, the coefficients of generalized hypergeometric functions can be
determined uniquely using inner products in the Bargmann spaces.

The states of a quantum system with $n$ degrees of freedom are usually
described by functions either in the configuration space $(q_{1},q_{2},\dots
q_{n})$ or in the momentum space $(p_{1},p_{2},\dots p_{n})$. The complex
combinations of these variables have proven to be effective both at quantum
and classical levels. Moreover, they appear in the study of harmonic
oscillator and Bose particle field theories as the creation and annihilation
operators.

We define the complex conjugate operators as

(19) $\displaystyle\eta_{k}=2^{-1/2}\left(q_{k}-ip_{k}\right),$ (20)
$\displaystyle\xi_{k}=2^{-1/2}\left(q_{k}+ip_{k}\right).$

If $q_{k},p_{k}$ are self-adjoint operators satisfying the canonical
commutation relations (with Planck constant $\hbar=1$)

(21)
$\displaystyle[q_{k},p_{l}]=i\delta_{kl},\;[q_{k},q_{l}]=[p_{k},p_{l}]=0,$

Then it follows that

(22)
$\displaystyle\xi_{k}=\eta^{\star}_{k},\;\eta_{k}=\xi_{k}^{\star},\;[\xi_{k},\eta_{l}]=\delta_{kl},\;[\xi_{k},\xi_{l}]=[\eta_{k},\eta_{l}]=0.$

In 1928, Fock introduced the operator solution
$\xi_{k}=\partial/\partial\eta_{k}$ of the commutation relation
$[\xi_{k},\eta_{k}]=1$ in analogy with $p_{k}=-i\partial/\partial q_{k}$ of
the relation $[q_{k},p_{k}]=i$ and applied it to quantum field theory222V. A.
Fock, _Verallgemeinerung und Loesungder Diracschen statistischen Gleichung_ ,
Z.Phys. 49,339 (1928)..

Consider a quantum harmonic oscillator with a single-mode described in terms
of the coordinates $q,p$ in the phase space. The creation and annihilation
operators are

(23) $\displaystyle\eta=2^{-1/2}\left(q-ip\right),$ (24)
$\displaystyle\xi=2^{-1/2}\left(q+ip\right).$

with canonical commutation relation $[\xi,\eta]=1$. The Hamiltonian operator
is

(25) $\displaystyle H=\eta\frac{\partial}{\partial\eta}+\frac{1}{2},$

with energy eigenvectors equal to $\frac{\eta^{n}}{\sqrt{n!}}$.

It is convenient as it will appear later in this work to represent spin
operators in terms of uncoupled quantum oscillators. Here, we use the Jordan-
Schwinger map of the $\mathfrak{su}(2)$ algebra to deal with particles with
arbitrary spin number by means of creation and annihilation operators. For two
uncoupled quantum oscillators with annihilation operators $a$ and $b$, the
spin operators read,

(26) $\displaystyle\sigma_{x}=\left(a^{\dagger}b+b^{\dagger}a\right),$ (27)
$\displaystyle\sigma_{y}=-i\left(a^{\dagger}b-b^{\dagger}a\right),$ (28)
$\displaystyle\sigma_{z}=\left(a^{\dagger}a-b^{\dagger}b\right).$

The canonical commutation relations between the spin operators are

(29)
$\displaystyle[\sigma_{i},\sigma_{j}]=i\;\epsilon_{ijk}\;\sigma_{k},\;\;\;[\sigma_{i},\sigma^{2}]=0,$

where $i=x,y,z$. The squared spin operator reads

(30) $\sigma^{2}=\frac{N}{2}\left(\frac{N}{2}+1\right)$

where $N=N_{1}+N_{2}$ is the total number operator with $N_{1}=a^{\dagger}a$
and $N_{2}=b^{\dagger}b$. Normalized states with occupation numbers $n_{1}$
and $n_{2}$ can be obtained by applying the creation operators on the vacuum
state i.e. $|0,0\rangle$,

(31)
$\displaystyle|n_{1},n_{2}\rangle=\frac{(a^{\dagger})^{n_{1}}}{\sqrt{n_{1}!}}\frac{(b^{\dagger})^{n_{2}}}{\sqrt{n_{2}!}}|0,0\rangle.$

With $j=\frac{n_{1}+n_{2}}{2}$ and $m=\frac{n_{1}-n_{2}}{2}$ runs from $-j$ to
$j$ in integer steps.

In Bargmann representation, the spin operators 26, 27 and 28 read

(32) $\displaystyle\sigma_{x}=\left(z_{1}\frac{\partial}{\partial
z_{2}}+z_{2}\frac{\partial}{\partial z_{1}}\right),$ (33)
$\displaystyle\sigma_{y}=-i\left(z_{1}\frac{\partial}{\partial
z_{2}}-z_{2}\frac{\partial}{\partial z_{1}}\right),$ (34)
$\displaystyle\sigma_{z}=\left(z_{1}\frac{\partial}{\partial
z_{1}}-z_{2}\frac{\partial}{\partial z_{2}}\right).$

In this case, the normalized states are

(35)
$\displaystyle|n_{1},n_{2}\rangle=\frac{z_{1}^{n_{1}}}{\sqrt{n_{1}!}}\frac{z_{2}^{n_{2}}}{\sqrt{n_{2}!}}\equiv
f_{n_{1}n_{2}}$

since

(36) $\displaystyle\langle
m_{1},m_{2}|n_{1},n_{2}\rangle=\frac{1}{\pi^{2}}\int_{\mathbb{C}^{2}}\overline{f_{m_{1}m_{2}}}\;f_{n_{1}n_{2}}\;e^{-|z|^{2}-|w|^{2}}\;dz\;dw=\delta_{m_{1},n_{1}}\delta_{m_{2},n_{2}}.$

##  3\. Holomorphic Representation of Logic Gates

In this section, we formulate the logic gates as differential operators
written in terms of complex variables and their partial derivatives. Such
representation is possible both in classical and quantum regimes. To see this,
consider a hypothetical material which allows red light to pass through it and
absorbs black light. In this case, if we assign the variable $z_{1}$ for the
red light and $z_{2}$ for the black light. We may represent the action of the
material by the differential operator $\frac{\partial}{\partial z_{2}}$.
Generally, it is possible to find equivalent expressions for propositional
logic in terms of holomorphic variables and their partial derivatives. The
fundamental logic gates that operates between two analytic functions are the
Pauli gates plus the identity gate $(I,\vec{\sigma})$[17]. Any given two-input
gate can be written in terms of the identity operator and the Pauli vector.
Pauli gates are defined as complex differential operators that act on analytic
functions with two-variables

(37) $\displaystyle
X=\mathrm{NOT}=\sigma_{x}=\left(z_{1}\frac{\partial}{\partial
z_{2}}+z_{2}\frac{\partial}{\partial z_{1}}\right),$ (38) $\displaystyle
Y=\sigma_{y}=-i\left(z_{1}\frac{\partial}{\partial
z_{2}}-z_{2}\frac{\partial}{\partial z_{1}}\right),$ (39) $\displaystyle
Z=\sigma_{z}=\left(z_{1}\frac{\partial}{\partial
z_{1}}-z_{2}\frac{\partial}{\partial z_{2}}\right).$

For spin-1/2 case, we have two distinct state functions: $|1,0\rangle=z_{1}$
which corresponds to spin-up, and $|0,1\rangle=z_{2}$ for spin-down. The
identity gate in the case of two-variable function is

(40) $\displaystyle I=\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}\right)$

For the $N$-dimensional case, the identity operator in the Bargmann space
reads

(41) $\displaystyle I=\sum_{i=1}^{N}z_{i}\frac{\partial}{\partial z_{i}}$

The Hadamard or Walsh-Hadamard gate acts on analytic functions with two-
variables in the following way

(42) $\displaystyle H=\frac{1}{\sqrt{2}}\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{1}\frac{\partial}{\partial z_{2}}+z_{2}\frac{\partial}{\partial
z_{1}}-z_{2}\frac{\partial}{\partial z_{2}}\right)$

The phase gate $S$ is

(43) $\displaystyle S=\left(z\frac{\partial}{\partial
z}+iw\frac{\partial}{\partial w}\right)$

The rotation operators about $x$-axis, $y$-axis and $z$-axis are

(44) $\displaystyle R_{x}(\theta)=\cos(\theta/2)z_{1}\frac{\partial}{\partial
z_{1}}-i\sin(\theta/2)z_{1}\frac{\partial}{\partial z_{2}}$
$\displaystyle-i\sin(\theta/2)z_{2}\frac{\partial}{\partial
z_{1}}+\cos(\theta/2)z_{2}\frac{\partial}{\partial z_{2}},$ (45)
$\displaystyle R_{y}(\theta)=\cos(\theta/2)z_{1}\frac{\partial}{\partial
z_{1}}-\sin(\theta/2)z_{1}\frac{\partial}{\partial z_{2}}$
$\displaystyle+\sin(\theta/2)z_{2}\frac{\partial}{\partial
z_{1}}+\cos(\theta/2)z_{2}\frac{\partial}{\partial z_{2}},$ (46)
$\displaystyle R_{z}(\theta)=\exp(-i\theta/2)z_{1}\frac{\partial}{\partial
z_{1}}+\exp(i\theta/2)z_{2}\frac{\partial}{\partial z_{2}}$

The controlled-Not gate acts on analytic functions with four-variables, it
consists of an identity operator with respect to a two-variables $z_{1},z_{2}$
and a Not gate operator between the remaining two,

(47) $\displaystyle\mathrm{CNOT}=\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}+z_{3}\frac{\partial}{\partial
z_{4}}+z_{4}\frac{\partial}{\partial z_{3}}\right)$

The swap , Toffoli and Fredkin gates are

(48) $\displaystyle\mathrm{SWAP}=\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{3}}+z_{3}\frac{\partial}{\partial
z_{2}}+z_{4}\frac{\partial}{\partial z_{4}}\right)$

(49) $\displaystyle\mathrm{Toffoli}=\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}+z_{3}\frac{\partial}{\partial
z_{3}}+z_{4}\frac{\partial}{\partial z_{4}}+z_{5}\frac{\partial}{\partial
z_{5}}+z_{6}\frac{\partial}{\partial z_{6}}+z_{7}\frac{\partial}{\partial
z_{8}}+z_{8}\frac{\partial}{\partial z_{7}}\right)$

(50) $\displaystyle\mathrm{Fredkin}=\left(z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}+z_{3}\frac{\partial}{\partial
z_{3}}+z_{4}\frac{\partial}{\partial z_{4}}+z_{5}\frac{\partial}{\partial
z_{7}}+z_{7}\frac{\partial}{\partial z_{5}}+z_{8}\frac{\partial}{\partial
z_{8}}\right)$

Other possible logic gates can be written as complex differential operators in
a way similar to the procedure we followed in writing the previous gates. If
the input function is of the form 35, the outputs are again normalized states.
The expectation values of $\langle n_{1},n_{2}|L|n_{1},n_{2}\rangle$, where
the differential operator $L$ is any logic gate operator such as $X,Y,Z,H$
etc., and the inner product is taken over the Bargmann space, are identical to
the expectation values of $\langle\psi|\hat{L}|\psi\rangle$ with a wave
function $\psi$, gate operator $\hat{L}$ and an inner product taken over the
vector Hilbert space. Moreover, considering the inputs to be any analytic
functions is also possible. However, in this case, the normalization condition
may be lost, but it could be recovered by dividing with a suitable
normalization constant. Below, we give a general procedure for obtaining the
differential and integral representations of logic gates in the two-
dimensional Bargmann space.

Differential operator representation of logic gates: Let $M$ be any $2\times
2$ matrix in $\mathbb{C}^{2}$, the mapping to analytical form in Bargmann
space is

(51) $M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\rightarrow{\bf
M}(z_{1},z_{2})=a\;z_{1}\frac{\partial}{\partial
z_{1}}+b\;z_{1}\frac{\partial}{\partial
z_{2}}+c\;z_{2}\frac{\partial}{\partial
z_{1}}+d\;z_{2}\frac{\partial}{\partial z_{2}}$

where $a,b,c$ and $d\in\mathbb{C}$. Let $f(z_{1},z_{2})$ be an analytic
function of two variables $z_{1}$ and $z_{2}$. The expansion of
$f(z_{1},z_{2})$ is

(52) $\displaystyle
f(z_{1},z_{2})=\sum_{nm=0}^{\infty}C_{nm}z_{1}^{n}z_{2}^{m}$

where the coefficients $C_{nm}$ are given by

(53) $\displaystyle C_{nm}=\frac{1}{n!m!}f^{(n)}(0)f^{(m)}(0)$

The inner product of $f(z_{1},z_{2})$ with itself is

(54) $\displaystyle\langle
f(z_{1},z_{2})|f(z_{1},z_{2})\rangle=\sum_{n^{\prime}m^{\prime}=0}^{\infty}\sum_{nm=0}^{\infty}\;C^{\star}_{n^{\prime}m^{\prime}}\;C_{nm}\;n!\;m!\;\delta_{n^{\prime},n}\delta_{m^{\prime},m}$

The action of ${\bf M}(z_{1},z_{2})$ on $f(z_{1},z_{2})$ is

(55) $\displaystyle{\bf
M}(z_{1},z_{2})f(z_{1},z_{2})=a\;n\;f(z_{1},z_{2})+d\;m\;f(z_{1},z_{2})$
$\displaystyle+b\;m\;\sum_{nm}^{\infty}C_{nm}z_{1}^{n+1}z_{2}^{m-1}+c\;n\;\sum_{nm}z_{1}^{n-1}z_{2}^{m+1}$

The generalization of the above recipe to higher-order logic gates could be
done analogously in a straightforward manner.

Integral representation of logic gates:Let $f(z)$ be an analytic function in
simply-connected domain $D$, and let $\Gamma$ be positively-oriented loop in
$D$. The integral representation of the derivative at point $z_{0}$ (any point
in the interior of $\Gamma$) is [15]

(56) $\displaystyle f^{(n)}(z_{0})=\frac{n!}{2\pi
i}\oint_{\Gamma}\frac{f(z)}{(z-z_{0})^{n+1}}\;dz$

which is known as the generalized Cauchy formula. Consequently, the integral
representation of the $X$-gate for example reads

(57) $\displaystyle Xf(z_{1},z_{2})=\frac{z_{1}}{2\pi
i}\oint_{\Gamma_{2}}\frac{f(z_{1},\xi_{2})}{\xi_{2}-z_{2}}d\xi_{2}-\frac{z_{2}}{2\pi
i}\oint_{\Gamma_{1}}\frac{f(\xi_{1},z_{2})}{\xi_{1}-z_{1}}d\xi_{1},$

Similarly, the $Y,Z$ and $I$ gates are

(58) $\displaystyle
Yf(z_{1},z_{2})=-\frac{z_{1}}{2\pi}\oint_{\Gamma_{2}}\frac{f(z_{1},\xi_{2})}{\xi_{2}-z_{2}}d\xi_{2}+\frac{z_{2}}{2\pi}\oint_{\Gamma_{1}}\frac{f(\xi_{1},z_{2})}{\xi_{1}-z_{1}}d\xi_{1},$
(59) $\displaystyle Zf(z_{1},z_{2})=\frac{z_{1}}{2\pi
i}\oint_{\Gamma_{1}}\frac{f(\xi_{1},z_{2})}{\xi_{1}-z_{1}}d\xi_{1}-\frac{z_{2}}{2\pi
i}\oint_{\Gamma_{2}}\frac{f(z_{1},\xi_{2})}{\xi_{2}-z_{2}}d\xi_{2},$ (60)
$\displaystyle If(z_{1},z_{2})=\frac{z_{1}}{2\pi
i}\oint_{\Gamma_{1}}\frac{f(\xi_{1},z_{2})}{\xi_{1}-z_{1}}d\xi_{1}+\frac{z_{2}}{2\pi
i}\oint_{\Gamma_{2}}\frac{f(z_{1},\xi_{2})}{\xi_{2}-z_{2}}d\xi_{2},$

##  4\. Models of Computation

The computational model consists of a computing machine $M$ connected to an
environment $E$. At each time $t$, we have input channel(s) $S(t)$ (for
stimulus) and output channel(s) $R(t)$ (for response) [18]. To describe how
the computing machine $M$ works, we need to specify the outputs as a function
of the inputs. Since the transmission of signals requires some time, the
output must be at a time later than the time of the input channel. We assume
the machine response $R(r+1)$ because of $S(t)$ to depend on the state of the
machine $Q(t)$ [18]

(61) $\displaystyle R(t+1)=F(Q(t),S(t))$

and the state of the machine $M$ depends on the stimulus $S(t)$ and the state
of the machine at previous time,

(62) $\displaystyle Q(t+1)=G(Q(t),S(t)).$

###  4.1. Memristive systems

The memristive systems are dynamical systems defined by [19]

(63) $\displaystyle\dot{x}=f(x,u,t),$ (64) $\displaystyle y=g(x,u,t)u,$

where $u$ and $y$ are the input and output of the system, and $x$ denotes the
state of the system. The connection between 63, 64 and 61, 62 is achieved by
assigning $R(t+1)=y(t+1)$, $Q(t+1)=x(t+1)$, and $u=S(t)$. Therefore, for each
value of $t$, we can realize the memristive system as a computational model.

###  4.2.  Chain of harmonic oscillators

The harmonic oscillator satisfy a second-order differential equation of the
form [20]

(65) $\displaystyle \ddot{x}+\omega^{2}x=0 $

The general solution is

(66) $\displaystyle x(t)=A\sin(\omega t)+B\cos(\omega t),$

where $A$ and $B$ are constants.
A single harmonic oscillator does not define a computational model by itself,
some interactions with the environment are needed to define a computational
model. Consider a chain of $N$ identical harmonic oscillators as the machine
$M$, the interaction of a single harmonic oscillator with other oscillators
would affect the whole system. In other words, it provides some stimulus $S$
to the whole system, regardless of whether this stimulus is large or
negligible. The response $R$ could be described as the changing of some
measurable quantities in this chain due to the interaction with this single
harmonic oscillator, such as the frequency modes, etc. The new displacement
$x_{n}$ of the $n$th oscillator defines the new state of the system $Q$. It is
sufficient to take just one oscillator within the chain and study its change
in dynamics due to the added extra oscillator to show the connection between
this system and the computational model described by equations 61 and 62.

Let $|{\bf
f}\rangle=\left(|f_{1}\rangle,|f_{2}\rangle,\dots|f_{N}\rangle\right)$ be a
set of analytic complex functions with several variables ( the stimulus $S$),
and let $M({\bf L})$ be a computing machine with ${\bf L}$ being a vector of
complex differential operators. The computation process $\mathfrak{C}$ is
described by the action of ${\bf L}$ on the inputs $|{\bf f}\rangle$

(67) $\mathfrak{C}:|{\bf f}\rangle\rightarrow{\bf L}|{\bf f}\rangle,$

and the expectation values of the outputs (the response $R$) are

(68) $\displaystyle\frac{\langle{\bf f}|{\bf L}|{\bf
f}\rangle}{\sqrt{\langle{\bf f}|{\bf f}\rangle}}$

where we divided the last expression by the normalization constant
$\sqrt{\langle{\bf f}|{\bf f}\rangle}$ since the inputs are arbitrary complex
functions. If both the inputs and outputs are discrete and not correlated
between themselves i.e. can not be written in superposition states like
quantum states, then the computation is classical. Quantum computations are
characterized by the possibility of having superposition states during the
computation process. The machine $M({\bf L})$ could be in principle any
dynamical system from tiny particles to the universe as a whole. Any change in
the dynamical degrees of freedom associated with $M$ is in principle due to
the interaction of $M$ with the surrounding environment i.e. all possible
inputs. However, this change might be very slow and may not affect the
macroscopic characteristics of the system immediately. It is worth mentioning
that dynamical systems are not the materials or objects around us but rather
any changing system such as stock prices, ecological models, and in principle
any changing system described by a set of differential equations.

$M({\bf L})$$|{\bf f}\rangle$Inputs${\bf L}|{\bf f}\rangle=|{\bf
f}^{\prime}\rangle$Outputs Figure 1. The computational model consists of a
computing machine ( any physical object) $M$ described by a set of complex
differential operators ${\bf L}$, $|{\bf f}\rangle$ and ${\bf L}|{\bf
f}\rangle$ are the inputs and outputs analytic complex functions.

To quantify the computation process, we compute the change of entropy taking
into account the fact that entropy associated with any physical processes is
always increasing[21, 22]

(69) $\displaystyle\Delta
S=S_{\mathrm{out}}-S_{\mathrm{in}}=-\sum_{i}p^{\mathrm{out}}_{i}\log
p^{\mathrm{out}}_{i}+\sum_{i}p^{\mathrm{in}}_{i}\log p_{i}^{\mathrm{in}}\geq
0$

where $p_{i}^{\mathrm{in}}=\frac{|f_{i}\rangle\langle f_{i}|}{\langle{\bf
f}|{\bf f}\rangle}$ and
$p^{\mathrm{out}}_{i}=\frac{|f^{\prime}_{i}\rangle\langle
f^{\prime}_{i}|}{\langle{\bf f}^{\prime}|{\bf f}^{\prime}\rangle}$ with $|{\bf
f}^{\prime}\rangle={\bf L}|{\bf f}\rangle$.

The relative entropy (Kullback–Leibler divergence) is

(70) $\displaystyle
D(P||Q)=\sum_{i}p_{i}\log(\frac{p_{i}}{q_{i}})=-\sum_{i}\frac{|f_{i}\rangle\langle
f_{i}|}{\langle{\bf f}|{\bf f}\rangle}\log\frac{|g_{i}\rangle\langle
g_{i}|}{\langle{\bf g}|{\bf g}\rangle}-S(p_{i})$

with $p_{i}=\frac{|f_{i}\rangle\langle f_{i}|}{\langle{\bf f}|{\bf
f}\rangle}$, $q_{i}=\frac{|g_{i}\rangle\langle g_{i}|}{\langle{\bf g}|{\bf
g}\rangle}$ and $S(p_{i})=-\sum_{i}p_{i}\log p_{i}$. Consequently, other
quantities such as the mutual information can be determined in terms of
holomorphic functions [22].

##  5\. Physical Systems

All physical systems process information. In this section, we study in some
sort of detail the computation mechanisms in a few basic systems. However, it
is obvious that our construction works for all physical systems in nature.

###  5.1.  Coupled oscillators

Simple harmonic motion is omnipresent in many mechanical and electrical
systems. For example, electrical circuits with inductance $L$ connected across
a capacitance $C$ carrying charge $Q$ obey the equation [20]

(71) $\displaystyle L\ddot{Q}+\frac{Q}{C}=0$

with frequency $\omega^{2}=\frac{1}{LC}$.
The general equation of motion for any simple harmonic motion with $N$
oscillators reads,

(72) $\displaystyle\ddot{{\bf x}}+\mathbf{\omega}^{2}{\bf x}=0$

where ${\bf x}=\left(x_{1},x_{2},\dots,x_{N}\right)$ is the $N$-dimensional
state vector and $\omega$ is $N\times N$ frequency matrix.

$\ell$$\ell$$s$$y$$x$ Figure 2. Two identical pendulums with mass $M$
connected by a spring with stiffness $s$.

To define the basic logic gates in such systems, one needs at least two
coupled oscillators at each time step $t_{0}$ 333On the contrary, one could
apply the basic logic gates on a single quantum particle due to the
superposition property (which leads to the quantum entanglement) of these
particles, which distinguishes quantum particles from classical ones. .
Consider two identical pendulums, each having a mass $M$ suspended on a light
rigid rod of length $\ell$. The masses are connected by a light spring of
stiffness $s$. The natural length of the spring is equal to the distance
between the two masses at equilibrium. The equation of motion are

(73) $\displaystyle M\ddot{x}=-Mg\;\frac{x}{\ell}-s\;(x-y),$ (74)
$\displaystyle M\ddot{y}=-Mg\;\frac{y}{\ell}+s\;(x-y).$

The above set of equations could be written as

(75) $\displaystyle\ddot{x}+\omega_{0}^{2}\;x=-\frac{s}{M}\;(x-y),$ (76)
$\displaystyle\ddot{y}+\omega_{0}^{2}\;y=\frac{s}{M}\;(x-y).$

where $\omega_{0}=\sqrt{\frac{g}{l}}$ is the natural frequency of each
pendulum. To solve 75 and 76, we introduce the normal coordinates

(77) $\displaystyle X=x+y,$ (78) $\displaystyle Y=x-y.$

Then, we have the following set of equations

(79) $\displaystyle\ddot{X}+\omega_{0}^{2}X=0,$ (80)
$\displaystyle\ddot{Y}+\left(\omega_{0}^{2}+2\frac{s}{M}\right)Y=0.$

One possible solution to the above set of equations is

(81) $\displaystyle X=A\sin(\omega_{0}t+\phi)=\alpha(z_{1}-\overline{z}_{1}),$
(82) $\displaystyle Y=B\sin(\omega t+\varphi)=\beta(z_{2}-\overline{z}_{2}),$

where $z_{1}=e^{(i\omega_{0}t+\phi)}$, $z_{2}=e^{(i\omega t+\varphi)}$, and
$\omega=\sqrt{\omega_{0}^{2}+2s/M}$. It is sufficient to take the complex part
only during our study of the logic gates for coupled oscillators. The state
function of the coupled oscillator is

(83) $\displaystyle|f\rangle=f(z_{1},z_{2})=\alpha\beta\;z_{1}z_{2}$

The action of logic gates on this state function is

(84) $\displaystyle
Xf=\mathrm{NOT}f=\alpha\beta\left(z^{2}_{1}+z^{2}_{2}\right),$ (85)
$\displaystyle Yf=-i\alpha\beta\left(z_{1}^{2}-z_{2}^{2}\right),$ (86)
$\displaystyle Zf=\alpha\beta\left(z_{1}z_{2}-z_{2}z_{1}\right)=0,$ (87)
$\displaystyle
If=\alpha\beta\left(z_{1}z_{2}+z_{2}z_{1}\right)=2\;\alpha\;\beta\;f,$

where the factor 2 in the last equation can be absorbed safely choosing
suitable normalization constant. The Hadamard gate is

(88) $\displaystyle
Hf=\frac{\alpha\;\beta}{\sqrt{2}}\left(z_{1}z_{2}+z_{1}^{2}+z_{2}^{2}-z_{2}z_{1}\right)=\frac{\alpha\;\beta}{\sqrt{2}}\left(z_{1}^{2}+z_{2}^{2}\right)=\frac{1}{\sqrt{2}}Xf$

Note that since our state function is classical (product state), we were able
to write the action of the Hadamard gates in the previous equation in terms of
the action of the $X$ gate. The inner products $\langle f|X|f\rangle=\langle
f|Y|f\rangle=\langle f|H|f\rangle=0$. If one considers a more general model of
higher-order oscillations, we may use the ansatz

(89) $\displaystyle
f(z_{1},z_{2})=\frac{z_{1}^{n}}{\sqrt{n!}}\frac{z^{m}_{2}}{\sqrt{m!}}$

and the expectation values of the logic gates assume more general form. As an
example, we find for the $X$ gate

(90) $\displaystyle
X|f\rangle=m\frac{z_{1}^{n+1}}{\sqrt{n!}}\frac{z_{2}^{m-1}}{\sqrt{m!}}+n\frac{z_{1}^{n-1}}{\sqrt{n!}}\frac{z_{2}^{m+1}}{\sqrt{m!}},$
(91) $\displaystyle\langle
f^{\prime}|X|f\rangle=m\;\delta_{n^{\prime},n+1}\delta_{m^{\prime},m-1}+n\;\delta_{m^{\prime},m+1}\delta_{n^{\prime},n-1}.$

Similar relations could be found for other gates.

###  5.2. Turing Patterns

Turing Patterns are described by the diffusion-reaction equations [23, 24]

(92) $\displaystyle\frac{\partial a}{\partial t}=D_{a}\nabla^{2}a+f(a,b),$
(93) $\displaystyle\frac{\partial b}{\partial t}=D_{b}\nabla^{2}b+g(a,b),$

where $a$ and $b$ describe the concentrations of chemicals at time $t$. The
functions $f(a,b)$ and $g(a,b)$ represent the reaction terms, and
$D_{a},D_{b}$ are the diffusion coefficients. One could for example write the
reaction equations using the FitzHugh–Nagumo equation. In this case,
$f(a,b)=a-a^{3}-b+\alpha$ and $g(a,b)=\beta(a-b)$ where $\alpha$ and $\beta$
are constants. Solving equations 92 and 93 we obtain the concentrations of
chemicals as a function of position and time i.e. $\\{a(x,t),b(x,t)\\}$. Using
the Segal-Bargmann transform, we may write the solutions in terms of
holomorphic variables,

(94) $\displaystyle\hat{a}(z_{1},t)=(\pi
t)^{-d/4}\int_{\mathbb{R}^{d}}e^{\left(-z_{1}^{2}+2\sqrt{2}z_{1}\cdot
x-x^{2}\right)/2t}a(x)\;dx,$ (95) $\displaystyle\hat{b}(z_{2},t)=(\pi
t)^{-d/4}\int_{\mathbb{R}^{d}}e^{\left(-z_{2}^{2}+2\sqrt{2}z_{2}\cdot
x-x^{2}\right)/2t}b(x)\;dx.$

The state function of the system is

(96) $\displaystyle f(z_{1},z_{2})=\hat{a}(z_{1},t)\hat{b}(z_{2},t)$

which is a product state. Feeding the state function into logic circuit
architectures built from complex differential operators defines the
computational processes of the Turing patterns.

###  5.3. Neural Networks

In this section, we describe the mechanism for incorporating the developed
formalism into the study of neural networks. We will restrict our analysis to
simple neural networks. However, the generalization to other types of nets
such as convolutional neural networks is possible. The neuron $j$ may be
described mathematically by a set of two equations [25, 26, 27, 28, 29, 30],

(97) $\displaystyle u_{j}=\sum_{i=1}^{N}\omega_{ji}x_{i},$ (98) $\displaystyle
y_{i}=f(u_{j}+b_{j}),$

where $\omega_{j1},\omega_{j2},\dots,\omega_{jN}$ are the synaptic weights,
$f$ is the activation function and $b_{k}$ is the bias. The inputs are

(99) $\displaystyle x_{1}=\langle f_{1}|L_{1}|f_{1}\rangle,x_{2}=\langle
f_{2}|L_{2}|f_{2}\rangle,\dots,x_{N}=\langle f_{N}|L_{N}|f_{N}\rangle.$

The outputs $y_{j}$ should be written as inner products in the Bargmann space
similar to the inputs in 99. Now, following any neural computation algorithms
for weights such as perceptrons, backpropogations one could find the
convergence learning process with minimal loss functions.

##  6\. Universal Programming Language

Programming languages are systems of expressions mostly text-based formal
languages that allow users to communicate with the machine language [31, 32,
33]. The first question that could be asked is whether the universe uses one
way for communication between its different sub-systems or each has its
distinct way of communication. Logically, the universe must follow one general
way for communications between its embedded sub-systems. This is because the
laws of physics are universal and thus there is no way to include a sub-system
in the universe that works genuinely in a very different way. The universal
programming language (UPL) which I shall describe here is different from any
other known programming language . First, it is the most basic language since
all other languages such as the machine, compiler-based and interpreter-based
can be decomposed into a group of instructions in the UPL language. Second,
its space of possibilities is very high. One could for complex systems in
biophysics create many distinguish patterns and manipulate them.

Let
$\mathcal{S}=\\{\mathcal{S}_{1},\mathcal{S}_{2},\mathcal{S}_{3},\dots,\mathcal{S}_{N}\\}$
be the system that we want to study its computational properties. Each sub-
system $\mathcal{S}_{i}$, where $i\in\\{1,\dots,N\\}$, is described by a state
function $|f_{i}\rangle$. Let
${\bf|f\rangle}=(|f_{1}\rangle,|f_{2}\rangle,|f_{3}\rangle,\dots,|f_{N}\rangle)$
be the joint state function for the system $\mathcal{S}$. The general
procedure to find the computational properties of $\mathcal{S}$ is determined
by the following list of instructions
$\mathcal{L}=\\{L_{1},L_{2},L_{3},L_{4},L_{5},L_{6},L_{7},L_{8}\\}$:

  * •

$L_{1}$: Determine the state functions $\\{|f_{i}\rangle\\}$ of the sub-
systems $\\{\mathcal{S}_{i}\\}$ as a function of time.

  * •

$L_{2}:$ If the state functions are defined over the field of real numbers,
apply the Segal-Bargmann transform for each variable such that the final
answer is written in terms of holomorphic variables. 444In principle, one
could use any unitary integral transform from real variables onto holomorphic
variables.

  * •

$L_{3}:$ Determine the joint state functions $|{\bf f}\rangle$ of the system
$\mathcal{S}$.

  * •

$L_{4}:$ Apply all possible logic gates (differential operators) between the
sub-systems $\\{\mathcal{S}_{i}\\}$, i.e. ${\bf L}|{\bf f}\rangle$.

  * •

$L_{5}:$ Lis all produced patterns and classify them into classical and
quantum patterns.

  * •

$L_{6}:$ Compute the expectation values i.e. $\langle{\bf f}|{\bf L}|{\bf
f}\rangle$, with suitable normalization conditions.

  * •

$L_{7}:$ If the joint state function is not written in a closed analytical
form but rather in a numerical approximated form. Repeat the previous
procedures for other joint state functions at different time iterations.

  * •

$L_{8}:$ For sufficient time iterations, list all produced patterns and select
the distinguished patterns that corresponds to different expectation values in
the Bargmann space.

Obtaining such lists of computational patterns for each material is like
cracking the DNA of life since it allows for designing novel ways for
controlling and manipulating the properties of materials for efficient
computing tasks. Consequently, any program in UPL should be described by the
triplet $(\mathfrak{CS},\mathcal{S},\mathcal{L})$ where $\mathfrak{CS}$ is the
computation space. For a single-qubit $q_{1}$,
$\mathfrak{CS}=\mathcal{H}L^{2}(\mathbb{C}^{2},\mu_{\hbar})$,
$\mathcal{S}=\\{q_{1}\\}$ and $\mathcal{L}$ are the eight instructions defined
for the case of a single-qubit.

##  7\. Hierarchical Computing: The Concept of Layers

The universe operates at different time scales from elementary particles with
quantum signatures to objects at cosmic scale such as galaxies, supernovas
…etc. The computations are performed first at the level of elementary
particles (the basic ingredients of any material). The results of such
computations affect the physical properties of atoms or molecules composed of
these elementary particles. The computations at atomic level are done at
different layer with time scale that characterizes the physical processes of
molecules. Analogously, the results of computations are sent to the next level
with different time scale and larger ingredients. The first level is
represented by a layer and it must be of a quantum nature where each fermion
by itself represents an independent logic gate as we explained in previous
sections. The last layer is the universe as a whole. Practically, we take the
last layer to be the layer which contains the object under study with its
surrounding environment. For example, if the object under study is of physical
dimensions relative to humans size, then the structure of layers starts at the
quantum level of the object’s ingredients and ends at the level of biosphere.

Let $\mathfrak{L}_{i}(t_{i})$, where $i=1,\dots N$, denotes the layer of
computation at time scale $t_{i}$. The computation space $\mathfrak{CS}$ is

(100)
$\displaystyle\mathfrak{CS}:\mathfrak{L}_{1}(t_{1})\otimes\mathfrak{L}_{2}(t_{2})\otimes\dots\otimes\mathfrak{L}_{N}(t_{N})$

Properties: 1\. Each individual layer is Turing-complete [34, 35].555The
Church-Turing states “Every effectively calculable function can be computed by
a Turing-machine transducer.”
2\. The space products of layers are time-ordered. Events at layer with time
index $t_{N-1}$ happen before the events at layer with time index $t_{N}$.
This supports the arrow of time since each computational process are
associated with specific time and messages between layers are causal[36].

The concept of layers presents a model for hierarchal computing devices where
the computations are performed at different time scales with distinguish
physical properties. The first layer in the universe was formed immediatley
after the big bang and was of quantum gravity nature. It was associated with
extremely low entropy. Generally, the case $N=1$ corresponds to one layer of
computation and we shall call it a primitive computational model. On contrary,
having large number of layers corresponds to advanced computational model such
as the universe at present time.

The hierarchal computing is a multilayer computational model. Each layer has
distinct physical properties such as unique time scale. In this type of
computing, layers are organized according to their time scale from short to
long. The first layer, as we mentioned before, must be of quantum signature.
The results of computations at layer $i$ affects the computations performed
within layer $i+1$. Intuitively, multilayer computational models are
associated with higher efficiency in comparison with single layer models.
Similar situation is found in the study of multilayer neural networks in deep
learning [29]. However, the situation in multilayer neural networks is
different from the multilayer concept in hierarchal computing since all the
computations of multilayer neural networks in deep learning are performed in
one time scale and thus within the same layer in the hierarchal computing
scheme.

##  8\. Conclusions

We formulated the theory of computation in terms of holomorphic functions in
the Bargmann space. This was done first by writing the quantum logic gates as
complex differential operators that act on analytic functions. Classical logic
gates are embedded in the quantum gates and may be recovered immediately
considering the inputs to be of classical signature (no superposition states
are allowed). Moreover, the current formalism enjoys great flexibility since
it is connected with the theory of analytic functions. Building on the
developed formalism, the connection between materials or any physical system
in the universe with computation is straightforward. We presented a recipe for
constructing universal programming language used by the universe alongside
with the hierarchy computing scheme. Some physical examples are presented in
the paper. We suggest a deep study of the formalism presented here in neural
dynamics models such as Hodgkin–Huxley and FitzHugh–Nagumo models[37, 38].
Moreover, building a correspondence between logic gates and the different
oscillation states of Belousov–Zhabotinsky reactions can be used for building
non-conventional computing devices [39].

Data Availability Statement: Data sharing not applicable – no new data
generated.

Conflict Of Interest Statement: The author has no conflicts of interest to
declare.

## References

  * [1] Zuse, K., 1969. _Rechnender Raum_ (Calculating Space). Friedrich Vieweg und Sohn, Braunschweig.
  * [2] Landauer, R., 1991. “Information is Physical”. _Physics Today_ 44, 5, 23.
  * [3] Landauer, R., 1961. “Irreversibility and heat generation in the computing process”. _IBM journal of research and development_ , 5(3), pp. 183–191.
  * [4] Feynman, R.P., 1982. “Simulating physics with computers”. _Int. J. Theor. Phys._ 21, 467 (1982).
  * [5] Bennett, C. H., 1982. “The thermodynamics of computation a review”. _International Journal of Theoretical Physics_ , 21(12), pp. 905–940.
  * [6] Lloyd, S., 2000. ”Ultimate physical limits to computation”. _Nature_ 406(6799), 1047-1054
  * [7] Parrondo, J.M.R., 2022. ” Thermodynamics of Information”. _Encyclopedia of Condensed Matter Physics-2nd Edition_ , edited by T. Chakraborty.
  * [8] de Broglie, L., 1924. “ Recherches sur la théorie des quanta”. Ph.D. thesis, University of Paris.
  * [9] Winfree, A.T., 1980. _The Geometry of Biological Time_. Springer-Verlag, New York.
  * [10] Strogatz, S., 2004. _Sync: The Emerging Science of Spontaneous Order_. Penguin Adult.
  * [11] Hall, B.C., 1999. “Holomorphic Methods in analysis and Mathematical Physics”. _Contemporary Mathematics_ , Volume 260, pp. 1-59
  * [12] AlMasri, M.W. and Wahiddin M. R. B, 2022. “Bargmann representation of quantum absorption refrigerators” . _Rep. Math. Phys._ 89(2), Pages 185-198
  * [13] AlMasri, M.W. and Wahiddin M. R. B, 2022. “Quantum Decomposition Algorithm For Master Equations of Stochastic Processes: The Damped Spin Case”. _Mod. Phys. Lett. A_ 37 (32), 2250216
  * [14] AlMasri, M.W. and Wahiddin M. R. B, 2023. “ Integral Transforms and PT-symmetric Hamiltonians”. Chinese Journal of Physics, volume 85, Pages 127-134.
  * [15] Ahlfors, L., 1979. _Complex Analysis_. McGraw-Hill Education; 3rd edition.
  * [16] R. W. Hamming, R.W., 1973. _Numerical Methods for Scientists and Engineers_ 2nd edition, Dover Publications.
  * [17] Nielsen, M.A., and Chuang, I.L., 2001. _Quantum Computation and Quantum Information_. Cambridge University Press.
  * [18] Minsky, M.L., 1967. _Computation: Finite and Infinite Machines_. Prentice-Hall
  * [19] Chua, L.O., and Kang, S.M., 1976. “Memristive devices and systems,” _Proceedings of the IEEE_ , 64(2), 209-223
  * [20] Pain, J. H., 2005. _The Physics of Vibrations and Waves_. Six edition, John Wiley & Sons Ltd.
  * [21] Shannon, C. E., 1948. “A mathematical theory of communication”. The Bell system technical journal 27 (3), 379-423.
  * [22] Thomas, J.A., and Cover, T.M., 2006. _Information Theory_. second edition; John Wiley & Sons.
  * [23] Turing, A. M., 1952. “The Chemical Basis of Morphogenesis”. _Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences_ , 237 (641), pages 37-72.
  * [24] Vittadello, S.T., Leyshon, T., Schnoerr, D., Stumpf, M.P.H., 2021. “ Turing pattern design principles and their robustness”. _Phil. Trans. R. Soc. A_ 379: 20200272.
  * [25] McCulloch, W.S., Pits, W., 1943. “A Logical Calculus of the Ideas Immanent in Nervous Activity”. Bull. Math. Biophys. 5, 115–133
  * [26] Rosenblatt, F., 1958. “The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain”. Cornell Aeronautical Laboratory. Psychological Review 65(6), 386–408
  * [27] Hopfield, J.J., 1982. “Neural networks and physical systems with emergent collective computational abilities”. _Proceedings of the national academy of sciences_ , 79(8), pp.2554-2558.
  * [28] Haykin, S.S., 1999. _Neural Networks: A Comprehensive Foundation_. Second Edition, Pearson Education
  * [29] LeCun,Y., Bengio, Y., and Hinton, G., 2015. “Deep learning”. _Nature_ , 521 ,p. 436–444 .
  * [30] AlMasri, M.W., 2024. “Multi-Valued Quantum Neurons”. Int.J.Theor.Phys 63, 39.
  * [31] Knuth, D.E., 1997. _The Art of Computer Programming_. Addison-Wesley
  * [32] McCarthy, J., 1960. “Recursive Functions of Symbolic Expressions and their Computation by Machine, part I”. _Communications of the ACM._ 3(4) : 184–195.
  * [33] Kernighan, B.W., and Ritchie, D.M, 1988. _The C Programming Language_. Second edition, Bell Telephone Laboratories.
  * [34] Church, A., 1936. ”An Unsolvable Problem of Elementary Number Theory.” Amer. J. Math. 58, 345-363.
  * [35] Turing, A.M., 1950. “Computing Machinery and Intelligence”. _Mind_ , Volume LIX, Issue 236, Pages 433–460.
  * [36] Dieter Zeh, H., 2007. _The Physical Basis of The Direction of Time_. 4th edition, Springer Berlin, Heidelberg.
  * [37] Koch, C., 1999. _Biophysics of Computation_. Oxford University Press
  * [38] Izhikevich, E. M. , 2007. _Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting_. MIT Press
  * [39] Kuramoto, Y., 1984. _Chemical Oscillations, Waves, and Turbulence_. Springer-Verlag Berlin Heidelberg