<s>
Reservoir	B-Algorithm
computing	I-Algorithm
is	O
a	O
framework	O
for	O
computation	O
derived	O
from	O
recurrent	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
theory	O
that	O
maps	O
input	O
signals	O
into	O
higher	O
dimensional	O
computational	O
spaces	O
through	O
the	O
dynamics	O
of	O
a	O
fixed	O
,	O
non-linear	O
system	O
called	O
a	O
reservoir	O
.	O
</s>
<s>
The	O
concept	O
of	O
reservoir	B-Algorithm
computing	I-Algorithm
stems	O
from	O
the	O
use	O
of	O
recursive	O
connections	O
within	O
neural	B-Architecture
networks	I-Architecture
to	O
create	O
a	O
complex	O
dynamical	O
system	O
.	O
</s>
<s>
It	O
is	O
a	O
generalisation	O
of	O
earlier	O
neural	B-Architecture
network	I-Architecture
architectures	O
such	O
as	O
recurrent	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
,	O
liquid-state	B-Algorithm
machines	I-Algorithm
and	O
echo-state	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Reservoir	B-Algorithm
computing	I-Algorithm
also	O
extends	O
to	O
physical	O
systems	O
that	O
are	O
not	O
networks	O
in	O
the	O
classical	O
sense	O
,	O
but	O
rather	O
continuous	O
systems	O
in	O
space	O
and/or	O
time	O
:	O
e.g.	O
</s>
<s>
The	O
resultant	O
complexity	O
of	O
such	O
recurrent	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
was	O
found	O
to	O
be	O
useful	O
in	O
solving	O
a	O
variety	O
of	O
problems	O
including	O
language	O
processing	O
and	O
dynamic	O
system	O
modeling	O
.	O
</s>
<s>
However	O
,	O
training	O
of	O
recurrent	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
is	O
challenging	O
and	O
computationally	O
expensive	O
.	O
</s>
<s>
Reservoir	B-Algorithm
computing	I-Algorithm
reduces	O
those	O
training-related	O
challenges	O
by	O
fixing	O
the	O
dynamics	O
of	O
the	O
reservoir	O
and	O
only	O
training	O
the	O
linear	O
output	O
layer	O
.	O
</s>
<s>
Recent	O
advances	O
in	O
both	O
AI	O
and	O
quantum	O
information	O
theory	O
have	O
given	O
rise	O
to	O
the	O
concept	O
of	O
quantum	B-Device
neural	I-Device
networks	I-Device
.	O
</s>
<s>
In	O
2018	O
,	O
a	O
physical	O
realization	O
of	O
a	O
quantum	O
reservoir	B-Algorithm
computing	I-Algorithm
architecture	O
was	O
demonstrated	O
in	O
the	O
form	O
of	O
nuclear	O
spins	O
within	O
a	O
molecular	O
solid	O
.	O
</s>
<s>
However	O
,	O
the	O
nuclear	O
spin	O
experiments	O
in	O
did	O
not	O
demonstrate	O
quantum	O
reservoir	B-Algorithm
computing	I-Algorithm
per	O
se	O
as	O
they	O
did	O
not	O
involve	O
processing	O
of	O
sequential	O
data	O
.	O
</s>
<s>
Rather	O
the	O
data	O
were	O
vector	O
inputs	O
,	O
which	O
makes	O
this	O
more	O
accurately	O
a	O
demonstration	O
of	O
quantum	O
implementation	O
of	O
a	O
random	O
kitchen	O
sink	O
algorithm	O
(	O
also	O
going	O
by	O
the	O
name	O
of	O
extreme	B-Algorithm
learning	I-Algorithm
machines	I-Algorithm
in	O
some	O
communities	O
)	O
.	O
</s>
<s>
In	O
2020	O
,	O
realization	O
of	O
reservoir	B-Algorithm
computing	I-Algorithm
on	O
gate-based	O
quantum	B-Architecture
computers	I-Architecture
was	O
proposed	O
and	O
demonstrated	O
on	O
cloud-based	O
IBM	O
superconducting	O
near-term	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
The	O
'	O
reservoir	O
 '	O
in	O
reservoir	B-Algorithm
computing	I-Algorithm
is	O
the	O
internal	O
structure	O
of	O
the	O
computer	O
,	O
and	O
must	O
have	O
two	O
properties	O
:	O
it	O
must	O
be	O
made	O
up	O
of	O
individual	O
,	O
non-linear	O
units	O
,	O
and	O
it	O
must	O
be	O
capable	O
of	O
storing	O
information	O
.	O
</s>
<s>
Virtual	O
reservoirs	O
are	O
typically	O
randomly	O
generated	O
and	O
are	O
designed	O
like	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
Virtual	O
reservoirs	O
can	O
be	O
designed	O
to	O
have	O
non-linearity	O
and	O
recurrent	O
loops	O
,	O
but	O
,	O
unlike	O
neural	B-Architecture
networks	I-Architecture
,	O
the	O
connections	O
between	O
units	O
are	O
randomized	O
and	O
remain	O
unchanged	O
throughout	O
computation	O
.	O
</s>
<s>
The	O
readout	O
is	O
a	O
neural	B-Architecture
network	I-Architecture
layer	O
that	O
performs	O
a	O
linear	O
transformation	O
on	O
the	O
output	O
of	O
the	O
reservoir	O
.	O
</s>
<s>
The	O
weights	O
of	O
the	O
readout	O
layer	O
are	O
trained	O
by	O
analyzing	O
the	O
spatiotemporal	O
patterns	O
of	O
the	O
reservoir	O
after	O
excitation	O
by	O
known	O
inputs	O
,	O
and	O
by	O
utilizing	O
a	O
training	O
method	O
such	O
as	O
a	O
linear	B-General_Concept
regression	I-General_Concept
or	O
a	O
Ridge	O
regression	O
.	O
</s>
<s>
An	O
early	O
example	O
of	O
reservoir	B-Algorithm
computing	I-Algorithm
was	O
the	O
context	O
reverberation	O
network	O
.	O
</s>
<s>
In	O
this	O
architecture	O
,	O
an	O
input	O
layer	O
feeds	O
into	O
a	O
high	O
dimensional	O
dynamical	O
system	O
which	O
is	O
read	O
out	O
by	O
a	O
trainable	O
single-layer	O
perceptron	B-Algorithm
.	O
</s>
<s>
Two	O
kinds	O
of	O
dynamical	O
system	O
were	O
described	O
:	O
a	O
recurrent	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
with	O
fixed	O
random	O
weights	O
,	O
and	O
a	O
continuous	O
reaction	O
–	O
diffusion	O
system	O
inspired	O
by	O
Alan	O
Turing	O
’s	O
model	O
of	O
morphogenesis	O
.	O
</s>
<s>
At	O
the	O
trainable	O
layer	O
,	O
the	O
perceptron	B-Algorithm
associates	O
current	O
inputs	O
with	O
the	O
signals	O
that	O
reverberate	O
in	O
the	O
dynamical	O
system	O
;	O
the	O
latter	O
were	O
said	O
to	O
provide	O
a	O
dynamic	O
"	O
context	O
"	O
for	O
the	O
inputs	O
.	O
</s>
<s>
The	O
Tree	O
Echo	B-Algorithm
State	I-Algorithm
Network	I-Algorithm
(	O
TreeESN	O
)	O
model	O
represents	O
a	O
generalization	O
of	O
the	O
reservoir	B-Algorithm
computing	I-Algorithm
framework	O
to	O
tree	O
structured	O
data	O
.	O
</s>
<s>
reservoir	O
)	O
of	O
a	O
Chaotic	O
Liquid	B-Algorithm
State	I-Algorithm
Machine	I-Algorithm
(	O
CLSM	O
)	O
,	O
or	O
chaotic	O
reservoir	O
,	O
is	O
made	O
from	O
chaotic	O
spiking	O
neurons	O
but	O
which	O
stabilize	O
their	O
activity	O
by	O
settling	O
to	O
a	O
single	O
hypothesis	O
that	O
describes	O
the	O
trained	O
inputs	O
of	O
the	O
machine	O
.	O
</s>
<s>
The	O
extension	O
of	O
the	O
reservoir	B-Algorithm
computing	I-Algorithm
framework	O
towards	O
Deep	B-Algorithm
Learning	I-Algorithm
,	O
with	O
the	O
introduction	O
of	O
Deep	O
Reservoir	B-Algorithm
Computing	I-Algorithm
and	O
of	O
the	O
Deep	O
Echo	B-Algorithm
State	I-Algorithm
Network	I-Algorithm
(	O
DeepESN	O
)	O
model	O
allows	O
to	O
develop	O
efficiently	O
trained	O
models	O
for	O
hierarchical	O
processing	O
of	O
temporal	O
data	O
,	O
at	O
the	O
same	O
time	O
enabling	O
the	O
investigation	O
on	O
the	O
inherent	O
role	O
of	O
layered	O
composition	O
in	O
recurrent	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Quantum	O
reservoir	B-Algorithm
computing	I-Algorithm
may	O
use	O
the	O
nonlinear	O
nature	O
of	O
quantum	O
mechanical	O
interactions	O
or	O
processes	O
to	O
form	O
the	O
characteristic	O
nonlinear	O
reservoirs	O
but	O
may	O
also	O
be	O
done	O
with	O
linear	O
reservoirs	O
when	O
the	O
injection	O
of	O
the	O
input	O
to	O
the	O
reservoir	O
creates	O
the	O
nonlinearity	O
.	O
</s>
<s>
Although	O
they	O
can	O
nowadays	O
be	O
created	O
and	O
manipulated	O
in	O
,	O
e.g	O
,	O
state-of-the-art	O
optical	O
platforms	O
,	O
naturally	O
robust	O
to	O
decoherence	O
,	O
it	O
is	O
well-known	O
that	O
they	O
are	O
not	O
sufficient	O
for	O
,	O
e.g.	O
,	O
universal	O
quantum	B-Architecture
computing	I-Architecture
because	O
transformations	O
that	O
preserve	O
the	O
Gaussian	O
nature	O
of	O
a	O
state	O
are	O
linear	O
.	O
</s>
<s>
Normally	O
,	O
linear	O
dynamics	O
would	O
not	O
be	O
sufficient	O
for	O
nontrivial	O
reservoir	B-Algorithm
computing	I-Algorithm
either	O
.	O
</s>
<s>
It	O
is	O
nevertheless	O
possible	O
to	O
harness	O
such	O
dynamics	O
for	O
reservoir	B-Algorithm
computing	I-Algorithm
purposes	O
by	O
considering	O
a	O
network	O
of	O
interacting	O
quantum	O
harmonic	O
oscillators	O
and	O
injecting	O
the	O
input	O
by	O
periodical	O
state	O
resets	O
of	O
a	O
subset	O
of	O
the	O
oscillators	O
.	O
</s>
<s>
With	O
a	O
suitable	O
choice	O
of	O
how	O
the	O
states	O
of	O
this	O
subset	O
of	O
oscillators	O
depends	O
on	O
the	O
input	O
,	O
the	O
observables	O
of	O
the	O
rest	O
of	O
the	O
oscillators	O
can	O
become	O
nonlinear	O
functions	O
of	O
the	O
input	O
suitable	O
for	O
reservoir	B-Algorithm
computing	I-Algorithm
;	O
indeed	O
,	O
thanks	O
to	O
the	O
properties	O
of	O
these	O
functions	O
,	O
even	O
universal	O
reservoir	B-Algorithm
computing	I-Algorithm
becomes	O
possible	O
by	O
combining	O
the	O
observables	O
with	O
a	O
polynomial	O
readout	O
function	O
.	O
</s>
<s>
The	O
reservoir	O
is	O
then	O
excited	O
by	O
radiofrequency	O
electromagnetic	O
radiation	O
tuned	O
to	O
the	O
resonance	B-Application
frequencies	O
of	O
relevant	O
nuclear	O
spins	O
.	O
</s>
<s>
The	O
most	O
prevalent	O
model	O
of	O
quantum	B-Architecture
computing	I-Architecture
is	O
the	O
gate-based	O
model	O
where	O
quantum	B-Architecture
computation	I-Architecture
is	O
performed	O
by	O
sequential	O
applications	O
of	O
unitary	O
quantum	O
gates	O
on	O
qubits	O
of	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
A	O
theory	O
for	O
the	O
implementation	O
of	O
reservoir	B-Algorithm
computing	I-Algorithm
on	O
a	O
gate-based	O
quantum	B-Architecture
computer	I-Architecture
with	O
proof-of-principle	O
demonstrations	O
on	O
a	O
number	O
of	O
IBM	O
superconducting	O
noisy	O
intermediate-scale	O
quantum	O
(	O
NISQ	O
)	O
computers	O
has	O
been	O
reported	O
in	O
.	O
</s>
