# Theoretical limits of Virtual Reality

Francesco Sisini, Valentina Sisini, Laura Sisini
Tekamed srl

## Abstract

In recent years there has been a strong development of the concept of virtual
reality (VR) which from the first video games developed in the 60s has reached
the current immersive systems. VR and the consequent deception of perception
pose an interesting question: is it possible to deceive the mind to the point
of not being able to recognize whether the perceived reality is real or
simulated? In addition to this question, another question arises
spontaneously: is it possible to simulate a non-reality in which the physical
laws do not apply?
The answer to the first question is that it would theoretically be possible to
deceive the mind to the point of not being able to recognize whether the
perceived reality is real or simulated, furthermore it is also possible to
simulate a non-real, i.e. magical, world. However, this possibility is based
on hiding degrees of freedom from the observer, therefore it requires the
complicity of the observer to be realised.
It can be said that a VR system can simulate both a real and non-real
experience.
For an observer, a virtual reality experience is still an experience of
reality. This experience can be exchanged for a different experience, that is,
for a different reality that can be real or not real.

## 1 Introduction

In recent years there has been a strong development of the concept of virtual
reality (VR) which from the first video games developed in the 60s has reached
the current immersive systems. The human being perceives the surrounding
reality through the senses. To date, the known senses are sight, hearing,
touch, taste and smell. The perception of reality can be deceived in
particular situations in which one is induced to misclassify what is perceived
with the senses, for example two parallel segments can be perceived as curved
lines if they are superimposed on a bundle of straight lines with an origin
close to them . The view is deceived by what we call optical illusions, but in
other cases it is not the view that is deceived, but our judgment as in the
well-known ”Fata Morgana” effect. Similarly to what happens for the ”Fata
Morgana”, VR technologies are focused not on the deception of sight but rather
on the deception of judgment. The strategy of deception of the VRs is
articulated in two different actions: the first consists in the obscuring of
the external reality and the second in the administration of an alternative
reality. Similarly to sight, the other senses can also be deceived by VR.
Current technologies are mainly focused on sight, hearing and touch, while
those for smell and taste, however important, have so far been only marginally
developed.
VR and the consequent deception of perception pose an interesting question: is
it possible to deceive the mind to the point of not being able to recognize
whether the perceived reality is real or simulated? In addition to this
question, another question arises spontaneously: is it possible to simulate a
reality in which the physical laws do not apply? In this research we have
given a definition of reality simulation and we have established the
theoretical limits that limit the possibilities of VR.

### 1 Relationship between physical systems and computation

Consider an ordinary computer during a computation, i.e. running an
application. From a physical point of view what happens in the computer is the
dynamic evolution of an electronic circuit, i.e. a flow of electric charges
which occurs in accordance with the laws of physics which describe the forces
acting on the system111The initial conditions of the system can be seen as the
application program.

Any computation is therefore a physical process which can be described in
formal terms by means of a set of variables representing the degrees of
freedom of the system in question.

This vision of computation understood as dynamic evolution has been well
described by David Deutsch in a seminal paper on quantum computing[Deutsch
(1985)]. Several authors have accepted this point of view (computation is a
physical process) and have further developed it by posing the inverse
question: for which physical systems and under which conditions can it be
considered that the dynamic evolution (such as the oscillation of a
pendulum[Sisini(2021)]) represents a computation?[Horsman(2014)]. We will make
use of this point of view, more physical than mathematical, to analyze the
problem posed in this paper and establish what are the physical limits for VR.
In particular, we will exploit the description of the physical computing
system through its degrees of freedom. In general, a computing system is a
physical system with $N$ degrees of freedom, where $N$ is the cardinality of
the set obtained by merging the set of degrees of freedom used for the input
with those used for output and those used for computation only. Computation is
therefore the sequence of states through which the system passes during its
dynamic evolution.

### 1 Degrees of freedom of a physical system

Physical systems, both material (particles, strings, planets, etc.) and
immaterial (light, etc.), are described mathematically through the concept of
degree of freedom (dof). The number of degrees of freedom of a system is the
number of variables required to fully describe its physical state. If the
system has no constraints, the degrees of freedom of the system coincide with
the coordinates of all the components of the system222Even quantum concepts
such as spin are described by a degree of freedom. Similarly fields, both
classical and quantum, are described by degrees of freedom. Otherwise, if
there are relations between the components of the system which constrain the
values that the degrees of freedom can assume (constraints) then the degrees
of freedom are less than the coordinates of the system and all these can be
obtained as functions of the degrees of freedom themselves.
If we consider a system composed of a certain number $n$ of particles free
from constraints (considered dimensionless) and integrating with each other,
then the number of degrees of freedom will be $3\times n$. Once the equations
of dynamics have been solved, it is noted that the motion of each particle
occurs along a curve and is parameterized by time. Thus the motion of each
particle can be described on this curve using only one parameter (time).
Consider, for example, a particle in a centrally symmetrical force field. For
suitable initial conditions, said particle will move on a circular trajectory.
Therefore, although the degrees of freedom of the system are 3, its motion can
be described using a single parameter which describes the progress along the
curve corresponding to the trajectory. In practice, during the evolution of a
physical system, the number of free parameters necessary to describe the
motion is reduced to one, ie the passage of time, while the various components
of the system move along trajectories determined by the equations of dynamics.
This aspect is crucial in the practical implementation of simulations for VR,
so it is good to keep it in the right consideration from the beginning. (image
of the parabolic trajectory)

### 1 Simulation process

In common parlance, two possible meanings of the term simulation can be
distinguished. The first refers to the use of numerical methods to evaluate
the equations of a physical system at a given instant of time. An example is
the simulation of atmospheric events that allows weather forecasts.

The second instead refers to a computer system that has the purpose of
creating a deception of human perception in order to simulate a different
reality from the one that would be perceived in the absence of the simulation.
The result of the simulation is the instant by instant production of a virtual
reality, which is transmitted to a person through specific tools that act on
his senses. An example is video games.

These two types of simulation can also be present in a single process, as is
the case with flight simulators used for pilot training.

## 2 Conceptual non-interactive simulation systems

### 2 Dynamic evolution of a physical system

Let us consider a builder who designs and assembles simple physical systems
that we will call computers (progressively computer $\mathcal{A}$,
$\mathcal{B}$, …) and an observer who observes these computers. The builder
shows his calculators to the observer. The builder agrees with the observer
which degrees of freedom of the system constitute the input, which the output
and which are to be considered computational. The output degrees of freedom
are those that the observer can observe.333An example could be the agreement
between the oculus manufacturer and the wearer, to wear them in the correct
way to view the headset output.

The observer is equipped with a Cartesian reference system and a clock, so he
can measure and keep track of what he observes.

  * •

The measuring instruments of the observer have a finite and non-zero
sensitivity. item The builder declares to the observer that the computer he
built simulates a specific physical system.

The observer observes the calculator and measures for a certain period of time
the values assumed by the variables that represent its output. Using the
measurements obtained, the observer draws a curve parameterized by time which
represents the trajectory of the representative point of the observed system.
It is possible that there exists a set of differential equations which admits
such a curve as a solution. If this system of equations is defined in the same
set of variables (or in a set obtained through an invertible transformation)
that is used to describe the output of the system, then the observer will
declare that the observed system is a real physical system[Einstein (1935)].
Furthermore, if the system of equations found coincides with the equations
describing the physical system that the builder claims to simulate, then the
observer will agree with the builder.

If the observer does not find this system of equations, he could resort to one
of the following alternatives to bring his own curve back to a system of
differential equations:

  * •

Add dynamic variables to equations that are not present in the calculator
output

  * •

Add rheonomic constraints whose shape could also change between two different
observations

In both these cases the observer will declare that the system simulated by the
computer is a non-real physical system and will not agree with the
manufacturer. Below we consider a series of calculators made by the
manufacturer and see which of them the observer agrees with the manufacturer.

### 2 Computer $\mathcal{A}$

Let us consider a calculator $\mathcal{A}$ composed of a box with very massive
walls and a bottom in which a disk can slide without friction.

The physical system of the $\mathcal{A}$ calculator can be described using
only two degrees of freedom and limiting the dynamics to the space enclosed by
the edges of the box. The $\mathcal{A}$ calculator can be thought of as a
computational system whose degrees of freedom are the $x$ and $y$ coordinates
of the disk. An observer can observe the disk moving and bouncing against the
edges of the box and can measure the coordinates of the disk at any instant of
time. The manufacturer declares that the system simulates the free motion of a
disk in a square space with side $l$. For the $\mathcal{A}$ calculator, the
constructor establishes that there are no input degrees of freedom and that
the diskette coordinates $x$ and $y$ constitute the system output. Once this
agreement is accepted by the observer he agrees that from his observation, the
$\mathcal{A}$ calculator is effectively simulating a physical system in which
a disk moves with inertial motion in a space bounded by four constraints of
the type $x<l/2$, $x>-l/2$, $y<l/2$ and $y>-l/2$ as there is a set of
differential equations which, together with four scleronomic constraints,
describes what has been observed without the need to introduce hidden
variables or further constraints in addition to those declared by the
constructor.

Figure 1: Computer $\mathcal{A}$ A disk of mass $m$ is free to move inertially
with little friction in a container with very massive walls.

### 2 Computer $\mathcal{B}$

Let’s consider a second calculator that we will call $\mathcal{B}$, similar to
the $\mathcal{A}$ calculator but which contains two diskettes instead of just
one.

If the set $\mathbf{O}$ of the output degrees of freedom declared by the
constructor corresponds to the coordinates $x_{1},\ x_{2},\ y_{1}$ and
$y_{2}$, then the same considerations made for the
$\mathcal{calculatorarevalidA}$, ie the observer agrees that the $\mathcal{B}$
calculator is simulating a physical system consisting of two free-moving
disks. However, the constructor declares that for the $\mathcal{B}$ calculator
the degrees of freedom dedicated to the output are defined by the set
$\mathbf{O}=\\{x_{1},y_{1}\\}$ therefore the output is limited to degrees of
freedom $x_{1}$ and $y_{1}$, i.e. to the first disk only, while the second
disk has a computational role only, and is part of the set
$\mathbf{C}=\\{x_{2},y_{2}\\}$ of the degrees system computations. To get a
simple mental picture of such a calculator, imagine that you are looking at
the $\mathcal{B}$ calculator in the dark and that you can only see one disk
because a light source has been placed on it.

In the dark, the observer will only be able to see the luminous disk and will
see it proceed in inertial motion until it meets the second disk, then will
see it change trajectory abruptly. Not being able to see the second disk, he
will not be able to identify a system of differential equations whose solution
is the trajectory of the motion of the luminous disk without introducing a set
of hidden variables or rheonomic constraints. It should also be noted that for
each possible computation of the computer, the manufacturer could modify the
initial conditions which fix the position and the initial speed of the disks.
Under these conditions, the additional constraints that the observer must
impose are not only rheonomic but also arbitrary, since their functional form
can vary from time to time.

Figure 2: Computer $\mathcal{B}$ Two diskettes, both of mass $m$, are free to
move by inertial motion with little friction in a container provided with very
massive walls.

### 2 Computer $\mathcal{C}$

Let us consider a third calculator $\mathcal{C}$ in which there is only one
disk but this is connected to two opposite walls with two springs invisible to
the observer. The manufacturer declares that the system output consists of the
disk which is the only visible detail of the system. The manufacturer declares
that the system is simulating a harmonic motion. The observer agrees with what
the manufacturer affirmed, in fact he is able to identify a system of
differential equations whose solution represents the trajectory of the motion
of the observed disk.

Figure 3: Computer $\mathcal{C}$ A disk of mass $m$ is connected to two
springs which are connected to the two opposite faces of a container. The
diskette moves by harmonic motion with little friction in a container provided
with very massive walls.

### 2 Computer $\mathcal{D}$

The $\mathcal{D}$ calculator is obtained from the $\mathcal{C}$ by inserting a
dividing wall in the box and a second disk free to move by inertial motion in
the half of the box in which the motion of the first disk does not occur. So
adding two computational degrees of freedom not visible to the observer. The
manufacturer declares that the system is simulating a harmonic motion. The
observer again agrees with what the manufacturer said, in fact he is able to
identify a system of differential equations whose solution represents the
trajectory of the motion of the observed disk without being disturbed by the
second disk which does not interact with what he observed.

Figure 4: Calcolatore $\mathcal{D}$ A diskette is free to move by inertial
motion inside a container while another diskette moves by harmonic motion
alongside it. The two disks are kept separate by a wall between them.

### 2 Computer $\mathcal{E}$

This calculator consists of a disc hinged at its center and free to rotate. In
one point of the disk a light source is applied and it is identified with the
output of the system, while the rest of the disk is invisible to the observer.
The manufacturer declares that the system is simulating a motion due to a
centrally symmetrical force field with module $F$ described by a certain
constant $k$ and by the charge $a$ placed at a distance a $r$ from the center
of symmetry:

$F=k\frac{a^{2}}{r^{2}_{12}}$ (1)

Using the measurements made on the output of the calculator, the observer
traces the curve which represents the trajectory of the representative point
of the system and observes that it is a circumference parametrized by time by
means of a relation which proportionally links time to the subtended angle
from puto with a fixed axis. This curve represents the solution of the
differential equations of the system described by the manufacturer, therefore
the observer agrees with the manufacturer. It is noted that with this
calculator for the first time the manufacturer seeks agreement with the
observer by declaring that he simulates a different reality from the one he is
actually calculating.

Figure 5: Computer $\mathcal{E}$ A disk of mass $m$ is hinged at its center to
a pivot so that it can rotate about its center of mass. A light source at a
distance $r$ from the center is applied to it. The mass of the light source is
considered to be negligible compared to the mass of the disk.

## Interactive conceptual systems, i.e. Virtual Reality (VR)

The calculators seen in the previous section expect that the observer is
limited to observing the output of the simulation. In practice, the sensory
activity of the observer is reduced to passive perception only: he cannot
touch.
Now we introduce a new type of calculator that we will call VR which provides
for a greater participation of the observer. This type of calculator also
provides input degrees of freedom that we will denote with the set
$\mathbf{I}$. With this calculator the observer is not limited to observing
the output but can act with forces on the input degrees of freedom and
therefore he becomes active and becomes part of the computation system, i.e.
of the calculator itself.

### 2 Computer $\mathcal{F}$

This calculator is built starting from the $\mathcal{A}$ model, but a rod with
mass $m_{c}<<m_{d}$ of the disk is applied to the disk. This rod constitutes
the controller with which the observer can apply an arbitrary force to the
puck. Acting on the disk with the controller, he verifies that the disk
accelerates according to Newton’s second law and therefore again agrees with
the manufacturer’s statement that the $\mathcal{F}$ calculator represents a VR
of a free disk.

Figure 6:  A disk of mass $m$ is free to move inertially with little friction
in a container with very massive walls. A small rod is hinged to the disk
which allows a force to be applied to it.

### 2 Computer $\mathcal{G}$

This calculator is built starting from the $\mathcal{B}$ model, where now only
one of the two disks present has a control rod applied. As happens for the
$\mathcal{B}$ calculator, also with the $\mathcal{G}$ calculator the observer
does not agree with the VR proclaimed by the manufacturer.

Figure 7: Computer $\mathcal{G}$ A disk of mass $m$ is free to move inertially
with little friction in a container with very massive walls. A small rod is
hinged to the disk which allows a force to be applied to it. A second puck is
free to move without the action of external forces.

### 2 Computer $\mathcal{H}$

The $\mathcal{H}$ calculator is obtained by applying a controller to the
luminous point $p$ of the rotating disk presented in the calculator E. By
applying a force to the point $p$ the observer verifies that the observed
system is not described by a potential with central symmetry as declared by
the manufacturer, in fact he verifies that he can stop the motion of the point
$p$ without its trajectory collapsing towards the center as it would happen if
it were the declared physical system.

Figure 8: Computer $\mathcal{H}$ A disk of mass $m$ is hinged at its center to
a pivot so that it can rotate about its center of mass. A light source at a
distance $r$ from the center is applied to it. The mass of the light source is
considered to be negligible compared to the mass of the disk. A small rod is
hinged in correspondence with the light source which allows you to apply a
force at will.

### 2 Computer $\mathcal{X}$

This calculator consists of a toothed linear element which alternately comes
into contact with two toothed wheels which rotate by inertia, one clockwise
and the other anti-clockwise. The moment of inertia of the two wheels is
assumed to be much greater than that of the linear element which, as a first
approximation, is considered as having mass equal to zero. Each wheel is hung
on a spring. Between the two motions there is a phase shift such that they are
not found to act simultaneously on the linear element. The result is that the
linear element moves along the axis $x$ with uniform rectilinear motion in an
alternating direction. On this element there is a light source which is
considered the output of the system. The manufacturer declares that the system
has two degrees of freedom, i.e. the coordinates $x$ and $y$ of the light
source and also declares that it moves by inertial motion.

Figure 9: Computer $\mathcal{X}$ A linear toothed element is scrolled
horizontally through a gear system that controls its movement. A light source
is placed on this element.

## 3 Theoretical limits of conceptual interactive systems (VR)

In the previous sections we have seen that a computing system (computer) is a
physical system described by a certain set $\mathbf{C}$ of degrees of freedom.
Similarly to the computation system, the output subsystem is also described by
a certain set $\mathbf{O}\subseteq\mathbf{C}$ of degrees of freedom and the
same is true for the input subsystem $\mathbf{I}\ subseteq\mathbf{C}$. Having
made these considerations, it is possible to draw considerations of a general
nature which we see below.

### 3 Simulation systems without input devices

These systems can simulate both a physical system (calculator $\mathcal{A}$)
and a non-physical system (calculator $\mathcal{B}$), i.e. a system that is
not describable through observable degrees of freedom but is possibly
traceable to a physical description if we consider the presence of hidden
variables.

It should be noted that if the observer can observe all the degrees of freedom
of the system, i.e. $\mathbf{O}\equiv\mathbf{C}$ then he always identifies the
computer with a physical system and agrees with what is declared by the
manufacturer only if the constructor declares a physical system compatible
with the equations obtained by the observer.

Instead, if the system has computational degrees of freedom that are not
observable $\mathbf{O}\subset\mathbf{C}$ the observer’s judgment depends on
the relationship existing between the computational degrees of freedom. If the
unobservable degrees of freedom $\mathbf{U}$
$(\mathbf{U}=\mathbf{C}-\mathbf{O})$ do not interact with the observable ones
$(\mathbf{O})$ the system will always be classified as physical by the
observer (calculator $\mathcal{D}$). Otherwise, if the degrees of freedom
$\mathbf{U}$ interact with the degrees of freedom $\mathbf{O}$ then the
observer will classify the non-physical system because, as already observed,
he will not be able to deduce a system of equations based only on the
variables $x_{i}\in\mathbf{O}$ (calculator $\mathcal{B}$).

It should be noted that even in the case where $\mathbf{O}\equiv\mathbf{C}$ it
can happen that the manufacturer claims to simulate a physical system
different from the one actually implemented, so in these cases the observer
agrees with a reality wrong physics, but still plausible (calculator
$\mathcal{E}$). With reference to what has been said in the 1.2 section, this
likelihood is generated because the computer is not analogous to the system it
is simulating but, in a condition of inertial motion, each point on it follows
the same trajectory followed by the representative point of the system
declared by the manufacturer (particle in a field with central symmetry).
Therefore: the system declared by the manufacturer foresees two degrees of
freedom but admits uniform circular motion as a solution. The system actually
implemented by the manufacturer is a system with a single degree of freedom
which in inertial motion moves with uniform circular motion.

### 3 Simulation systems with input devices

As we have seen, the presence of controllers in computers gives the observer
the ability to act on the computer input degrees of freedom $\mathbf{I}$ by
applying a force to it. This ability gives him yet another tool for
classifying the underlying physical system. In case the controller can act on
all $\mathbf{C}$ degrees of freedom of the system, then the observer will
always be able to classify the system correctly, as happens with the
$\mathcal{H}$ calculator when the observer realizes that the system is
different from what was declared by the manufacturer. While otherwise, the
observer will keep the same dilemmas.

## 4 Concrete systems of Virtual Reality

The calculators that have been described in the previous paragraphs are based
on the mechanics of the material point (or of the rigid body). In the real
world, VR is produced by computer systems that are not based on the mechanics
of the material point that we have used in this dissertation, but on digital
electronics, i.e. on the physics of classical electromagnetism444In reality,
the LED devices used to make video screens they require both mechanical
explanations, but the same result could be obtained using a cathode ray tube
instead of LEDs, so that we can say that current VR systems can already be
easily explained with classical electromagnetism.. However, they are physical
systems whose dynamics can be described by a certain number of degrees of
freedom which, as for mechanical computers, can be divided into degrees of
input, output and computation. So even for modern VR systems we can say that
the same considerations elaborated for mechanical computers that we have dealt
with up to now are valid.

In real examples, the output degrees of freedom are in the pixels projected by
the oculus and the sound frequencies produced by the earphones.

As an example, in the first proposed example of the floppy disk bouncing in a
box (computer $\mathcal{A}$) it can be described as an ordinary computer
connected to a screen. In a nutshell, the screen is an electronic device to
which a matrix of LEDs is connected. A few-instruction software is running on
the computer. Each instruction is encoded as an electrical charge stored in a
capacitor array. The disk is displayed by executing the code which calculates
its coordinates and transforms them into physical addresses of the LEDs
connected to the screen. Code execution is governed by computer CPU cycles:
electrical oscillations. The observer visualizes the disk, that is a subset of
the LEDs activated on the basis of what is computed by the CPU. As you can
easily understand, even what is perceived as an image on a screen is still the
set of a certain number of degrees of freedom of a physical system, those
defined for the output. So even in the electronic version of the bouncing
disk, the observer is observing part of the degrees of freedom of an evolving
dynamic system.

Having made this necessary premise to link the conceptual reasoning set out
above to practical considerations involving current systems, let us come to
the discussion of the results.

## 5 Discussione

The results presented in the previous section allow us to outline a picture of
the theoretical limits of the possibilities offered by VR.

Starting from the various cases exposed, one can reason inductively and affirm
that a physical system of $N$ degrees of freedom can be simulated using a
physical system with a number $M$ degrees of freedom which can be greater,
less than or equal to $N$. In more direct terms, wearing oculus and earphones,
one can witness a simulated reality, i.e. one can observe a reality that is
not happening where we believe but is taking place in the computer in the form
of a physical process that passes through a set of states similar to that of
perceived reality.

In practice, therefore, a computerized system can actually simulate a physical
system, that is, it can mislead an observer into agreeing on the likelihood of
what he is observing with respect to what is declared by the producer of the
simulation.
However, the simulation can take place in different ways:

  1. 1.

the system to be simulated consists of $N$ degrees of freedom and is simulated
using a calculator with $P$ degrees of freedom of which $M$ degrees of freedom
output with $M<N$. In this case the simulating calculator is not physically
analogous to the system to be simulated but its output can be (calculator
$\mathcal{E}$).

  2. 2.

The system to be simulated consists of $N$ degrees of freedom and is simulated
using a calculator with $N$ degrees of freedom all labeled as output degrees.
In this case the simulating calculator coincides with the system to be
simulated (calculators $\mathcal{A}$ and $\mathcal{C}$).

  3. 3.

the system to be simulated consists of $N$ degrees of freedom and is simulated
by a computer with $M>N$ degrees of freedom of which $N$ are used as output.
If the $N$ of $M$ output degrees of freedom are physically independent of the
remaining $M-N$ degrees of freedom (calculator $\mathcal{D}$), then we return
to the previous case, otherwise this means that the degrees observable degrees
of freedom (ouput) are dependent on hidden degrees of freedom ($\mathcal{B}$
and $\mathcal{X}$ calculators).

In the first simulation mode we see that a computer with limited resources can
simulate a more complex reality as it does not simulate the interaction
between the system components but only the trajectories of the system
components which generally take place on parameterized curves. This allows for
example to transform a system with two degrees of freedom (point subject to a
central potential) into a system with only one degree of freedom (point on a
rotating circle). This type of simulation remains plausible only if the
observer simply observes. If instead the observer has an input system towards
the computer (ie a controller) then he realizes that the reality he has joined
does not correspond to the physics of the system, therefore the simulation
does not hold up to experimental verification.
In the second way the computer and the system to be simulated coincide. In
this case the simulation is a “staging” and is absolutely resistant to the
action of the observer who can act with the controller on all the degrees of
freedom of the system without running into any inconsistencies. The limit of
this way of simulating is evident in that to simulate a given reality it is
necessary to create it in all its details by transforming the simulation into
a copy of reality

The third way is the most used and consists in using more degrees of freedom
than the observable ones in the simulation. This mode allows to simulate
reality as long as the observer limits himself to observing, but if the
observer has a controller then he, acting on a given degree of freedom, will
experience that it is linked to other hidden degrees of freedom, therefore he
will the consistency of the simulation.

At this point we are able to formulate the answers to the initial questions.
The answer to the first question is affirmative: it is possible to deceive the
mind to the point of not being able to recognize whether the perceived reality
is real or simulated. However, this deception only holds up as long as the
simulation is passively undergone but it does not hold up if it is possible
for the observer to interact with the simulated reality.

We have also formulated an answer for the second question. In fact, just as it
is possible to simulate reality, it is also possible to simulate a non-real,
i.e. magical world, therefore: flying fairies, objects that remain suspended
from the ground in defiance of the law of gravity, objects that appear out of
nowhere or that vanish without conservation of mass etc. However, this
possibility is based on hiding degrees of freedom from the observer, therefore
it requires the complicity of the observer to be realised, exactly like the
simulation of reality.
In conclusion it can be said that a VR system can simulate both reality and
magic. To completely simulate reality, however, it is necessary to use more
degrees of freedom than those describing the reality to be simulated. This is
an important limitation because obviously there is a conceptual limit to this
process.
In summary, for an observer, a virtual reality experience is still an
experience of reality. This experience can be exchanged for a different
experience, that is, for a different reality that can be real or not real.

## References

  * [Deutsch (1985)] Deutsch, D. (1985). Quantum theory, the Church–Turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818), 97-117.
  * [Einstein (1935)] A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47 (1935)
  * [Horsman(2014)] Horsman Clare, Stepney Susan, Wagner Rob C. and Kendon Viv 2014. When does a physical system compute?Proc. R. Soc. A.4702014018220140182 http://doi.org/10.1098/rspa.2014.0182
  * [Sisini(2021)] Sisini F, Sisini V. The Turing machine of a harmonic oscillator: from the code to the dynamic system. https://doi.org/10.48550/arXiv.2110.06119

## Acknowledgement

Thanks to Annalisa Pazzi for the kind realization of the images presented in
this work.


# Fast-forward adiabatic quantum dynamics of XY spin model on three spin
system

Iwan Setiawan1 Riska Ekawita2 Ryan Sugihakim3 and Bobby Eka Gunara3 1
Department of Physics Education, University of Bengkulu, Kandang Limun,
Bengkulu 38371, Indonesia  2 Department of Physics, University of Bengkulu,
Kandang Limun, Bengkulu 38371, Indonesia  3 Department of Physics, Institut
Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
<EMAIL_ADDRESS>
###### Abstract

We discussed a method to accelerate an adiabatic quantum dynamics of $XY$ spin
model by using the fast-forward method proposed by Masuda and Nakamura. The
Accelerated scheme is constructed by adding the driving Hamiltonian to the
original Hamiltonian and speeding it up with a large time-scaling factor and
an adiabatic parameter that realizes adiabatic quantum dynamics in a shortened
time. Accelerated adiabatic dynamics start by assuming the candidate of
driving Hamiltonian consists of the pair-wise exchange interaction and
magnetic field. The driving Hamiltonian terms multiplied by the velocity
function together with the original Hamiltonian give fast-forward driving for
adiabatic states. We apply our method to $XY$ spin model by considering three
spin systems on the Kagome lattice. In this model, we obtained the $XY$ pair-
wise exchange interaction of nearest neighbors and next-nearest neighbors
should be added to the original Hamiltonian as a driving interaction to
accelerate the adiabatic motion. This pair-wise driving interaction in the
fast-forward scheme guarantees the complete fidelity of accelerated states.

##  1 Introduction

The method to optimize and control the quantum evolution system has greatly
improved quantum technology. Some techniques have been developed to obtain an
optimum and robust quantum state. One of the techniques that attract much
attention is adiabatic quantum evolution. In this technique, the eigenstate of
the system would not change during a very small change of Hamiltonian, and it
needs a longer time. Some methods have been developed to accelerate adiabatic
quantum evolution. The method of shortcut to adiabaticity (STA) has received
much more attention [6, 7, 8, 13, 14, 15, 16, 17, 18, 19]. The design of STA
can be achieved using a variety of techniques[15, 19]. A universal approach is
provided by counterdiabatic driving [8], also known as transitionless quantum
driving. These methods have been developed to design a shortcut between two
states without knowing the adiabatic relation between them [19]. Instead of
this method’s work in quantum mechanics, related schemes appear in terms of
classical mechanics [9] and stochastic dynamics [10]. Masuda and Nakamura [1]
constructed another method so-called fast-forward method. They consider to
accelerate standard quantum dynamics by using the additional phase and driving
potential to accelerate the wave function. The fast-forward method can speed
up any given quantum system and obtain the target state on the shortened time
scale by speeding up the reference quantum dynamics. Khujakulov and Nakamura
[2] proposed that the fast-forward method is useful to improve the quantum
tunneling power. This theory was further developed to speed up the quasi-
static or adiabatic quantum dynamics [3, 4, 5] by finding the regularization
term and driving potential. Some works on how to accelerate the adiabatic spin
dynamics are also proposed [20, 22, 23, 24, 25] and extended to control the
dynamics of many-body systems governed by the transverse Ising [11] and
Lipkin-Meshkov-Glick [12] Hamiltonians. The fast-forward scheme of quantum
entangled states [26] is proposed by obtaining the driving interaction that
consists of pair-wise interaction and magnetic field. The three body
interactions for up to 4 spin systems [29] should be added instead of pairwise
interaction and magnetic field. The driving interactions in this paper depend
on the geometry formed by the spin clusters. Application of the fast-forward
method to three-spin of quantum annealing systems also have been reported
[28]. This model is devoted to the triangle spin model and obtained the
driving interaction that consists of pairwise interaction, magnetic field, and
three-body interaction. The method to accelerate spin system on $XY$ model has
been investigated[20]. This paper considers an $XY$ spin model for two-spin
and one-dimensional isotropic of many spin systems. By considering the spin
interaction between the nearest neighbor of $X$ and $Y$ direction, they
obtained the driving Hamiltonian to accelerate the dynamics of the system.
Another method to accelerate the system using an $XY$ model in the regime in
which the adiabatic theorem cannot be applied also have been reported[21].

In this present work, we shall investigate the scheme of fast forward of
adiabatic dynamics in $XY$ spin model by applying the scheme to three spin
systems in a Kagome lattice. This spin system has a different time-dependent
spin interaction between the nearest neighbors and the next-nearest neighbors.
Using the fast-forward method, we obtain a driving Hamiltonian to guarantee
accelerated dynamics. In Section 2, we shall briefly summarize the scheme of
fast forward of adiabatic quantum spin dynamics. In Section 3, we consider the
fast-forward Hamiltonian for $XY$ spin model for three spin systems in Kagome
lattice. Section 4 is devoted to the conclusion.

##  2 Fast-forward of adiabatic spin dynamics

In this section, we shall discuss a brief review of the theory of adiabatic
spin dynamics proposed by Masuda and Nakamura [26, 27]. Let us assume the
adiabatic state of time-dependent Schrödinger equation (TDSE) as

$\Psi_{0}(R(t))=\begin{pmatrix}C_{1}(R)\\\ \vdots\\\
C_{n}(R)\end{pmatrix}e^{-\frac{i}{\hbar}\int_{0}^{t}E(R(t^{\prime}))dt^{\prime}}e^{i\xi(R(t))},$
(1)

where $C(R)$ is a spinor component, available from time independent
Schrödinger equation. Here

$R\equiv R(t)=R_{0}+\epsilon t$ (2)

is the adiabatically-changing parameter with $\epsilon\ll 1$, and $\xi$ is the
adiabatic phase[8]. To obtain the adiabatic wave function as in Eq.(1), we
should modify the Hamiltonian as regularized Hamiltonian as given by

$H_{0}^{reg}=H_{0}+\epsilon\mathcal{\tilde{H}},$ (3)

where $\mathcal{\tilde{H}}$ is the regularization term. The wave function as
in Eq.(1) and the regularized Hamiltonian will satisfy the time-dependent
Schrödinger equation

$i\hbar\frac{\partial}{\partial_{t}}\Psi_{0}(R(t))=H_{0}^{reg}\Psi_{0}(R(t)).$
(4)

By solving Eq.(4) we obtain the equation to solve the regularization terms
($\mathcal{\tilde{H}}$) in $O(\epsilon^{1})$ as

$\displaystyle\mathcal{\tilde{H}}_{n}\begin{pmatrix}C_{1}(R)\\\ \vdots\\\
C_{n}(R)\end{pmatrix}$ $\displaystyle=$ $\displaystyle
i\hbar\partial_{R}\begin{pmatrix}C_{1}(R)\\\ \vdots\\\ C_{n}(R)\end{pmatrix}$
(5) $\displaystyle-\begin{pmatrix}C_{1}(R)^{\dagger}\\\ \vdots\\\
C_{n}(R)^{\dagger}\end{pmatrix}\partial_{R}\begin{pmatrix}C_{1}(R)\\\
\vdots\\\ C_{n}(R)\end{pmatrix}$

which is the core equation of this scheme. Here $\mathcal{\tilde{H}}_{n}$ is
the $n$-th state-dependent regularization term. The adiabatic wave function in
Eq.(1) will be obtained at the initial and the final time of evolution by
using the regularization term as in Eq.(5). Obtaining this adiabatic wave
function needs a very long time, which will give less advantage to realize in
the laboratory in particular when we want to see the time evolution of the
wave function. To fast-forward such adiabatic motion, we introduce the time
scaling factor ($\Lambda(t)$). The time scaling factor is a variable that can
indicate an event occurring in a shorter time. By using the time scale factor,
an event or motion of an object can be arranged according to the desired time.
Several types of time scaling factors can be used for fast-forward schemes
including spin cases such as cosine functions, hyperbolic functions, or
Gaussian functions. This kind of function shortens the distance between two
positions. In this three-spin system, the time scaling factor in the form of a
cosine function is used. Here $\Lambda(t)$ which is defined as

$\Lambda(t)=\int_{0}^{t}\alpha(t^{\prime})dt^{\prime}$ (6)

Here $\alpha$ is an arbitrary time magnification factor that is typically
given by [3]

$\alpha(t)=\bar{\alpha}-(\bar{\alpha}-1)\cos\left(\frac{2\pi}{T_{FF}}t\right)$
(7)

Time-dependent Schrödinger equation for fast-forward state is written by

$i\hbar\frac{\partial}{\partial_{t}}\Psi_{FF}=H_{FF}\Psi_{FF},$ (8)

where

$\Psi_{FF}=\begin{pmatrix}C_{1}(R(\Lambda(t)))\\\ \vdots\\\
C_{n}(R(\Lambda(t)))\end{pmatrix}e^{-\frac{i}{\hbar}\int_{0}^{t}E(R(\Lambda(t^{\prime})))dt^{\prime}}e^{i\xi(R(\Lambda(t)))}.$
(9)

By taking time derivative of $\Psi_{FF}$ in Eq.(9), we obtain the equation for
fast-forward Hamiltonian as [26]

$\displaystyle i\hbar\frac{\partial\Psi_{FF}}{\partial t}$ $\displaystyle=$
$\displaystyle\left(v(t)\mathcal{\tilde{H}}_{n}(R(\Lambda(t)))+H_{0}(R(\Lambda(t)))\right)\Psi_{FF}$
(10) $\displaystyle\equiv$ $\displaystyle H_{FF}\Psi_{FF},$

where $v(t)$ is velocity function taken from the asymptotic limit, limit
$\bar{\alpha}\rightarrow\infty$ and $\epsilon\rightarrow 0$ under the
constraint that $\bar{\alpha}\dot{\epsilon}\equiv\bar{v}$. Here
$R(\Lambda(t))$ is given by [27]

$R(\Lambda(t))=R_{0}+2\bar{v}\left(\frac{t}{2}-\frac{T_{FF}\sin\left(\frac{2\pi
t}{T_{FF}}\right)}{4\pi}\right).$ (11)

$H_{FF}$ is the driving Hamiltonian, $\mathcal{\tilde{H}}_{n}$ is the
regularization term obtained from Eq.(5) to generate the fast-forward scheme
in spin system. As an example, we apply this fast-forward scheme to two spin
systems with $XY$ spin model described in Appendix A.

##  3 XY Spin model with Kagome Lattice

To see how the above scheme works well, we also apply our method to three spin
system models by considering the Kagome lattice [29, 31, 30]. The
configuration of this model is shown in Fig.(1). Here, the time-dependent spin
interactions between nearest neighbors, $J_{1}$ and next nearest neighbors,
$J_{2}$ are shown by single and double bonds, respectively. This spin model
has a different time-dependent spin interaction for nearest neighbor and next
nearest neighbor which is described by a Kagome lattice, a two-dimensional
lattice pattern found in the crystal structure of many natural minerals. Some
minerals, namely jarosites and herbertsmithite, contain two-dimensional layers
or three-dimensional Kagome lattice arrangements of atoms in their crystal
structure. Here we consider a simple Kagome lattice for up to three spin
systems. Furthermore, this method can be applied to accelerate the dynamics of
microparticles on crystal structure that will be very useful for understanding
and manipulating correlated quantum materials [32].

Figure 1: Three spin systems of Kagome lattice

.

The Hamiltonian for three spin systems is given by

$H_{0}=J_{1}(\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{2}^{x}\sigma_{3}^{x})+J_{2}(\sigma_{3}^{y}\sigma_{1}^{y})+\frac{1}{2}(\sigma_{1}^{z}+\sigma_{2}^{z}+\sigma_{3}^{z})B_{z}(R(t)),$
(12)

with $J_{1}=J_{1}(R(t))$, $J_{2}=J_{2}(R(t))$, and $B_{z}=B_{z}(R(t))$ By
using these basis : $\Ket{\uparrow\uparrow\uparrow}$,
$\Ket{\uparrow\uparrow\downarrow}$, $\Ket{\uparrow\downarrow\uparrow}$,
$\Ket{\downarrow\uparrow\uparrow}$, $\Ket{\uparrow\downarrow\downarrow}$,
$\Ket{\downarrow\uparrow\downarrow}$, $\Ket{\downarrow\downarrow\uparrow}$ and
$\Ket{\downarrow\downarrow\downarrow}$. The matrix representation of Eq.(12)
is written as

$H_{0}=\begin{pmatrix}\frac{3B_{z}}{2}&0&0&J_{1}&0&-J_{2}&J_{1}&0\\\
0&\frac{B_{z}}{2}&J_{1}&0&J_{2}&0&0&J_{1}\\\
0&J_{1}&\frac{B_{z}}{2}&0&J_{1}&0&0&-J_{2}\\\
J_{1}&0&0&-\frac{B_{z}}{2}&0&J_{1}&J_{2}&0\\\
0&J_{2}&J_{1}&0&\frac{B_{z}}{2}&0&0&J_{1}\\\
-J_{2}&0&0&J_{1}&0&-\frac{B_{z}}{2}&J_{1}&0\\\
J_{1}&0&0&J_{2}&0&J_{1}&-\frac{B_{z}}{2}&0\\\
0&J_{1}&-J_{2}&0&J_{1}&0&0&-\frac{3B_{z}}{2}\end{pmatrix}.$ (13)

From the time-dependent eigenvalue of this Hamiltonian, as shown in Fig.(2),
we have numerically shown that there is no level crossing of the ground states
$E_{0}$ of this model in the fast-forward time scale.

Figure 2: Time dependence of eigenvalue for $XY$ model in three spin system,
with $J_{1}=J_{0}-R(\Lambda(t))$, $J_{2}=R(\Lambda(t))$,
$B_{z}$=$B_{0}-R(\Lambda(t))$, $\bar{v}$ = 100, $T_{FF}=0.1$,
$J_{0}(R_{0}=0)=10$, and $B_{0}=0$.

The ground state $E_{0}$ of this model is obtained as

$E_{0}=-\frac{1}{2}\sqrt{3}\left(\frac{\beta}{\sqrt[3]{\Gamma}}-\sqrt[3]{\Gamma}\right)-\frac{\beta}{2\sqrt[3]{\Gamma}}+\frac{B_{z}}{6}-\frac{\sqrt[3]{\Gamma}}{2}+\frac{J_{2}}{3},$
(14)

with
$\Gamma=\sqrt{\left(-\alpha-\frac{3B_{z}^{3}}{16}-\frac{1}{27}\left(\frac{B_{z}}{2}+J_{2}\right)^{3}+\eta+\frac{J_{2}^{3}}{2}\right)^{2}-\beta^{3}}+\gamma+\Delta$,
$\alpha=\frac{1}{6}\left(\frac{B_{z}}{2}+J_{2}\right)\left(\frac{5B_{z}^{2}}{4}-B_{z}J_{2}+4J_{1}^{2}+J_{2}^{2}\right)$,
$\beta=\frac{5B_{z}^{2}}{12}-\frac{B_{z}J_{2}}{3}+\frac{1}{9}\left(\frac{B_{z}}{2}+J_{2}\right)^{2}+\frac{4J_{1}^{2}}{3}+\frac{J_{2}^{2}}{3}$,
$\gamma=-\frac{1}{8}3B_{z}^{2}J_{2}-B_{z}J_{1}^{2}+\frac{B_{z}J_{2}^{2}}{4}-2J_{1}^{2}J_{2}$,
$\eta=\frac{3B_{z}^{2}J_{2}}{8}+B_{z}J_{1}^{2}-\frac{B_{z}J_{2}^{2}}{4}+2J_{1}^{2}J_{2}$,
and
$\Delta=\alpha+\frac{3B_{z}^{3}}{16}+\frac{1}{27}\left(\frac{B_{z}}{2}+J_{2}\right)^{3}-\frac{J_{2}^{3}}{2}$.

The derivation of regularization term is given in Appendix C. From the matrix
of the candidate of regularization term in Eq.(Appendix C. The derivation of
regularization term for three spin systems), the equation for regularization
term is written as

$\displaystyle i\hbar\frac{\partial C_{1}}{\partial
R}=\frac{3\tilde{B}_{z}}{2}C_{1}-4i\tilde{W}_{1}C_{4}-2i\tilde{W}_{2}C_{6}$
$\displaystyle i\hbar\frac{\partial C_{4}}{\partial
R}=2i\tilde{W}_{1}C_{1}-\frac{B_{z}}{2}C_{4},$ $\displaystyle
i\hbar\frac{\partial C_{6}}{\partial
R}=2i\tilde{W}_{2}C_{1}-\frac{\tilde{B}_{z}}{2}C_{6},$ (15)

which gives the solution

$\tilde{B}_{z}=0,$ (16)
$\tilde{W_{1}}=-\frac{i\left(aC_{4}C_{1}+3bC_{1}^{2}-bC_{6}^{2}+cC_{4}C_{6}\right)}{2C_{1}\left(3C_{1}^{2}-2C_{4}^{2}-C_{6}^{2}\right)},$
(17)

and

$\tilde{W_{2}}=-\frac{i\left(aC_{6}C_{1}+2bC_{4}C_{6}+3cC_{1}^{2}-2cC_{4}^{2}\right)}{2C_{1}\left(3C_{1}^{2}-2C_{4}^{2}-C_{6}^{2}\right)}.$
(18)

with $a=i\hbar\frac{\partial C_{1}}{\partial R}$, $b=i\hbar\frac{\partial
C_{4}}{\partial R}$, and $c=i\hbar\frac{\partial C_{6}}{\partial R}$. Noting
Eq.(B.45), we find that $B_{z}=0$. Comparing with two spin models, on this
spin model we obtained that two regularization terms should be added to the
original Hamiltonian i.e: $\tilde{W_{1}}$ and $\tilde{W_{2}}$ as an exchange
parameter of $X$ and $Y$ direction of nearest neighbor and next nearest
neighbor interaction respectively.

By numerically solving the TDSE in Eq.(10) with parameter $R(\Lambda(t))$ in
Eq.(11), and $J_{1}=J_{0}-R(\Lambda(t))$, $J_{2}=R(\Lambda(t))$,
$B_{z}$=$B_{0}-R(\Lambda(t))$, $\bar{v}$ = 10, $T_{FF}=1$, $J_{0}(R_{0}=0)=10$
and $B_{0}=0$, we see the dynamics of amplitude at the initial state as a
linear combination of $\Ket{\uparrow\uparrow\uparrow}$,
$\Ket{\downarrow\uparrow\uparrow}$, $\Ket{\downarrow\uparrow\downarrow}$, and
$\Ket{\downarrow\downarrow\uparrow}$. As $J_{1}$ and $B_{x}$ decreased from a
positive value towards $0$ and $J_{2}$ increased from $0$ toward a positive
value, the initial states rapidly change to nonentangled state i.e:
$\Ket{\uparrow\uparrow\uparrow}$ state at the final time ($T_{FF}$) as shown
in Fig.(3). The solution of TDSE as shown in Fig.(3) has a complete fidelity
with time-dependent eigenvector as in Eq.(B.36, B.37 and B.38), during the
fast forward time range $0\leq t\leq T_{FF}$.

(a)  (b)

Figure 3: Time dependence of $|C_{1}^{FF}|^{2}$ (solid line),
$|C_{4}^{FF}|^{2}$= $|C_{7}^{FF}|^{2}$(dashed line), and $|C_{6}^{FF}|^{2}$
(dot-dashed line: a. Obtaining by solving TDSE b. Obtaining from the
eigenvector)

Time dependence of regularization term is shown in Fig.(4)

(a)  (b)

Figure 4: Time dependence of regularization term : (a) $\tilde{W}_{1}$; (b)
$\tilde{W}_{2}$.

To conclude, for three spin systems in Kagome lattice, where the original
Hamiltonian is an $XY$ interaction of the nearest neighbor with interaction
parameter $J_{1}$ and $XY$ interaction of the next nearest neighbor with
interaction parameter $J_{2}$, we obtain the regularization terms. These
regularization terms consist of $\tilde{W}_{1}$ and $\tilde{W}_{2}$ as an
exchange parameter of nearest neighbor (NN) interaction
$(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x})+(\sigma_{2}^{x}\sigma_{3}^{y}+\sigma_{2}^{y}\sigma_{3}^{x})$
and next nearest neighbor interaction
(NNN)$(\sigma_{1}^{x}\sigma_{3}^{y}+\sigma_{1}^{y}\sigma_{3}^{x})$
respectively. This regularization term should be added to the original
Hamiltonian to guarantee the adiabatic motion and to speed up the dynamics of
the spin.

##  4 Conclusion

We presented a scheme of fast-forward adiabatic spin dynamics proposed by
Masuda and Nakamura to an $XY$ spin model in the Kagome lattice. We consider
three spin systems with the original Hamiltonian, including time-dependent
pair-wise exchange interaction parameters of nearest neighbor (NN), next
nearest neighbor (NNN), and magnetic fields. We constructed the quasi-
adiabatic dynamics by adding the regularization terms to the original
Hamiltonian and then accelerated it using a large time-scaling factor.
Assuming the candidate of regularization term that is experimentally
realizable consisting of the exchange interactions and magnetic field, we
solved the regularization terms. We obtained for this model, the
regularization term are $\tilde{W}_{1}$ as an $xy$ exchange interaction
parameter of
$(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x})+(\sigma_{2}^{x}\sigma_{3}^{y}+\sigma_{2}^{y}\sigma_{3}^{x})$
and $\tilde{W}_{2}$ for
$(\sigma_{1}^{x}\sigma_{3}^{y}+\sigma_{1}^{y}\sigma_{3}^{x})$. The candidate
of regularization term for three spin system of an $XY$ model in Kagome
lattice can be obtained from the candidate of regularization term in two spin
system by adding an $XY$ pair-wise exchange interaction in the nearest
neighbor (NN) and next-nearest neighbor (NNN) as
$\mathcal{\tilde{H}}=\sum_{(i,j)\in
all}\tilde{W}_{ij}^{NN}(\sigma_{i}^{x}\sigma_{j}^{y}+\sigma_{i}^{y}\sigma_{j}^{x})+\sum_{(i,j)\in
all}\tilde{W}_{ij}^{NNN}(\sigma_{i}^{x}\sigma_{j}^{y}+\sigma_{i}^{y}\sigma_{j}^{x})$.
We also confirmed the complete fidelity of the wave function during the fas-
forward motion. Furthermore, this method will be beneficial to accelerate the
dynamics of microparticles on crystal structures that form a Kagome geometry.
A further investigation of many $XY$ spin systems of Kagome lattice remains a
challenge.

I.S is grateful to Prof. Katsuhiro Nakamura, Rudy Kusdiantara, Ardian Nata
Atmaja, Andy Octavian Latif, and Fiki Taufik for a valuable discussion. I.S is
supported by PDUPT grant from Kemendikburistek 2022. B.E.G acknowledged Hibah
PPD Kemdikbud Ristek for financial support.

## Appendix A. An $XY$ spin model of two spin systems

This model is a Heisenberg spin interaction with $X$ and $Y$ directions and an
external magnetic field in $Z$ direction. Furthermore, the model can be
applied to two-qubit systems that are very important in developing a quantum
computer. The Hamiltonian for this model is written as [20]

$H_{0}=J_{1}(R(t))\sigma_{1}^{x}\sigma_{2}^{x}+J_{2}(R(t))\sigma_{1}^{y}\sigma_{2}^{y}+\frac{1}{2}(\sigma_{1}^{z}+\sigma_{2}^{z})B_{z}(R(t)),$
(A.19)

On these two spin systems, the spin interacts with the nearest neighbor where
$J_{1}$ and $J_{2}$ are the time-dependent interaction parameters, and
$\sigma^{x}$ and $\sigma^{y}$ are Pauli matrices. By using the basis on two
spin systems : $\Ket{\uparrow\uparrow}$, $\Ket{\uparrow\downarrow}$,
$\Ket{\downarrow\uparrow}$, and $\Ket{\downarrow\downarrow}$, matrix
representation of Eq.(A.19) can be written as

$H_{0}=\left(\begin{array}[]{cccc}B_{z}&0&0&J_{1}-J_{2}\\\
0&0&J_{1}+J_{2}&0\\\ 0&J_{1}+J_{2}&0&0\\\ J_{1}-J_{2}&0&0&-B_{z}\\\
\end{array}\right).$ (A.20)

We can see from the Hamiltonian in Eq. (A.20), time-dependent of eigenvalue
for this Hamiltonian as shown in Fig.(A5)

Figure A5: Time dependence of eigenvalue (solid line for the ground state) for
$XY$ model in two spin systems, with $J_{1}=J_{0}-R(\Lambda(t))$,
$J_{2}=R(\Lambda(t))$, $B_{z}$=$B_{0}-R(\Lambda(t))$, $\bar{v}$ = 10,
$T_{FF}=1$, $J_{0}(R_{0}=0)=10$, and $B_{0}=0$.

By decreasing $J_{1}$ and $B_{z}$ from a positive value towards $0$ and
increasing $J_{2}$ from $0$ towards a positive value, the ground state of the
system as indicated by a solid line in Fig.(A5), will have no transition from
the initial time and the final time of the evolution. Here we consider
applying the fast-forward scheme to the ground state of Hamiltonian in
Eq.(A.20). This ground state is obtained as
$E_{0}=-\sqrt{B_{z}^{2}+J_{1}^{2}+J_{2}^{2}-2J_{1}J_{2}}$. Here we shall
obtain the driving Hamiltonian for these two spin systems by using the
equation of regularization term in Eq.(5). To solve the regularization term,
we use a candidate of regularization term as written by

$\displaystyle\mathcal{\tilde{H}}=\tilde{W}(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x})+\frac{1}{2}(\sigma_{1}^{z}+\sigma_{2}^{z})\tilde{B}_{z},$
(A.21)

The detailed derivation to obtain the regularization term is given in Appendix
C. We obtained the solution for regularization term equation (as written in
Eq.(A.35)) as

$\tilde{B}_{z}=0,$ (A.22)

and

$\tilde{W}=-\frac{i\left(aC_{4}+bC_{1}\right)}{2\left(C_{1}^{2}-C_{4}^{2}\right)},$
(A.23)

with $a=i\hbar\frac{\partial C_{1}}{\partial_{R}}$ and $b=i\hbar\frac{\partial
C_{4}}{\partial_{R}}$. Here by noting Eq.(A.33) we find $B_{z}=0$. The matrix
expression for regularization term is given by

$\mathcal{\tilde{H}}=\begin{pmatrix}0&0&0&-i\tilde{W}\\\ 0&0&0&0\\\ 0&0&0&0\\\
i\tilde{W}&0&0&0\end{pmatrix},$ (A.24)

with $\tilde{W}$ is the exchange interaction parameter for
$\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x}$. This
regularization term is analogous to the result obtained by Takahashi [20]. By
using an explicit expression of eigenvector ($C_{1}$ and $C_{2}$) in Eq.(A.27)
and Eq.(A.29), together with its $R$ derivative, the regularization term
$\tilde{W}$ can be rewritten as

$\tilde{W}=\frac{B_{z}(\dot{J_{1}}-\dot{J_{2}})+\dot{B_{z}}(J_{2}-J_{1})}{2\left(B_{z}^{2}+(J_{1}-J_{2})^{2}\right)}.$
(A.25)

Using Eq.(10), the fast forward Hamiltonian is given by

$\displaystyle H_{FF}$ $\displaystyle=$ $\displaystyle
J_{1}(\Lambda(t))\sigma_{1}^{x}\sigma_{2}^{x}+J_{2}(\Lambda(t))\sigma_{1}^{y}\sigma_{2}^{y}$
(A.26) $\displaystyle+$
$\displaystyle\frac{1}{2}(\sigma_{1}^{z}+\sigma_{2}^{z})B_{z}(\Lambda(t))+v(t)\tilde{W}.$

We then numerically solved the fast forward Schrödinger equation in Eq.(10)
with the additional regularization term as in Eq.(A.25) instead of the
original Hamiltonian in Eq.(A.19) with parameter $R(\Lambda(t))$ in Eq.(11).
Here we used parameter $J_{1}=J_{0}-R(\Lambda(t))$, $J_{2}=R(\Lambda(t))$,
$B_{z}$=$B_{0}-R(\Lambda(t))$, $\bar{v}$ = 10, $T_{FF}=1$, $J_{0}(R_{0}=0)=10$
and $B_{0}=0$. We see by adding the regularization term to the original
Hamiltonian and decreasing $J_{1}$ and $B_{z}$ from a positive value towards
$0$ and increasing $J_{2}$ from $0$ towards a positive value, the dynamics of
the state at the initial time as a linear combination of
$\Ket{\uparrow\uparrow}$ and $\Ket{\downarrow\downarrow}$ states change
rapidly to nonentangled state at the final time ($T_{FF}$). The dynamics of
wave function $\Psi_{FF}$ of fast forward Schrödinger equation is exactly the
same as the dynamics of wave function in the standard Schrödinger equation
(without fast forwarding), i.e the eigenstate of the Hamiltonian as in
Eq.(A.27) and Eq.(A.29). This fact is shown in Fig.(A6). This result showed
that by adding the regularization term, we can get the desired adiabatic wave
function in a shorter time. The time dependence of regularization term
($\tilde{W}$) is shown in Fig.(A7)

(a)  (b)

Figure A6: Time dependence of $|C_{1}^{FF}|^{2}$ (solid line),
$|C_{4}^{FF}|^{2}$ (dotted line) with the same parameter as in Fig.(A5) :(a)
obtained by Solving TDSE in Eq.10 (b) obtained from the eigenvector Figure A7:
Time dependence of of regularization term, with $J_{1}=J_{0}-R(\Lambda(t))$,
$J_{2}=R(\Lambda(t))$, $B_{z}$=$B_{0}-R(\Lambda(t))$, $\bar{v}$ = 10,
$T_{FF}=1$, $J_{0}(R_{0}=0)=10$, and $B_{0}=0$.

We confirmed that we obtained the driving Hamiltonian for two spin systems of
$XY$ model. By adding the regularization term $\tilde{W}$ to the original
Hamiltonian, we obtain the complete fidelity. This guarantee that the
regularization term $\tilde{W}$ can accelerate the system adiabatically.

## Appendix B. The derivation of regularization term for two spin systems

The normalized component of eigenvector for the ground state is written as

$C_{1}=\kappa\Bigg{(}\sqrt{\frac{J_{1}^{2}-2J_{1}J_{2}+J_{2}^{2}}{2B_{z}^{2}+2J_{1}^{2}+2J_{2}^{2}-4J_{1}J_{2}-2B_{z}\omega}}\Bigg{)},$
(A.27) $C_{2}=C_{3}=0,$ (A.28)

and

$C_{4}=\sqrt{\frac{J_{1}^{2}-2J_{1}J_{2}+J_{2}^{2}}{2B_{z}^{2}+2J_{1}^{2}+2J_{2}^{2}-4J_{1}J_{2}-2B_{z}\omega}},$
(A.29)

with

$\kappa=-\frac{\sqrt{B_{z}^{2}+J_{1}^{2}+J_{2}^{2}-2J_{1}J_{2}}-B_{z}}{J_{1}-J_{2}},$
(A.30)

and

$\omega=\sqrt{B_{z}^{2}+J_{1}^{2}-2J_{1}J_{2}+J_{2}^{2}}.$ (A.31)

By using Eq.(5) and the component of eigenvector, the equation to obtain
regularization term can be rewritten as

$\displaystyle i\hbar\frac{\partial C_{1}}{\partial
R}=\tilde{\mathcal{H}}_{11}C_{1}+\tilde{\mathcal{H}}_{14}C_{4}$ (A.32)
$\displaystyle i\hbar\frac{\partial C_{2}}{\partial
R}=\tilde{\mathcal{H}}_{21}C_{1}+\tilde{\mathcal{H}}_{24}C_{4}=0$
$\displaystyle i\hbar\frac{\partial C_{3}}{\partial
R}=\tilde{\mathcal{H}}_{31}C_{1}+\tilde{\mathcal{H}}_{34}C_{4}=0$
$\displaystyle i\hbar\frac{\partial C_{4}}{\partial
R}=\tilde{\mathcal{H}_{41}}C_{1}+\tilde{\mathcal{H}}_{44}C_{4}.$

From the normalization $C_{1}^{2}+C_{4}^{2}=1$, we see that

$C_{1}\frac{\partial C_{1}}{\partial_{R}}+C_{4}\frac{\partial
C_{4}}{\partial_{R}}=0,$ (A.33)

and then adiabatic phase $\xi$= 0. To solve the regularization term in
Eq.(A.32), we use a candidate of regularization term as written in Eq. (A.21),
with $\tilde{W}$ is pair-wise exchange parameter of
$(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x})$. The matrix
representation of Eq.(A.21) is obtained as

$\mathcal{\tilde{H}}=\begin{pmatrix}\tilde{B}_{z}&0&0&-i2\tilde{W_{1}}\\\
0&0&0&0\\\ 0&0&0&0\\\ i2\tilde{W}_{1}&0&0&-\tilde{B}_{z}\end{pmatrix}.$ (A.34)

By substituting Eq.(A.34) to Eq.(A.32), we obtain

$\displaystyle i\hbar\frac{\partial C_{1}}{\partial
R}=\tilde{B}_{z}C_{1}-2i\tilde{W}C_{4}$ (A.35) $\displaystyle
i\hbar\frac{\partial C_{4}}{\partial R}=2i\tilde{W}C_{1}-\tilde{B}_{z}C_{4},$

## Appendix C. The derivation of regularization term for three spin systems

From the eigen vector of the ground state, we see that $C_{1}\neq 0$,
$C_{2}=C_{3}=C_{5}=C_{8}=0$, $C_{4}=C_{7}\neq 0$, and $C_{6}\neq 0$. The
normalized component of eigenvector for the ground state is written as

$C_{1}=B_{1}\zeta,$ (B.36)

$C_{4}=C_{7}=B_{4}\zeta,$ (B.37)

$C_{6}=B_{6}\zeta,$ (B.38)

where,

$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-\frac{-B_{z}^{2}+16J_{1}^{2}+4J_{2}^{2}}{8B_{z}J_{1}}$
$\displaystyle+$
$\displaystyle\frac{\left(\frac{1}{2}\sqrt{3}i\left(\frac{\beta}{\sqrt[3]{\Gamma}}-\sqrt[3]{\Gamma}\right)+\frac{\beta}{2\sqrt[3]{\Gamma}}-\frac{B_{z}}{6}+\frac{\sqrt[3]{\Gamma}}{2}-\frac{J_{2}}{3}\right)^{2}}{2B_{z}J_{1}}$
$\displaystyle-$
$\displaystyle\frac{\frac{1}{2}\sqrt{3}i\left(\frac{\beta}{\sqrt[3]{\Gamma}}-\sqrt[3]{\Gamma}\right)+\frac{\beta}{2\sqrt[3]{\Gamma}}-\frac{B_{z}}{6}+\frac{\sqrt[3]{\Gamma}}{2}-\frac{J_{2}}{3}}{2J_{1}},$
$B_{4}=1,$ (B.40) $\displaystyle B_{6}$ $\displaystyle=$
$\displaystyle\frac{3B_{z}^{2}-8B_{z}J_{2}+16J_{1}^{2}+4J_{2}^{2}}{8B_{z}J_{1}}$
$\displaystyle-$
$\displaystyle\frac{\left(\frac{1}{2}\sqrt{3}i\left(\frac{\beta}{\sqrt[3]{\Gamma}}-\sqrt[3]{\Gamma}\right)+\frac{\beta}{2\sqrt[3]{\Gamma}}-\frac{B_{z}}{6}+\frac{\sqrt[3]{\Gamma}}{2}-\frac{J_{2}}{3}\right)^{2}}{2B_{z}J_{1}}$
$\displaystyle-$
$\displaystyle\frac{\frac{1}{2}\sqrt{3}i\left(\frac{\beta}{\sqrt[3]{\Gamma}}-\sqrt[3]{\Gamma}\right)+\frac{\beta}{2\sqrt[3]{\Gamma}}-\frac{B_{z}}{6}+\frac{\sqrt[3]{\Gamma}}{2}-\frac{J_{2}}{3}}{2J_{1}},$

and

$\zeta=\frac{1}{\sqrt{B_{1}^{2}+B_{4}^{2}+B_{6}^{2}+B_{7}^{2}}}.$ (B.42)

By using Eq.(5) and component of eigenvector, the equation for regularization
term can be rewritten as

$\displaystyle i\hbar\frac{\partial C_{1}}{\partial
R}=\tilde{\mathcal{H}}_{11}C_{1}+(\tilde{\mathcal{H}}_{14}+\tilde{\mathcal{H}}_{17})C_{4}+\tilde{\mathcal{H}}_{16}C_{6}$
$\displaystyle i\hbar\frac{\partial C_{2}}{\partial
R}=\tilde{\mathcal{H}}_{21}C_{1}+(\tilde{\mathcal{H}}_{24}+\tilde{\mathcal{H}}_{27})C_{4}+\tilde{\mathcal{H}}_{26}C_{6}=0,$
$\displaystyle i\hbar\frac{\partial C_{3}}{\partial
R}=\tilde{\mathcal{H}}_{31}C_{1}+(\tilde{\mathcal{H}}_{34}+\tilde{\mathcal{H}}_{37})C_{4}+\tilde{\mathcal{H}}_{36}C_{6}=0,$
$\displaystyle i\hbar\frac{\partial C_{4}}{\partial
R}=\tilde{\mathcal{H}}_{41}C_{1}+(\tilde{\mathcal{H}}_{44}+\tilde{\mathcal{H}}_{47})C_{4}+\tilde{\mathcal{H}}_{46}C_{6},$
$\displaystyle i\hbar\frac{\partial C_{5}}{\partial
R}=\tilde{\mathcal{H}}_{51}C_{1}+(\tilde{\mathcal{H}}_{54}+\tilde{\mathcal{H}}_{57})C_{4}+\tilde{\mathcal{H}}_{56}C_{6}=0,$
$\displaystyle i\hbar\frac{\partial C_{6}}{\partial
R}=\tilde{\mathcal{H}}_{61}C_{1}+(\tilde{\mathcal{H}}_{64}+\tilde{\mathcal{H}}_{67})C_{4}+\tilde{\mathcal{H}}_{66}C_{6},$
$\displaystyle i\hbar\frac{\partial C_{7}}{\partial
R}=\tilde{\mathcal{H}}_{71}C_{1}+(\tilde{\mathcal{H}}_{74}+\tilde{\mathcal{H}}_{77})C_{4}+\tilde{\mathcal{H}}_{76}C_{6},$
$\displaystyle i\hbar\frac{\partial C_{8}}{\partial
R}=\tilde{\mathcal{H}}_{81}C_{1}+(\tilde{\mathcal{H}}_{84}+\tilde{\mathcal{H}}_{87})C_{4}+\tilde{\mathcal{H}}_{86}C_{6}=0.$
. (B.43)

From the normalization factor

$C_{1}^{2}+2C_{4}^{2}+C_{6}^{2}=1$ (B.44)

we see that

$C_{1}\frac{\partial C_{1}}{\partial R}+2C_{4}\frac{\partial C_{4}}{\partial
R}+C_{6}\frac{\partial C_{6}}{\partial R}=0.$ (B.45)

To solve this equation, a candidate of regularization as shown in Eq.(A.21) is
also proposed, i.e :

$\displaystyle\mathcal{\tilde{H}}$ $\displaystyle=$
$\displaystyle\tilde{W}_{1}[(\sigma_{1}^{x}\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{x})+(\sigma_{2}^{x}\sigma_{3}^{y}+\sigma_{2}^{y}\sigma_{3}^{x})]$
(B.46) $\displaystyle+$
$\displaystyle\tilde{W}_{2}(\sigma_{3}^{x}\sigma_{1}^{y}+\sigma_{3}^{y}\sigma_{1}^{x})+\frac{1}{2}(\sigma_{1}^{z}+\sigma_{2}^{z}+\sigma_{3}^{z})\tilde{B}_{z},$

where $\tilde{W}_{1}$ is a time dependent $XY$ interaction parameter of
nearest neighbor and $\tilde{W}_{2}$ is for next nearest neighbor The matrix
representation of the candidate of regularization term in Eq.(B.46) is given
by

$\displaystyle\mathcal{\tilde{H}}=\begin{pmatrix}\frac{3\tilde{B}_{z}}{2}&0&0&-2i\tilde{W}_{1}&0&-2i\tilde{W}_{2}&-2i\tilde{W}_{1}&0\\\
0&\frac{\tilde{B}_{z}}{2}&0&0&0&0&0&-2i\tilde{W}_{1}\\\
0&0&\frac{\tilde{B}_{z}}{2}&0&0&0&0&-2i\tilde{W}_{2}\\\
2i\tilde{W}_{1}&0&0&-\frac{\tilde{B}_{z}}{2}&0&0&0&0\\\
0&0&0&0&\frac{B_{z}}{2}&0&0&-2i\tilde{W}_{1}\\\
2i\tilde{W}_{2}&0&0&0&0&-\frac{\tilde{B}_{z}}{2}&0&0\\\
2i\tilde{W}_{1}&0&0&0&0&0&-\frac{\tilde{B}_{z}}{2}&0\\\
0&2i\tilde{W}_{1}&2i\tilde{W}_{2}&0&2i\tilde{W}_{1}&0&0&-\frac{3\tilde{B}_{z}}{2}\end{pmatrix}.$

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