###### Abstract

Artificial intelligence (AI) systems of autonomous systems such as drones,
robots and self-driving cars may consume up to 50% of total power available
onboard, thereby limiting the vehicle’s range of functions and considerably
reducing the distance the vehicle can travel on a single charge. Next-
generation onboard AI systems need an even higher power since they collect and
process even larger amounts of data in real time. This problem cannot be
solved using the traditional computing devices since they become more and more
power-consuming. In this review article, we discuss the perspectives of
development of onboard neuromorphic computers that mimic the operation of a
biological brain using nonlinear-dynamical properties of natural physical
environments surrounding autonomous vehicles. Previous research also
demonstrated that quantum neuromorphic processors (QNPs) can conduct
computations with the efficiency of a standard computer while consuming less
than 1% of the onboard battery power. Since QNPs is a semi-classical
technology, their technical simplicity and low, compared with quantum
computers, cost make them ideally suitable for application in autonomous AI
system. Providing a perspective view on the future progress in unconventional
physical reservoir computing and surveying the outcomes of more than 200
interdisciplinary research works, this article will be of interest to a broad
readership, including both students and experts in the fields of physics,
engineering, quantum technologies and computing.

###### keywords:

artificial intelligence; autonomous vehicles; fluid dynamics; nonlinear
dynamics; neuromorphic computing; quantum reservoir computing.

1 1 0 https://doi.org/ Classical and Quantum Physical Reservoir Computing for
Onboard Artificial Intelligence Systems: A Perspective Classical and Quantum
Physical Reservoir Computing for Onboard Artificial Intelligence Systems: A
Perspective A. H. Abbas , Hend Abdel-Ghani and Ivan S. Maksymov∗
Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

The distance between the modern world and the world depicted in science
fiction has narrowed dramatically in the last two decades. However, despite
the impressive progress in the development of unmanned ground vehicles (UGVs),
unmanned aerial vehicles (UAVs) and underwater remotely operated vehicles
(ROVs) Boylen (2020); Davies et al. (2022); Kunertova (2023), safe and
efficient fully autonomous AI-controlled machines and robots remain elusive
Giannaros et al. (2023); Zhang et al. (2024).

One of the major problems faced by the developers of autonomous platforms is
high power consumption by hardware and computers de Vries (2023); Verdecchia
et al. (2023) that enable AI systems to communicate with the environment in a
manner that mimics human reasoning and perception of the world Takeno (2013).
Indeed, the rapid progress in the development of AI systems is expected to
further increase the global energy consumption since approximately 1.5 million
AI server units are expected to be commissioned per year by 2027. Combined
together, these devices will consume approximately 85.4 terawatt-hours of
electric power annually de Vries (2023); Verdecchia et al. (2023).

Several strategies have been suggested to resolve the problem of high energy
consumption by AI systems. In particular, it has been demonstrated that AI
systems themselves can optimise the energy consumption by forecasting supply
and demand, helping prevent grid failures and increase reliability and
security of energy networks that support the operation of large data centres
Ghimire et al. (2023); Rozite et al. (2023). There is also a consensus in the
research and engineering communities that many AI systems of the future will
operate adhering to the principles of Green AI designed to reduce the climate
impact of AI Verdecchia et al. (2023).

However, even though a singe self-driving car would need to store and process
small datasets compared with large amounts of data processed in data centres,
the hardware of its computer may consume up to 50% of total power available
onboard of the vehicle, thus dramatically reducing the distance the vehicle
can travel on a single charge Faghihian and Sargolzaei (2023); Grant (2024);
Zewe (2023). Subsequently, novel approaches to the development of AI systems
for autonomous vehicles need to be developed, taking into account such
restrictions as the weight and size of onboard hardware and batteries Othman
(2022); Rauf et al. (2024). The latter is especially the case of specialised
flying and underwater drones that must be able to reliably carry sophisticated
and often power consuming equipment for long distances in complex natural
environments and adversarial conditions Boylen (2020).

Figure 1: (a–c) Schematic illustration of the physical processes—turbulence,
water waves and vibrations caused by surface roughness—that onboard AI systems
can employ as a means of energy-efficient computation. The so-envisioned AI
systems can be used in UGVs, UAVs and ROVs as well as human-operated
aeroplanes, boats and cars. (d) Identified as a separate category, quantum-
mechanical physical systems can be used as onboard AI systems. Superior
computational performance and low power consumption of quantum systems
compared with traditional computers render them especially useful for
applications in lightweight and long-range drones.

##  2 What is this review article about?

In this article, we approach the problem of energy-efficient AI systems from a
different point of view. We suggest a novel approach to the development of
energy-efficient AI systems for autonomous vehicles and discuss the recent
relevant academic publications. As schematically illustrated in Figure 1a–c,
we propose to create AI systems that would exploit physical phenomena that
either surround human-made machines, including aeroplanes, boats and cars, or
are induced by their movement. For example, such AI systems would do certain
computations employing turbulence in the atmosphere that forms behind an
aeroplane or UAV Yang et al. (2018); Sun et al. (2023). Similarly, in the case
of a boat or a submersible ROV, an AI system could use water waves and other
fluid-mechanical phenomena such as the formation of bubbles Yusvika et al.
(2020); Ju and Choi (2022) and vortices Arndt et al. (2015); Yu et al. (2023)
in water. We also demonstrate that the physical phenomena illustrated in
Figure 1a–c can enable unconventional Adamatzky (2017, 2019), neuromorphic
Shastri et al. (2020); Marcucci et al. (2020); Marković et al. (2020); Suárez
et al. (2021); Rao et al. (2022); Sarkar et al. (2022); Schuman et al. (2022);
Krauhausen et al. and approximate Mittal (2016); Liu et al. (2020); Henderson
et al. (2022); Ullah and Kumar (2023); Maksymov et al. (2021) computing
systems that are expected to play an increasingly important role in the
development of autonomous vehicles Nomani et al. (2020).

Unconventional computing is an approach to computer engineering that exploits
the physical properties of mechanical Adamatzky (2017), fluid-mechanical
Adamatzky (2019); Maksymov (2023) and living Adamatzky et al. (2023); Sharma
et al. (2024) systems to perform computations. This category also includes
computer architectures that are based on electron devices that are not
employed in mainstream digital and analogue computers and intended to enrich
the traditional Von Neumann computer architecture and the Turing machine
approaches Adamatzky (2017, 2019).

Neuromorphic computers can be regarded as a subclass of unconventional
computers that are built using some of the operating principles of a
biological brain Maass et al. (2002); Jaeger and Haas (2004); Lukoševičius and
Jaeger (2009); Nakajima and Fisher (2021). While a neuromorphic computer may
not be as universal as a traditional digital computer, it can solve certain
practically important problems using readily affordable computational
resources. Such an advantage originates from an inherent scalability,
parallelisation and collocation of data processing and memory attainable in
neuromorphic architectures Schuman et al. (2022). Since a neuromorphic
computer mimics the operation of a biological brain, it operates mostly when
input data are available, optimising energy consumption and decreasing the
cost of computations Maass et al. (2002); Jaeger and Haas (2004).

The advantages of neuromorphic computers make them ideally suitable for
applications in onboard AI systems Miller and Lohn (2023). Indeed,
traditionally designed onboard AI systems have certain constraints that can
introduce significant limitations to the vehicle design, creating a gap
between the AI backed by high-performance computers and AI deployed in an
autonomous vehicle. For example, while high-performance AI systems rely on
energy-consuming multi-processor computations, systems designed for onboard
use can do mostly basic calculations. Yet, even though an increase in the
computational power of onboard systems is achievable by increasing the number
of power supply batteries, this approach is impracticable for lightweight
systems such as small and long-range drones Okulski and Ławryńczuk (2022). On
the contrary, neuromorphic computers can do approximate and energy efficient
computations using limited resources Mittal (2016); Liu et al. (2020);
Henderson et al. (2022); Ullah and Kumar (2023); Maksymov et al. (2021), also
exploiting computational errors as a mechanism of enhancing the efficiency and
additionally saving energy stored in the batteries Schuman et al. (2022).

Whereas the forthcoming discussion mostly focuses on AI systems based on the
paradigm of reservoir computing Lukoševičius and Jaeger (2009); Tanaka et al.
(2019); Nakajima (2020); Nakajima and Fisher (2021); Cucchi et al. (2022), a
computational framework derived from recurrent neural network Maass et al.
(2002); Jaeger and Haas (2004), the ideas and models reviewed by us can be
applied to other kinds of neural network models and AI systems, including
those based on deep neural network architectures Maksymov (2024). In
particular, a large section of this article is dedicated to quantum
neuromorphic computing systems (schematically illustrated in Figure 1d) since
their ability to make accurate forecasts using just a few artificial neurons
Mujal et al. (2021); Govia et al. (2021); Suzuki et al. (2022); Govia et al.
(2022); Dudas et al. (2023); Götting et al. (2023); Llodrà et al. (2023);
Čindrak et al. (2024), alongside a small footprint and low energy consumption,
perfectly aligns them with our vision of the application of approximate
neuromorphic computing in mobile AI systems.

The remainder of this article is organised as follows. We first review the
principles of operation of reservoir computing systems, focusing on the
computational algorithm known as physical reservoir computing. Then, building
our discussion around the concepts illustrated in Figure 1, we provide a
number of specific examples of onboard AI systems that exploit turbulence,
water waves, surface roughness and quantum-mechanical effects.

Figure 2: Schematic representation of (a) a traditional algorithmic RC system
and (b) a computational system with a physical reservoir constructed using the
physical effects that take place in the environment that surrounds moving
vehicles. The mathematical meaning of the vectors and matrices mentioned in
this figure is explained in the main text.

##  3 Reservoir computing

###  3.1 Traditional reservoir computing approach

Neural networks employed in AI systems usually consist of thousands or
millions of interconnected processing units Veelenturf (1995); Haykin (1998);
Galushkin (2007). Each node has connections of varying strengths to other
nodes in the network. The strengths of the connections is updated when the
network learns to analyse large datasets.

This algorithm enables the AI system to mimic the operation of a biological
brain that exploits a large and complex network of neural connections. Since
the brain is also a dynamical system that exhibits complex nonlinear and
sometime chaotic behaviour McKenna et al. (1994); Korn and Faure (2003) (a
dynamical system is a system, of either artificial, physical or biological
origin, that constantly changes its state in time Marinca and Herisanu
(2012)), it has been demonstrated that a neural network can be constructed
using the principles of nonlinear dynamics Maass et al. (2002); Jaeger and
Haas (2004); Krauhausen et al. . Since in the realm of mathematics a dynamical
system also employs differential equations, different kinds of nonlinear
differential equations have been adopted to describe the variation of the
connection strength between units of an artificial neural network Maass et al.
(2002); Jaeger and Haas (2004); Lukoševičius and Jaeger (2009).

This approach to the design of artificial neural network is called reservoir
computing (RC) Tanaka et al. (2019); Nakajima (2020); Nakajima and Fisher
(2021); Cucchi et al. (2022); Maksymov (2023); Yan et al. (2024). In a
traditional RC algorithm (Figure 2a), the differential equation that governs
the nonlinear dynamics of the system of randomly connected nodes can be
written as Jaeger (2005); Lukoševičius and Jaeger (2009); Lukoševičius (2012):

$\displaystyle{\bf x}_{n}=(1-\alpha){\bf x}_{n-1}+\alpha\tanh({\bf W}^{in}{\bf
u}_{n}+{\bf W}{\bf x}_{n-1})\,,$ (1)

where $n$ is the index denoting entries corresponding to equally-spaced
discrete time instances $t_{n}$, ${\bf u}_{n}$ is the vector of $N_{u}$ input
values, ${\bf x}_{n}$ is a vector of $N_{x}$ neural activations, the element-
wise operator $\tanh(\cdot)$ is a sigmoid activation function Haykin (1998),
${\bf W}^{in}$ is the input matrix consisting of $N_{x}\times N_{u}$ randomly
generated elements, ${\bf W}$ is the recurrent weight matrix containing
$N_{x}\times N_{x}$ randomly generated elements and $\alpha\in(0,1]$ is the
leaking rate that controls the speed of the temporal dynamics.

Using Eq. (1), one calculates the output weights ${\bf W}^{out}$ by solving a
system of linear equations ${\bf Y}^{target}={\bf W}^{out}{\bf X}$, where the
state matrix ${\bf X}$ and the target matrix ${\bf Y}^{target}$ are
constructed using, respectively, ${\bf x}_{n}$ and the vector of target
outputs ${\bf y}_{n}^{target}$ as columns for each time instant $t_{n}$.
Usually, the solution is obtained in the form ${\bf W}^{out}={\bf
Y}^{target}{\bf X^{\top}}({\bf X}{\bf X^{\top}}+\beta{\bf I})^{-1}$, where
${\bf I}$ is the identity matrix, $\beta$ is a regularisation coefficient and
${\bf X^{\top}}$ is the transpose of ${\bf X}$ Lukoševičius (2012). Then, one
uses the trained set of weights ${\bf W}^{out}$ to solve Eq. (1) for new input
data ${\bf u}_{n}$ and compute the output vector ${\bf y}_{n}={\bf
W}^{out}[1;{\bf u}_{n};{\bf x}_{n}]$ using a constant bias and the
concatenation $[{\bf u}_{n};{\bf x}_{n}]$ Lukoševičius and Jaeger (2009);
Lukoševičius (2012).

RC systems have been successfully applied to solve many important problems,
including the prediction of highly nonlinear and chaotic time series,
modelling the variation of climate, understanding trends in financial markets
and optimisation of energy generation Lukoševičius and Jaeger (2009);
Lukoševičius (2012); Bala et al. (2018); Tanaka et al. (2019); Nakajima
(2020); Cucchi et al. (2022); Damicelli et al. (2022); Zhang and Vargas
(2023); Maksymov (2023). Importantly, similarly to the efficiency of the brain
of some insects, which has a low number of neurons compared with a human
brain, RC systems require just several thousands of neurons to undertake
certain tasks more efficiently than a high-performance workstation computer
running sophisticated software Maksymov et al. (2021); Maksymov (2023). This
property is ideally suitable for the development of AI systems for mobile
platforms, where the control unit must consume low power while delivering
practicable machine learning, vision and sensing capabilities in a real-time
regime Lee et al. (2023).

###  3.2 Physical reservoir computing

Many physical systems and processes exhibit nonlinear dynamical behaviour and,
therefore, can also be described by differential equations Marinca and
Herisanu (2012). Hence, as illustrated in Figure 2b, it was suggested that a
computationally efficient reservoir can be created using a real-life, either
experimental or theoretical, nonlinear dynamical systems Lukoševičius and
Jaeger (2009). Following this idea, subsequent research works demonstrated
computational reservoirs based on spintronic devices Riou et al. (2019); Watt
and Kostylev (2020); Allwood et al. (2023), electronic circuits Cao et al.
(2022); Liang et al. (2024), photonic and opto-electronic devices Sorokina
(2020); Rafayelyan et al. (2020) as well as mechanical Coulombe et al. (2017)
and liquid-based Kheirabadi et al. (2023); Gao et al. (2022); Maksymov (2023);
Marcucci et al. (2023) physical systems. In the theoretical works, the
demonstration of the functionality of a physical reservoir required the
researchers to replace Eq. (1) by a differential equation that describes the
physical system of interest (e.g., a nonlinear oscillator Coulombe et al.
(2017); Maksymov et al. (2021)). In turn, in the experimental works, the
output of Eq. (1) was effectively replaced by measured traces of temporal
evolution of the physical system (e.g., curves of microwave power absorption
caused by the excitation of spin waves in a ferromagnetic material Watt and
Kostylev (2020) or optically-detected profiles of solitary waves developed on
the liquid surface Maksymov and Pototsky (2023)). Naturally, compared with the
theoretical works on physical RC systems, the experimental demonstration of
physical reservoirs required an extra step that included such processing
procedures as signal amplification, noise filtering and data sampling Maksymov
and Pototsky (2023). However, both theoretical and experimental works
unanimously demonstrated substantial advantages of the physical RC systems
over the algorithmic ones, including low poser consumption and high accuracy
of forecasts while using a reservoir with a relatively small number of
artificial neurons Maksymov (2023).

Figure 3: (a) Photographs of the prototype of ROV designed to test AI systems
that employ water waves and other disturbances caused by the motion of the
drone as a means of computation. (b) Schematic illustration of the difference
between the wave patterns produced by the ROV that has received the left,
forward and right commands from the operator.

##  4 Reservoir computing using the physical properties of fluids

###  4.1 State-of-the-art

One of the first experimental demonstrations of a physical reservoir computer
was a system that exploited the ripples on the surface of water contained in a
tank placed on an electric motor-driven platform Fernando and Sojakka (2003)
(for a review see, e.g., Refs. Tanaka et al. (2019); Adamatzky (2019);
Maksymov (2023)). The ripples caused by judiciously controlled vibrations were
recorded using a digital camera. Then, the images were processed on a computer
and then the processed data were used as the input of the reservoir computing
algorithm.

Although that pioneering work clearly demonstrated the potential of liquids to
perform RC calculations, for about two decades it was considered to be mostly
of fundamental interest to researchers working on physics-inspired AI systems
Nakajima and Aoyagi (2015); Nakajima (2020); Tanaka et al. (2019). A similar
idea was exploited in the recent theoretical Marcucci et al. (2020, 2023) and
experimental works Maksymov and Pototsky (2023), where it was suggested that
the ripples created by motors can be replaced by the dynamical properties of
solitary waves—nonlinear, self-reinforcing and localised wave packets that can
move for long distances without changing their shape Remoissenet (1994).

Importantly, those novel works demonstrated the ability of water-based RC
system to do complex calculations relying only on low computational power
microcontrollers (e.g., Arduino models Maksymov (2024)) that can, in
principle, continuously operate for several months without the need to
recharge the battery. Therefore, such systems are especially attractive for
application in drones, robots and other autonomous platforms.

Figure 4: (a) Illustration of the vortex shedding taking place when a fluid
such as air or water flows past a cylinder. As theoretically shown in Ref.
Goto et al. (2021), modulating the flow velocity and monitoring the vortex
dynamics using a set of virtual sensors one can create an efficient physical
RC system. An experimental implementation of this computational approach was
discussed in Ref. Vincent et al. . (b) Photograph of vortices and other water
flow effects created by the ROV in a lab setting.

###  4.2 Towards Reservoir Computing with Water Waves Created by an ROV

Figure 3a shows the photographs of an experimental ROV designed to test
different kinds of autonomous AI systems. The hull of ROV consists of two
wing-like fins that hold two electric motors that rotate the propellers: one
in the counterclockwise direction and another one in the clockwise direction
Aguirre-Castro et al. (2019). The fins are connected to a cylindrical frame
that is also used as a compartment where sensing equipment, including lasers
and photodetectors, can be placed. As with many commercial ROV designs Bohm
(1997), the movement of the experimental ROV is controlled via an electric
cable.

When both propellers rotate at the same speed, the ROV moves forward (Figure
3b). However, the ROV will turn left (right) when the speed of the left
(right) motor is reduced. Thus, a moving ROV creates a wake wave whose pattern
depends on the relative speed of the two motors. The temporal dynamics of the
wave pattern can be sensed using two laser-photodiode pairs that measures the
light reflected from the waves created by the left and right propellers (a
similar detection mechanism was used in Ref. Maksymov (2024)). The resulting
optical signals can then be converted in electric signals that are transmitted
to the control unit via the cable. The waveform of the so-generated electric
signals will be a function of time and it will correlate with the commands
(left, right, forward and so on) given to the ROV by the operator. Indeed, as
schematically shown in Figure 3b, when the ROV moves forward the waves created
by both propellers are approximately the same. However, the wave patterns
change when the ROV turns left or right.

Similarly to the seminar work Ref. Fernando and Sojakka (2003), the
differences in the wave pattern of the moving ROV can be employed to perform
reservoir computations. Such computations can help predict an optimal
trajectory of the ROV based on the previous inputs of the human operator and
taking into account the environmental factors, including the speed of wind,
temperature and salinity of water as well as the presence of obstacles such as
rocks and debris Yang et al. (2023); Perrusquía and Guo (2024). Since the ROV
creates the waves that are used to predict the trajectory, the on-board
reservoir computer does not consume significant energy apart from the need to
power the sensors, thereby satisfying the requirements for autonomous AI
systems Henderson et al. (2022).

###  4.3 Physical Reservoir Computing using Fluid Flow Disturbances

The research work involving the ROV shown in Figure 3 is still in early
development. However, at the time of writing, to the best of our knowledge it
is the only experimental attempt to build a prototype of a water drone that
employs physical processes in its environment as a means of computation (a UGV
designed using similar principles is discussed in Section 5). Nevertheless,
similar ideas have been explored theoretically and also tested in laboratory
settings Goto et al. (2021); Maksymov et al. (2021); Vincent et al. ; Vincent
et al. ; Maksymov and Pototsky (2023); Maksymov (2024). Although the results
presented in the cited papers do not involve any autonomous vehicle, in the
following we will analyse them in the context of the ROV model discussed
above.

Before discussing those results, we also highlight the proposals of physical
RC systems designed to predict the trajectory of a flying drone Vargas et al.
(2015); Perrusquía and Guo (2024). Although the idea of using physical
processes taking place in the surrounding environment of the drone were not
expressed in Ref. Vargas et al. (2015); Perrusquía and Guo (2024), the
approach presented there can be extended to implement the concepts illustrated
in Figure 1a, b. In this context, we remind that the discipline of fluid
dynamics is concerned with the flow of both liquids and gases Sears (2011).
This means that AI systems designed for flying and underwater drones can, in
principle, exploit the same physical processes. In fact, as mentioned above,
both moving airborne and underwater vehicles create vortices as they move in
the atmosphere Saban et al. (2009); Hrúz et al. (2022); Nathanael et al.
(2022) and water Yu et al. (2023).

Vortices are ubiquitous in nature (e.g., they can be observed in whirlpools,
smoke rings and winds surrounding tropical cyclones and tornados) and they
exhibit interesting nonlinear dynamical that has been the subject of
fundamental and applied research Wu et al. (2006); Billah and Scanlan (1991);
Williamson (1996); Williamson and Govardhan (2004). For instance, it is well-
established that when fluid flows around a cylindrical object, the physical
effects known as von Kármán vortex street can be observed in wide a range of
flow velocities Williamson (1996); Wu et al. (2006). The shedding of such
vortices imparts a periodic force on the object. In many situations, this
force is not significant enough to accelerate the object. However, in some
practical cases the object can vibrate about a fixed position, undergoing
harmonic motion. Yet, when the frequency of the periodic driving force matches
the natural frequency of the oscillation of the object, the resonance
processes come into play and the amplitude of the oscillations can increase
dramatically Williamson and Govardhan (2004); Billah and Scanlan (1991).

Figure 5: Sketch of an ROV, bubbles created by its propellers and highly
nonlinear acoustic waves emitted by them. As discussed in the main text, the
nonlinear dynamics of these physical processes opens up novel opportunities
for physical reservoir computing.

Therefore, it has been theoretically demonstrated that the physical properties
of vortices can be used in reservoir computing Goto et al. (2021); Vincent et
al. . In Ref. Goto et al. (2021), the authors conducted a rigorous numerical
analysis of a vortex-based RC system based on a von Kármán vortex street
(Figure 4a). A periodic pattern of numerical sensors located across the
computational domain was used to monitor the nonlinear dynamics and collect
data for further processing following the traditional RC algorithm. The flow
of fluid was used as the input. For example, to create an input that
corresponds to a signal that varies in time, the velocity of the flow was
modulated such that it follows the time-varying shape of the signal (which can
readily be done using an electric pump Maksymov and Pototsky (2023); Maksymov
(2024)).

The dynamics of the resulting computational reservoir depends on the value of
the Reynolds number ${\mathcal{R}e}$. Subsequently, different operating
regimes were tested using inputs corresponding to the values of
${\mathcal{R}e}$ below and above the threshold of the formation of a von
Kármán vortex street Goto et al. (2021). Those tests confirmed a high memory
capacity of the reservoir and its ability to learn from input data and
generalise them. Further tests also revealed the ability of the reservoir to
make accurate predictions of time series datasets.

Figure 6: False-colour map composed of a large number of theoretical frequency
spectra of a single gas bubble trapped in the bulk of water and excited with a
sinusoidal acoustic pressure wave. The $x$-axis corresponds to the normalised
frequency $f/f_{a}$, where $f_{a}$ is the frequency of the incident acoustic
wave. The $y$-axis denotes the peak pressure of the incident acoustic wave.
Using this figure, we can see the evolution of the frequecny spectrum of the
bubble as a function of the peak pressure of the incident wave. For example,
at 260 kPa we can identify the frequency peaks with the frequencies
$f/f_{a}=1,2,3$ and so on. At 320 kPa, we observe the appearance of the
subharmonic peak at $f/f_{a}=\frac{1}{2}$ and its ultraharmonic components
$f/f_{a}=\frac{3}{2}$, $\frac{5}{2}$ and so on.

We note that in an experimental attempt to test the theoretical vortex-based
RC system Vincent et al. the virtual sensors shown in Figure 4 can be replaced
by real ones. Alternatively, one can use a digital camera to film the vortex
and then process different pixels of the individual frames extracted from the
video file Maksymov and Pototsky (2023); Marcucci et al. (2023); Vincent et
al. ; Maksymov (2024). It is also noteworthy that propellers of the ROV can
also create vortices Asnaghi et al. (2021); Yu et al. (2023) that can be used
for reservoir computing purposes (Figure 4b).

###  4.4 Acoustic-based Reservoir Computing

Another promising approach to physical reservoir computing employs acoustic
waves, vibrations and adjacent physical processes Meffan et al. (2023);
Yaremkevich et al. (2023); Phang (2023). Although high-frequency (MHz-range)
acoustic waves have been used in the cited papers, the ideas presented in
those works can be implemented using the acoustic phenomena observed in a wide
range of frequencies. For example, the temporal dynamics of vortices and other
disturbances created by ROVs (Figure 4b) has a spectral signature in the
frequency range that spans from several tens of Hz to several hundreds of kHz.
Such acoustic signals can be detected using hydrophones, sonar technologies
and other well-established acoustic location techniques Wilson and Liu (2004);
Rubin (1999); Manneville et al. (1999); Digulescu et al. (2016) and then
processed using an RC algorithm Onasami et al. (2022).

As with the sound radiated by a moving ship Lidtke et al. (2015), ROVs and
similar autonomous vehicles can produce a tonal (related to the blade pass
frequency) acoustic disturbance and broadband noise associated with the
presence of unsteadiness in the flow. The tonal disturbances can be further
categorised into the contributions related to such technical parameters of the
propellers as the blade thickness parameter and blade loading Lidtke et al.
(2015). These acoustic processes also exhibit significant nonlinear effects
Made and Kurtz (1970) that can be exploited in a computational reservoir
Tanaka et al. (2019); Nakajima (2020); Maksymov (2023).

Moreover, as the propeller rotates it pushes the ROV through the water,
causing a positive pressure on the face of the blade and a negative pressure
on its back. The negative pressure causes any gas in solution in the water to
evolve into bubbles Plesset (1949); Brennen (1995); Lauterborn and Kurz
(2010); Maksymov et al. (2022a, b). These bubbles collapse via the process
called cavitation, causing hammer-like impact loads on the blades and damaging
their surface Lauterborn and Kurz (2010); Maksymov et al. (2022a, b). The
cavitation also causes significant acoustic noise that originates both from
oscillations and collapse of bubbles Lauterborn and Kurz (2010) and the
formation of vortices Arndt et al. (2015); Yang et al. (2023). This physical
picture is sketched in Figure 5.

In particular, the underwater acoustic noise is associated with the highly-
nonlinear oscillation of the bubble volumes that typically occur in the range
of frequencies from several hundred Hz to approximately 40 kHz Brennen (1995);
Lauterborn and Mettin (1999); Lauterborn and Kurz (2010); Maksymov and
Greentree (2019) (microscopic bubbles oscillate at higher frequencies Chen et
al. (2009); Suslov et al. (2012); Dzaharudin et al. (2013); however, the
physics of their acoustic response remains essentially the same). Indeed,
considering an idealised scenario of a sinusoidal acoustic pressure wave, when
a wave moves through water, its initial waveform changes so that its initial
monochromatic spectrum acquires higher harmonic frequencies Maksymov and
Greentree (2019). The more nonlinear the medium in which the wave propagates,
the stronger the enrichment of the spectrum with the peaks corresponding to
harmonics.

The degree of acoustic nonlinearity can be characterised by the acoustic
parameter $\beta=B/A$, which is the ratio of coefficients $B$ and $A$ of
quadratic and linear terms in the Taylor series expansion of the equation of
state of the medium (see Maksymov and Greentree (2019) and references
therein). The larger the value of $\beta$, the more nonlinear the medium, the
stronger a distortion of the acoustic spectrum from the initial monochromatic
state. For instance, water with $\beta=3.5$ is more acoustically nonlinear
than air with $\beta=0.7$ (we note that the degree of nonlinearity is
considered to be moderate in both media).

However, when air bubbles are present in water, the value of $\beta$ increases
to around 5000 Maksymov and Greentree (2019). This is because liquids are
dense and have little free space between molecules, which explains their low
compressibility. In contrast, gases are easily compressible. When an acoustic
pressure wave propagating in water reaches a bubble, due to the high
compressibility of the gas trapped in it, its volume changes dramatically,
causing substantial local acoustic wavefront deformations that result in
strong variation of the initial acoustic spectrum.

The evolution of the acoustic spectrum of an oscillating bubble trapped in the
bulk of water is illustrated in Figure 6 (for the computational details and
model parameters see, e.g., Suslov et al. (2012); Nguyen et al. (2021);
Maksymov et al. (2022a, b)). The units of the $x$-axis of this figure
correspond to the normalised frequency $f/f_{a}$, where $f_{a}$ is the
frequency of the incident sinusoidal acoustic pressure waves. The $y$-axis
corresponds to the peak pressure of the incident wave (in kPa units) but the
false colour encodes the amplitude (in dB units) of the acoustic pressure
scattered by the bubbles. The bright traces with the amplitude of
approximately 0 dB correspond to the frequency peaks in the spectra of the
bubble forced at one particular value of the peak pressure of the incident
wave. We can see that at a relatively low pressure of the incident wave the
spectrum contains the frequency peaks at the normalised frequencies
$f/f_{a}=1,2,3$ and so forth. However, an increase in the pressure results in
the generation of the subharmonic frequency $f/f_{a}=\frac{1}{2}$ and its
ultraharmonic components $f/f_{a}=\frac{3}{2}$, $\frac{5}{2}$ and so on.
Further increase in the peak pressure of the incident waves leads to a
cascaded generation of subharmonics frequency peaks, resulting in a comb-like
spectrum Maksymov et al. (2022b); Nguyen et al. (2021).

The nonlinear response of a cluster of oscillating bubbles trapped in water
was used to create a computational reservoir in theoretical work Ref. Maksymov
et al. (2021). Each bubble in the cluster oscillates at a certain frequency
when the entire cluster is irradiated with an acoustic wave. The oscillation
frequency of each bubble in the cluster depends on the equilibrium radius of
the bubble and the strength of its interaction with the other bubbles in the
cluster. Importantly, the cluster maintains its structural stability (i.e. the
bubble do not merge) when the pressure of the incident acoustic wave remains
relatively low Nguyen et al. (2021).

Thus, when the temporal profile of the incident pressure wave is modulated to
encode the input data (e.g., a time series that needs to be learnt and then
forecast by the RC system), the cluster of bubbles acts as a network of
artificial neurons. Since each oscillating bubble emits sound waves, these
waves can be detected using either a hydrophone or a laser beam Maksymov et
al. (2022a, b) and then process following to the traditional RC algorithm. As
demonstrated in Ref. Maksymov et al. (2021), this computational procedure
enables the RC system to predict the future evolution of highly nonlinear and
complex time series with the efficiency of the traditional RC algorithm while
using low energy consuming computational resources.

Figure 7: (a) Photograph of the UGV controlled by a physical RC system that
employs the road roughness and vibrations of a whiskered sensor as a means of
computation. (b) Whiskered sensor and the detectors used to monitor the
nonlinear vibrations of the spring caused by the rough terrain. (c) Schematic
diagram of the onboard reservoir computer that exploits the nonlinear dynamics
of the vibrating spring and processes data using a microcontroller.
Identifying the texture of the terrain, the microcontroller computes as
optimal trajectory of the vehicle. (d) Axial displacement and natural angular
frequency of the vibrational modes of the tapered spring obtained using a
theoretical model (top row) and rigorous numerical simulations by means of a
finite-elements method. Adapted from Ref. Yu et al. (2023) under the terms of
a Creative Commons Attribution 4.0 International License.

Subsequently, it is plausible that bubbles created by a moving ROV (Figure 5)
can be used to construct an onboard RC system. The input of such a reservoir
will be the local pressure variations caused by the propellers of the ROV. It
can be shown that these variations correlate with the control signals received
by the ROV from the operator and/or its onboard control unit Patterson et al.
(2020); Su et al. (2024). While the implementation of such a computational
scheme requires resolving several technological problems, it enables the
researchers to access a rich spectrum of fascinating nonlinear effects
associated with oscillating bubbles Prosperetti (1974); Keller and Miksis
(1980); Lauterborn and Kurz (2010); Suslov et al. (2012); Dzaharudin et al.
(2013); Maksymov and Greentree (2019). Interestingly enough, as demonstrated
in the following section, a similar, from the point of view of nonlinear
dynamics, idea was proposed in the domain of autonomous ground vehicles.

##  5 Physical Reservoir Computing for UGVs

The preceding discussion of the fluid flow disturbances and their application
in reservoir computing should also be applicable to cars Paul et al. (2019);
Nakamura et al. (2022), bicycles Miau et al. (2020) and other road vehicles
Garcia-Ribeiro et al. (2023) that create turbulence and give rise to other
physical effects that can be used in reservoir computing. However, the
physical contact of road vehicles with the ground often results in unique
nonlinear dynamical processes that can be employed in a physical reservoir
computer, as schematically illustrated in Figure 1c.

To date, different AI-based approaches to terrain identification have been
developed since this functionality is essential for autonomous vehicles and
robots that operate in extreme and unstructured environments Trautmann and Ray
(2011); Otsu et al. (2016); Christie and Kottege (2016); Valada and Burgard
(2017). Identifying the terrain surface and perceiving its texture, UGVs and
robots can dynamically adjust their initial trajectory, achieving safer and
more efficient navigation with the help of neural network models. Similarly to
flying and underwater drones, UGVs and robots must be able to perceive the
surrounding environment with high accuracy, consuming low power and using a
low computational onboard computer that can process data in a real-time
regime.

A range of physical sensing systems have been developed to fulfil the
aforementioned technical specifications. For example, computer vision
techniques have been used to enable robots to recognise different terrain
textures from a longer distance to modify the route and avoid obstacles
Santana et al. (2011); Nava et al. (2019). LIDAR (Light Detection and Ranging)
and hybrid optical-machine vision systems have also been proposed in Refs.
Konolige et al. (2009); Zhou et al. (2012). Nevertheless, despite the
encouraging results demonstrated in those works, the accuracy of terrain
monitoring enabled by visual methods is adversely affected by variations in
light intensity, limited visibility conditions caused, for example, by smoke,
and weather-related factors such as rain, snow, fog and fallen leaves.

Yet, the accuracy of many optical sensing methods deteriorates due to
vibrations. Of course, one can use different signal processing techniques to
mitigate the adverse effect of vibrations Engelsman and Klein (2023). However,
the implementation of such approaches increases the complexity of software
that controls the vehicle, which, in turn, requires a more powerful onboard
computer that consumes the energy stored in the vehicle’s batteries.

On the other hand, it has been suggested that acoustic effects Christie and
Kottege (2016) and vibrations Brooks and Iagnemma (2005) can be used as a
means of terrain classification. Indeed, it is plausible to assume that
vibrations of certain structural elements of a vehicle will vibrate following
the roughness of the terrain, thereby providing a valuable information from
which the structure of the terrain can be deduced. However, the extraction of
vibration profiles and integration of those data with the existing machine
learning techniques have proven technically difficult and computationally
demanding Giguere and Dudek (2011); Valada and Burgard (2017).

An innovative approach to the solution of this problem has been proposed in
Ref. Yu et al. (2023), where it was demonstrated that the nonlinear dynamical
response of the vehicle’s mechanical features can be used as the means of
computations. That is, instead of trying to either mitigate the effect of
vibration or process vibration-induced signals using an onboard AI system, it
was proposed that vibrations could serve as the input of a reservoir computer
(Figure 7a).

Previous research demonstrated that a structural part of a soft robot can be
used as a computational reservoir Nakajima et al. (2013). Following that idea,
in Ref. Yu et al. (2023) used a whisker sensor that mimics the tactile sensing
capabilities displayed by various living organisms, particularly insects and
mammals Wang et al. (2023). The particular whisker sensor geometry used in
Ref. Yu et al. (2023) is a tapered spring (Figure 7a, b) that vibrates as the
vehicle moves through a rough terrain, producing nonlinear signals that are
detected using three Hall sensors and permanent magnets (Figure 7b, c). The
so-measured signals are processes using a low-power consuming onboard
microcontroller that implements the basic reservoir computing functionality
(Figure 7c). Identifying the texture of the terrain, the microcontroller
steers the vehicle in the desired direction, varying the electric power
delivered to the motors. Essentially, this control scheme is similar to the
ideas expressed in the preceding sections of this articles: onboard sensors of
a moving vehicle detect nonlinear-dynamical processes occurring in the outer
environment and the signals produced by them are employed as a means of
reservoir computation aimed to identify an optimal route as well as to perform
other navigation and energy consumption optimisation functions.

Importantly, it has been demonstrated that the nonlinear dynamics of the
whiskered sensor can be captured using just three sensors places along the
tapered spring (Figure 7b). The optimal location of the sensors was computed
based on the information about the vibrational modes of the spring obtained
using a theoretical model and rigorous numerical simulations (Figure 7d). It
was shown that the sensors located closer to the base of the tapered spring
follows the vibration caused by the rough terrain. However, the second and
third sensors located in the middle of the spring and close to its tip,
respectively, mostly detect the higher-order vibrational modes. Such a sensor
arrangement was found to be optimal for the construction of an efficient
computational reservoir. Interestingly, this observation agrees with the
previous theoretical demonstration of the RC system based on nonlinear
oscillations of bubbles trapped in water Maksymov et al. (2021), where it
sufficed to take into account the fundamental mode and a few higher-order
nonlinear acoustic oscillation modes of the bubble to perform complex
forecasting tasks.

##  6 Adjacent technologies for onboard reservoir computations

Thus, mechanical whiskered sensors and adjacent sensor technologies open up
exciting opportunities for the detection of nonlinear variations in the
environment and their application in reservoir computing. In fact, different
kinds of whiskered sensor—optical, magnetic, resistive, capacitive and
piezoelectric—have also been used in underwater ROVs Wang et al. (2023), which
means that not only UGVs but the other types of autonomous vehicles can employ
the physics of whiskered sensors for computational purposes.

However, every technology has technical and fundamental limitations. For
example, magnetic whiskered sensor can be sensitive to temperature
fluctuations, which may renders them inefficient in certain practical
situations Wang et al. (2023). Moreover, from the fundamental point of view,
the operation of a whiskered sensor requires the application of relatively
strong magnetic fields Trigona et al. (2019). Subsequently, strictly speaking,
such sensors are not suitable for the detection of small magnetic field
variations in the surrounding environment.

Magnetic phenomena have been widely used in the domain of reservoir computing,
where a number of magnonic Furuta et al. (2018); Watt and Kostylev (2020) and
spintronic Taniguchi et al. (2022); Vidamour et al. (2023); Edwards et al.
(2023) RC systems have been demonstrated. Therefore, it is conceivable that a
sensor located onboard of an autonomous vehicle can take a series of
measurements to determine the magnetic field surrounding the vehicle, which
can then be compared to the already known magnetic field maps to provide
valuable information about the vehicle’s location and serve as a means for
reservoir computing.

While certain ultra-sensitive measurement techniques developed in the field of
microwave magnetism Ivanov et al. (1998); Maksymov and Kostylev (2015) could
be used to implementation this idea in practice, mostly quantum sensors that
can measure small magnetic field variations with high accuracy Degen et al.
(2017) have the potential to serve as an onboard quantum reservoir computer
(Figure 1d). Indeed, a number of onboard quantum sensors have been developed
and tested both in lab and real-life environments Jeske et al. (2016);
Templier et al. (2022); Tasker et al. (2024). Therefore, in the following
section, we discuss the emergent quantum reservoir computing architectures
that can be used to conduct complex real-time calculations onboard of
autonomous vehicles.

##  7 Quantum Reservoir Computing

Developing a functional quantum computer presents numerous challenges,
primarily due to the fragile nature of quantum bits (qubits). Qubits are
highly susceptible to decoherence and various forms of quantum noise, which
complicates the engineering of a high-fidelity processor capable of executing
quantum algorithms across an exponentially large computational space Preskill
(2018); Arute et al. (2019). Subsequently, robust mechanisms for precise error
correction are essential for maintaining computational accuracy. Yet, these
mechanisms must be suitable for applications at extremely low temperatures.

To resolve this problem, alternative avenues for harnessing the power of
quantum physics for computational tasks have been explored. Quantum reservoir
computing (QRC) is one of these approaches Fujii and Nakajima (2017). QRC
systems mimic the computational capabilities of the human brain by using
quantum dynamics, which is an innovative approach that holds a tremendous
potential to advance the research on quantum AI and neuromorphic computing
Schuman et al. (2022). One particular advantage of QRC is low training cost
and fast learning capabilities, which makes it especially efficient for
processing complex data Abbas and Maksymov (2024).

Essentially, QRC is a quantum dynamical system whose quantum states are
represented by the density matrices $\rho_{k}\in T(\mathcal{H})$, where
$T(\mathcal{H})$ denotes the space of Hermitian, positive semi-definite,
trace-one operators. Since quantum dynamics is inherently linear, the
achievement of nonlinearity in QRCs requires one to careful choose the quantum
observable that can produce a nonlinear input-output transformation. An
efficient computational reservoir should also fulfil the echo state and fading
memory conditions Jaeger and Haas (2004); Lukoševičius and Jaeger (2009);
Tanaka et al. (2019); Nakajima (2020), which implies that its temporal
dynamics must be dissipative and independent of the initial conditions Park et
al. (2024); Sannia et al. (2024).

In QRC, Fock states of the quantum system serve as the neural activations of
the reservoir Marković and Grollier (2020); Govia et al. (2021); Abbas and
Maksymov (2024). Created by a tensor product of quantum subsystems, Fock
states enable a unique way to model the activity of neurons. In such a quantum
system, external signals prompt the quantum states to evolve dynamically over
time. The evolution of these quantum states is governed by a unitary operator
that is determined by the Hamiltonian of the system, a fundamental concept in
quantum physics that describes the total energy of the system and dictates how
the quantum states change over time Griffiths (2004); Nielsen and Chuang
(2002). As the system evolves, it processes the input signals through its
inherent quantum dynamics.

The output from each neuron (i.e. the Fock state) is then collected and
processed using a linear function. This function aggregates the contributions
from individual neurons, providing a combined output with different weights
that represent the response of the system to the input signals. The weights
associated with each neuron’s output are then trained using linear regression
techniques, a statistical method that adjusts the weights to minimise the
error between the predicted and actual outputs. By training the weights
through linear regression, the QRC network can be fine-tuned to improve its
ability to process and interpret input signals, leading to more accurate and
efficient performance.

A variety of technological platforms were proposed to function as QRCs,
including trapped ions Zhang et al. (2017); Pino et al. (2021), nuclear
magnetic resonance (NMR) in molecules Negoro et al. (2018), quantum circuits
Chen et al. (2020); Dasgupta et al. (2022) and photonic devices Cai et al.
(2017); Nokkala et al. (2018). More recently, novel QRC systems have emerged,
including arrays of Rydberg atoms Bravo et al. (2022) and Josephson mixers
Dudas et al. (2023). In the theoretical domain, models involving the quantum
master equations have been developed to describe quantum spins with controlled
losses, enabling the studies of coherence, scalability and controllability of
the system Dudas et al. (2023); Abbas and Maksymov (2024).

Another major advantage of QRC is a large number of degrees of freedom
available in small quantum systems. To optimise the extraction of information
from these degrees of freedom, techniques like temporal multiplexing, which
improves the performance of QRC models, and spatial multiplexing, which
simultaneously uses multiple reservoir layers, have been proposed Nakajima et
al. (2019).

In the following subsections, we will survey the main categories of QRC
systems, exploring their key characteristics and discussing their
computational capabilities.

###  7.1 Spin-Network Based Reservoir

A spin-network system is a promising platform for implementing QRC algorithms.
Indeed, extensive research has delved into the dynamics of spin chain
networks, showcasing their potential for complex computational tasks Marzuoli
and Rasetti (2005); Tserkovnyak and Loss (2011). QRC systems utilising a
quantum network of randomly coupled spins Refs. Fujii and Nakajima (2017);
Chen and Nurdin (2019); Mujal et al. (2021); Martínez-Peña et al. (2023) have
also been demonstrated using a range of quantum technologies, including solid-
state systems like quantum dots Hanson and Awschalom (2008); Loss and
DiVincenzo (1998), superconducting qubits Wendin (2017); Clarke and Wilhelm
(2008) and trapped ions Blatt and Wineland (2008); Monroe and Kim (2013). Each
of these platforms offers distinct advantages in terms of scalability and
coherence time.

In the context of spin-network-based QRC, the reservoir consists of a network
of quantum spins organised in a particular topology (Figure 8). The
interactions between these spins govern the dynamics of the reservoir. These
quantum spins can represent qubits or higher-dimensional quantum systems,
providing a versatile computational platform for diverse information
processing tasks Fujii and Nakajima (2017); Mujal et al. (2021). Furthermore,
the network topology and the Hamiltonian parameters can be fine-tuned to
optimise the performance of the reservoir for specific applications. For
example, by adjusting the strength and nature of the interactions between the
spins, one can control the response of the reservoir to input signals and its
ability to perform different computational tasks Fujii and Nakajima (2017);
Tanaka et al. (2019); Mujal et al. (2021). Such a tuneability is crucial for
tailoring the reservoir to the requirements of various applications, ranging
from pattern recognition and time-series prediction to more sophisticated
quantum information processing tasks Suzuki et al. (2022).

Figure 8: Sketch of a spin-network-based QRC system. The system operates as
follows. First, the classical input data are injected into the spin network,
altering its quantum state. Then, the quantum dynamics of the network evolves
under the influence of the input data. Finally, the network produces an output
signal derived from certain observables of the system. Redrawn after Refs.
Fujii and Nakajima (2017); Mujal et al. (2021).

The concept of spin-network-based QRC system was introduced in Ref. Fujii and
Nakajima (2017), where the reservoir exploited the dynamical properties of a
nuclear magnetic resonance spin-ensemble architecture, using a multiplexing
technique to enhance its computational capabilities Nakajima et al. (2019).
The so-created QRC system utilised the intrinsic properties of a network of
qubits—the fundamental two-level quantum systems that reside in a two-
dimensional complex Hilbert space $\mathcal{H}^{2}$ Nielsen and Chuang (2002).
The state of a qubit can be described as a linear combination of linearly
independent bounded operators acting on $\mathcal{H}^{2}$. These operators are
formed from the tensor products of basic Pauli operators
$\\{I,\sigma_{z},\sigma_{x},\sigma_{y}\\}$ for each qubit. The state of a
single qubit can be visually represented on the Bloch sphere, a three-
dimensional space that represents the qubit’s state vector Nielsen and Chuang
(2002).

The dynamics of the reservoir layer is driven by the transverse-field Ising
model Singh (2020) and it forms a high-dimensional space. The Hamiltonian of
the transverse-field Ising model typically includes the terms that represent
the interaction between neighbouring qubits and an external transverse
magnetic field, and it is given by

$H=J\sum_{i\neq j}\sigma_{i}^{x}\sigma_{j}^{x}+h\sum_{i}\sigma_{i}^{z}\,,$ (2)

where $N$ is the number of qubits, $h$ is an external magnetic field and
$J_{ij}$ denotes the coupling strength between the qubits $i$ and $j$. The
evolution of such closed quantum systems is described by the time-dependent
Schrödinger equation

$\rho(t)=e^{-iHt}\rho(0)e^{iHt}\,,$ (3)

where $\rho$ is the reservoir state, $H$ is the Hamiltonian of the system
given by Eq. 2 and $\hbar$ is the reduced Planck constant.

The reservoir employs an amplitude encoding scheme to drive its quantum
states. In this approach, the state of each qubit is initialised by a set of
input signals $s_{k}$. The state of the qubit is reset at each time step to

$|\psi_{s_{k}}\rangle=\sqrt{1-s_{k}}|0\rangle+\sqrt{s_{k}}|1\rangle\,,$ (4)

represented by the density matrix
$\rho_{1}=|\psi_{s_{k}}\rangle\langle\psi_{s_{k}}|$. The updated reservoir
state $\rho^{\prime}_{k}$ at each time step is given by

$\rho^{\prime}_{k}=\rho_{1}\otimes\text{tr}_{1}(\rho_{k-1})\,,$ (5)

where $\text{tr}_{1}$ denotes the partial trace over the state of the first
qubit.

There exist various strategies that can be used to implement the input
protocol, including strong local dissipation followed by a quantum rotation
gate Sannia et al. (2022) or projective measurements over the first qubit
Lüders (1950); Von Neumann (2013). Subsequently, the system evolves under its
natural dynamics for a time interval $\Delta t$ to process the information:

$\rho_{k}=e^{-iH\Delta t}\rho^{\prime}_{k}e^{iH\Delta t}\,.$ (6)

The output of the reservoir is obtained by measuring specific observables of
the spin network. These observables are related to certain properties of the
system, such as the magnetisation of individual spins and the correlations
between spins at different locations within the network. The data obtained
from these measurements serve as the output of the reservoir. This output can
then be further processed using classical computational techniques such as
linear regression approaches Jaeger and Haas (2004); Lukoševičius and Jaeger
(2009).

It is well-known that quantum systems exhibit physically rich dynamical
regimes, involving the effects of localisation or thermalisation. An attempt
to address the impact of these dynamical phases on a quantum computational
reservoir was made in Refs. Xia et al. (2022); Martínez-Peña et al. (2023). In
those works, the Hamiltonian Eq. (3) was extended with the aim to investigate
the dynamical regimes Martínez-Peña et al. (2023), where the condition for the
homogeneous external field would be relaxed and a local magnetic field applied
to each spin $\sigma_{i}^{z}$. It is also established that variations of the
dynamical regimes of the reservoir influence its computational robustness
Jaeger and Haas (2004); Lukoševičius and Jaeger (2009). For instance, systems
presenting localisation can provide quantum memories at finite temperature
Ponte et al. (2015), also improving the trainability of parameterised quantum
Ising chains Altshuler et al. (2010). Yet, the thermal phase appears to be
naturally adapted to the requirements of QRC since the performance of the
reservoir increases at the thermalisation transition Martínez-Peña et al.
(2023).

###  7.2 Quantum Oscillator for Reservoir Computing

Errors caused by decoherence and noise remain a significant challenge in the
field of quantum information processing Horodecki et al. (2022). One potential
solution of this challenge consists in adapting noise-resilient classical
computing modalities to the quantum realm.

For example, the study conducted in Ref. Govia et al. (2021) proposed a
continuously-variable QRC system based on a single nonlinear oscillator,
demonstrating that such a quantum-mechanical system serves as a computational
reservoir that outperforms its classical counterparts in both performance and
reliability. In particular, the continuous-variable approach was implemented
using a single nonlinear oscillator with the Kerr nonlinearity Govia et al.
(2021) (the state of this system can be described by continuous variables
corresponding to its position $X$ and momentum $P$ quadrature). The advantage
of using a continuous-variable system lies in its ability to provide a richer
set of computational nodes due to the infinite-dimensional Hilbert space
associated with these continuous variables, which, in turn, reduces costly
repetitions required for accurate measurement of expectation values in
discrete-variable quantum machine learning approaches Qiu et al. (2019);
Peral-García et al. (2024).

The dynamics of a classical reservoir can be described by the differential
equation

$\dot{a}=-iK(a-2a^{*})-\frac{\kappa}{2}a-i\alpha u(t)\,.$ (7)

In the case of a quantum reservoir, the Hamiltonian of the Kerr-nonlinear
oscillator is

$\hat{H}(t)=K\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}+\alpha
u(t)(\hat{a}+\hat{a}^{\dagger})\,$ (8)

and its evolution is described by the Lindblad master equation Campaioli et
al. (2023)

$\dot{\rho}=-i[\hat{H}(t),\rho]+\kappa D[\hat{a}]\rho\,.$ (9)

It has been established that a single-oscillator QRC system outperforms its
classical counterpart in the task of sine wave phase estimation, primarily due
to the nonlinearity of quantum measurements that is known to provide an
intrinsic advantage by effectively performing nonlinear transformations on
output data Govia et al. (2021) (for a relevant discussion of nonlinear
transformation in the context of the traditional reservoir computing see Refs.
Bollt (2021); Gauthier et al. (2021); Maksymov (2024)). The QRC system also
demonstrated smaller average root mean square (RMS) errors and greater
reliability, with a lower sensitivity to parameter variations compared with a
classical reservoir tasked with similar benchmarking problems. The performance
of QRC enhanced with the increase in Hilbert space dimension, suggesting an
increase in the number of degrees of freedom and boosting of its computational
power Govia et al. (2021).

The impact of the size of the Hilbert space dimension on the computational
power of QRC was investigated in Ref. Kalfus et al. (2022). It was established
that the use of higher-dimensional quantum oscillators helps improve the
performance of the reservoir in both signal processing tasks and memory
capacity benchmarking problems. Furthermore, quantum reservoirs demonstrated
superior memory retention capabilities, additionally improving their
performance at stronger nonlinearity and higher dimensionality Kalfus et al.
(2022).

Another implementation of QRC using coherently coupled quantum oscillators was
suggested in Ref. Dudas et al. (2023). The approach proposed in that work
aimed to address certain limitations of the existing quantum neural networks
that rely on qubits, including the problem of limited connectivity between the
artificial neurons. The authors of Ref. Dudas et al. (2023) demonstrated that
one can use coupled quantum oscillators to achieve a large number of densely
connected neurons, thereby obtaining advantage over the traditional qubit-
based approaches.

Figure 9: (a) Sketch of a QRC using coherently coupled oscillators. The blue
and yellow circles represent arbitrarily connected neurons of the reservoir.
The black connections arrows denote the fixed weights but the red connections
correspond to the trained weights. (b) Illustration of a system of coupled
quantum oscillators driven at the frequencies $\omega_{a,b}$ and with the
amplitudes $\epsilon_{a,b}$. Adapted from Ref. Dudas et al. (2023) under the
terms of a Creative Commons Attribution 4.0 International License.

The system investigated in Ref. Dudas et al. (2023) can be experimentally
realised using superconducting circuits with resonators linked via parametric
elements such as Josephson mixers or SNAILs (Superconducting Nonlinear
Asymmetric Inductive eLements) employed as the source of tuneable nonlinearity
Frattini et al. (2018). Utilising Fock states as the functional neurons, the
theoretical analysis carried out in Ref. Dudas et al. (2023) revealed that a
quantum reservoir with up to 81 neurons based on two coupled oscillators
(Figure 9) can achieve a 99% accuracy rate on challenging benchmarking tasks.
To put this result into perspective, a similar accuracy was achieved using at
least 24 classical oscillators.

The authors of Ref. Dudas et al. (2023) also investigated the necessary
coupling and dissipation conditions for optimal performance of the quantum
reservoir proposed by them. They established that high dissipation rates lead
to greater errors and reduced memory span for the neural network, which
emphasised the need to employ high-quality-factor oscillators for
computational problems that require extensive memory. Conversely, stronger
coupling between the oscillators resulted in an increase in the reservoir’s
performance by enabling more significant data transformations between
different basis states, which is crucial for effective learning. The operation
in a strong coupling regime also facilitates a larger population of basis
state neurons, thereby boosting computational capabilities. Overall, it was
also demonstrated that the optimal performance of a quantum reservoir relies
on finding a balance between the dissipation rate and coupling strength
parameters, which maximises memory retention and data transformation
efficiency.

However, despite the substantial progress made in the field of quantum
computing, experimental realisation of quantum neural networks capable of
handling real-world classification tasks remains elusive. In particular, such
tasks require networks with millions of interconnected neurons to efficiently
process complex data. However, as already mentioned above, traditional qubit-
based approaches face connectivity limitations, making it challenging to
achieve the necessary network density. To resolve this problem, it has been
suggested that using 10 coupled quantum oscillators could enable the creation
of an analogue quantum neural network architectures with billions of neurons
Dudas et al. (2023). In the following subsection, we discuss alternative
strategies aimed to additionally decrease the demand for the high number of
artificial neurons in the reservoir.

###  7.3 Quantum Reservoir with Controlled-Measurement Dynamics

While QRC is a promising platform for quantum information studies, as of time
of writing the existing QRC models encounter execution time challenges due to
the need for a repeated system preparation and measurement at each time step.
To address this problem, improved QRC systems have been proposed relying on
repeated quantum non-demolition measurements employed to generate time-series
data and reduce the execution time Yasuda et al. (2023). The system described
in Ref. Yasuda et al. (2023) was experimentally implemented using IBM’s
quantum superconducting devices, and it demonstrated higher accuracy and
smaller execution time compared with conventional QRC methods.

The concept of Temporal Information Processing Capacity was introduced into
research practice to evaluate the computational abilities of QRC systems,
highlighting the fact that an optimal measurement strength needs to be found
to balance the retention and dissipation of information Yasuda et al. (2023).
In the cited paper, the proposed QRC method was applied to a soft robot over
1000 time steps, showcasing its practical utility. Initial experiments on a
120-qubit device were discussed, indicating the method’s potential for
scalability.

Moreover, addressing the challenge of integrating quantum measurement while
retaining processing memory and handling large Hilbert spaces, the authors of
Ref. Mujal et al. (2023) proposed additional measurement protocols aimed to
achieve optimal performance for both memory and forecasting tasks. The three
proposed measurement protocols for QRC are:

  * •

Restarting Protocol (RSP): Repeats the entire experiment for each measurement,
maintaining unperturbed dynamics but requiring significant resources.

  * •

Rewinding Protocol (RWP): Restarts dynamics from a recent past state (washout
time $\tau_{wo}$), optimising resources compared to RSP.

  * •

Online Protocol (OLP): Uses weak measurements to continuously monitor the
system, preserving memory with less back-action but introducing more noise.

The efficiency of these measurement protocols was evaluated by tasking a QRC
system to solve a number of standard benchmarking problems. It was established
that RSP and RWP require strong measurements to minimise statistical errors,
whereas OLP with weak measurements effectively preserves the state of the
reservoir while providing sufficient information for time-series processing
Mujal et al. (2023).

Nevertheless, the ambiguity of quantum mechanics and quantum measurements
continues to inspire research on the topic of quantum reservoir computing. A
novel quantum RC architecture that exploits the dynamics of an atom trapped in
a cavity was recently proposed in Ref. Abbas and Maksymov (2024). One
prominent feature of the proposed system is a coherent driving of the atom at
a certain driving rate with the possibility to observe transitions between
quantum states and effective “freezing” of the quantum evolution of the
system. Frequent observations of the atom eigenstates prevent the system from
undergoing significant changes, a phenomenon known as the Zeno effect
Harrington et al. (2017). Conversely, less frequent probing of the system
states enables the system to undergo Rabi oscillations Raimond et al. (2012);
Lewalle et al. (2023). Using these properties, the rate at which the atom is
driven becomes optimised, making the quantum dynamics of the reservoir
suitable for undertaking diverse classification and prediction tasks. This
approach also enables controlling and stabilising quantum states during a
computation, which is beneficial for such practical applications as mitigation
of decoherence Kondo et al. (2016), quantum information processing Monras and
Romero-Isart (2009), correction of quantum error and stabilisation of quantum
states Paz-Silva et al. (2012); Lewalle et al. (2023); Burgarth et al. (2014).

Figure 10: Sketch of the quantum reservoir computing system employing
measurement-controlled dynamics. The input, output and target data are defined
similarly to the notation used in Figure 2. The quantum reservoir is
constructed using the Fock states $|n,\sigma\rangle$ representing the quantum
states within the cavity. Input data points are encoded as signals modulating
the driving amplitude $\beta(t)$. The output of the reservoir is determined by
the expectation values $P(n,\sigma)$ of the occupancy of the basis states.

The theoretical QRC system proposed in Ref. Abbas and Maksymov (2024) exploits
the dynamics of a probed atom in a cavity (Figure 10). Coherently driving the
atom and inducing measurement-controlled quantum evolution, the authors of
Ref. Abbas and Maksymov (2024) demonstrated that the QRC system can exhibit
fast and reliable forecasting capabilities using fewer artificial neurons
compared with the traditional RC algorithms. The computational tests of the
system also showcased its potential to be employed in error-tolerant
applications, particularly in conditions of limited computational and energy
resources.

The model developed in Ref. Abbas and Maksymov (2024) is constructed around
the interaction of a two-level atom with a quantised electromagnetic field in
a cavity, enabling real-time observations and control of the quantum state of
the atom. The interaction between the atom and the cavity is governed by the
Hamiltonian

$\displaystyle\hat{H}_{i}=ga^{\dagger}a\sigma_{-}\sigma_{+}\,,$ (10)

where $g$ represents the strength of the atom-cavity coupling, $a$ is the
cavity annihilation operator and $\sigma^{-}$ and $\sigma^{+}$ are the
lowering and raising operators for the atom, respectively. The cavity is
coherently driven by an external signal, which is modelled by the Hamiltonian

$\displaystyle\hat{H}_{c}=-i\beta(a^{\dagger}-a)\,,$ (11)

but state of the atom undergoes continuous monitoring by means of coherent
measurement processes, which accounted for by the Hamiltonian

$\displaystyle\hat{H}_{z}=g_{z}(\sigma_{+}+\sigma_{-})\,,$ (12)

where $g_{z}$ denotes the amplitude of the coherent atomic driving that
controls the frequency at which the state of the atom is measured. The
dynamics of the entire system are described by the stochastic master equation

$\displaystyle\dot{\rho}=-i[\hat{H},\rho]+\hat{C}\rho{\hat{C}}^{{\dagger}}-\frac{1}{2}{\hat{C}}^{{\dagger}}\hat{C}\rho-\frac{1}{2}\rho{\hat{C}}^{{\dagger}}\hat{C}\,,$
(13)

where $\rho$ is the density matrix and $\mathcal{C}=\sqrt{\kappa}a$ is the
collapse operator associated with cavity decay.

Incorporating these components and equations, the model of the proposed QRC
system revealed the system’s capability to efficiently process input data,
adapt its dynamics through coherent driving and measurement and make accurate
predictions while using low computational and energy resources compared with
the traditional RC algorithms.

##  8 Conclusions and Outlook

Thus, in this article we have reviewed the recent advances in the fields of
classical and quantum reservoir computing systems. Although the potential
practical applications considered by us in this work are mostly limited to
onboard AI systems of autonomous vehicles, the neuromorphic computational
reservoirs discussed throughout the main text are expected to find numerous
applications in the broad area of science and technology.

Indeed, at present the world of AI has been enabled by well-established
technologies that are about to reach their fundamental operation limits and
already consume the power comparable with the annual electricity generation by
a medium-size country. Subsequently, since the computational power required
for sustaining the demand for novel AI units is doubling approximately every
three months, humankind will soon face the problem of finding a balance
between the progression of AI with the imperatives of sustainable development.

Of course, the problem is not exclusively limited to managing the rise of AI
and transitioning to green energy. In fact, since several billions of self-
driving car, drones and autonomous robotic systems are estimated to enter
service by 2050, each of those vehicles will require a source of energy
exclusively dedicated to powering their onboard AI. Subsequently, since no
current technology is able to satisfy such a high demand for lightweight and
high-capacity energy sources, humans will need to invent alternative, low
power consuming computational technologies.

In his famous talk “Plenty of Room at the Bottom” addressed to the American
Physical Society in Pasadena on December 1959 Feynman (1959), Richard P.
Feynman suggested exploring the immense possibilities enabled by
miniaturisation of devices, thus heralding the coming era of nanotechnologies
and quantum computing. A large section of this present review article has been
dedicated to computational approaches that directly benefit from the
suggestions made by Feynman in that talk. Yet, standing on the shoulders of
giants and rephrasing Feynman, in this article we invite researchers to enter
a new field of AI, pointing out that that there is plenty room and rich
physics around autonomous vehicles to employ them as a means of energy-
efficient computing.

The idea to write this review article belongs to all authors. I.S.M. wrote
Sections 1-3, 5 and 6. Section 4 was written by H.A.G and I.S.M. Section 7 was
written by A.H.A. All authors discussed the manuscript and approved the
submitted version of the paper.

This research received no external funding

Not applicable.

Not applicable.

This article has no additional data.

The authors declare no conflicts of interest.

Abbreviations The following abbreviations are used in this manuscript:

AI | artificial intelligence
---|---
LIDAR | Light Detection and Ranging
NMR | nuclear magnetic resonance
OLP | online protocol
QNP | quantum neuromorphic processor
QRC | quantum reservoir computing
RC | reservoir computing
ROV | remotely operated vehicle
RSP | restarting protocol
RWP | rewinding protocol
SNAILs | Superconducting Nonlinear Asymmetric Inductive eLements
UAV | unmanned aerial vehicle
UGV | unmanned ground vehicles

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