This thesis explores the use of machine learning, particularly supervised learning, combined with quantum control techniques to characterize and classify environmental noise in multi-level quantum systems. The research aims to develop robust strategies for understanding noise correlations and statistical properties, which is crucial for advancing quantum technologies.
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>> And u so we give you 25 minutes to make
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>> Okay. Um hi everyone. So I am Soshi
Mukharji. Um and first of all I thank
the committee and all for being present
here and uh I thank to my supervisor
professor Luigi Dan Nali. Um I thank you
for everything especially for being so
patient with me. And um I also thank to
my uh co-supervisors u professor Mao
Patnostro from University of Palmo and
Dr. Fabio Cherina from Leonardo SPA for
their cooperations
and u a special thanks uh goes to our
group leader professor Fali who has
always been very supportive and
encouraging. And at the end I um I am
also grateful to my family and the and
my colleagues here who made the state
who made my stay in Italy memorable.
So um okay today I'm going to talk about
my thesis titled as quantum sensing of
noise based on machine learning. So um
we know that uh exploiting coherence
properties of quantum systems such as
superposition, entanglement is essential
for achieving quantum advantage.
But the presence of the environmental
noise can disrupt these properties as it
causes decoherence and thereby reduces
the fidelity of uh quantum operations.
Um we do have some noise mitigation
strategies for example dynamical
decoupling or passive error avoidance
etc. But all these methods rely highly
on the knowledge of the noise itself or
on the characteristics of the noise. So
if we want reliable quantum technology
then noise characterization is
essential. Um even though single cubit
noise characterization have has been
well established um um which led to
quantum operations uh beyond uh error
correction threshold which have been
shown in many different works but still
it's a challenging task in multicubit
system because here the special and
temporal correlated noise can really uh
have some uh detrimental effects. In
fact, specially correlated noise um can
affect badly uh quantum error
correction, which is one of the pillars
of quantum computing.
Um and in the last decade, machine
learning has emerged as a great
diagnostic tool. And in this thesis
work, our goal is to enable a robust
noise classification or characterization
strategy based on on their temporal,
spatial and energy correlations in
multi-level quantum systems by combining
quantum control and machine learning
especially supervised learning.
Um so I I will mainly talk about uh
three main main works that uh we have
done during my PhD period. The first one
is classification of noise correlations
in a three-level system. The second is um
um
sorry the second is two cubit quantum
sensor for detection of noise
correlations. The third one is detection
of noise gausianity in quantum systems
and at the end I will show you the
conclusions and uh discuss some future perspectives.
perspectives.
So let's move to the first one uh which
is classification of noise correlations
in a three-level system with machine
learning. Here we considered a
three-level system with basis 0 1 and
two. The system is driven by two time
dependent pulses omega p the pump pulse
that couples the states 0 and one uh and
the stock's pulse omega s which couples
the states uh one and two. So we have a
lambda system.
It is also subjected to diagonal noise.
Um and the parameters here are delta P
and delta which are the single and two
photon D tunings respectively. And X1
and X2 are two stochastic processes that
represents the noise.
So basically the role of the noise here
is to induce fluctuations in the energy
levels and based on the characteristics
of these two stoastic processes we will
define different noise classes that will
eventually be distinguished by the
machine learning tool.
So um to control the dynamics of the
system here we exploit um stimulated
ramen diabetic passage tab or in quantum
dot system it is called sitta protocol
which is an diabetic protocol uh which
allows us to transfer the population
from state zero to state two without
populating state one. This protocol is
well known for its robustness and high
fidelity. And to achieve this efficient
population transfer, it is required the
two photon resonance condition which
implies that the parameter delta has to
be nearly equal to zero. And then in
this case, one of the instantaneous
again states states of the system is a
dark state which does not have a
component along state one.
This protocol is also based on the
application of counterintuitive pulse
sequence. Which means that first we
apply omega s pulse that couples the
states one and two and then while
gradually turning it off we turn on
omega pulse which couples the states uh
zero and one. And in this situation if
the system evolves a diabetically
starting from state zero then it will
always follow the dark state and that's
why the population will be transformed
from 0 to two without populating state one.
one.
Here I want to point out two things.
First um uh here we define the
efficiency of the protocol by the
parameter s which is the population of
state two at the end of the time
evolution of the system. This parameter
is important because later we will see
that we will use this to train our
machine learning model. And the second
point is um our main goal here is not to
achieve an efficient population transfer
but we are just exploiting this protocol
in order to get information about the
noise that affects the system. In fact
we will see that we will get more
information from the imperfections.
Um so our main goal here is to
distinguish among five different types
of noise uh with the help of machine
learning. Among these five types there
are three non-marovian quasistatic noise
and two marovian noise. Here quasyatic
implies that both the random variables
x1 and x2 remain constant during a
single trajectory but change their
values in the next trajectory.
And uh for the non-marovian uh noise
case uh based on their energy
correlations we have three subclasses.
First the correlated noise where x2 is
equal to ea x1. The correlation
parameter I here always is positive. The
second class is anti-correlated noise
where it always assumes negative values.
And the third class is uncorrelated
noise where x1 and x2 are completely
independent of each other.
And for the marovian noise uh it is
delta correlated in time and as per the
energy correlations we have correlated
and anti-correlated noise.
Now uh for the non-marovian
uh noise um
um okay now for the non-marovian noise
we can for a single noise realization we
can solve the dynamics of the system
using time dependent stringer equation
and we can calculate the steerup
efficiency by taking average over all
possible realizations of the noise and
for the marovian case um as the system
is ruled by zero mean and delta
correlated noise. In this case, we are
able to derive a lean blood master
equation and the efficiency can be
calculated using the formula below
and uh um so we propose the following uh
protocol. As we will use supervised
learning which requires label data set
and to generate this um we apply syup
protocol under three different driving
conditions. when omega pax is equal to
omega s max. When omega pax is greater
than omega s max and when omega pax is
less than omega s max where this omega
pax and s max are the amplitudes of the
pump and stokes pulses and the
efficiencies calculated under these
three driving conditions goes as input
to the neural network. And of course the
labels are the five different noise
classes that I already mentioned. And in
fact in the output of the neural network
these are the probabilities that the
noise belongs to any of the five classes
that we have.
Now um this is um this is uh the main
result here. Each diagonal entry
represents the percentage of accurate
classification where the of diagonal
entries represents the um the the uh
percentage of error. Um and in the
y-axis here actually we plotted the true
levels of the noise in and in the x-axis
we plotted the predicted levels and from
here actually we can get three messages.
First that the neural network is able to
distinguish very well the non-marrobian
and marovian noise and the second
message is it is also able to
distinguish um very well almost with
100% accuracy the subclasses of the
non-marovian noise the energy
correlations of the non-marovian noise
and the third message is it fails to
distinguish the energy correlations uh
of the marovian noise and in fact we can
get an overall classification accuracy
of 80%.
And um if we and I mean uh if we simply
merge the two marovian uh subclasses and
make it a single one then from the
figure right here we can see that again
we uh achieve an classification accuracy
around 99%. And in the figure below here
uh here we plotted the training history
of the neural network considering both
four and five noise classes uh with
respect to the number of epochs. In the
figure left here we plotted the accuracy
and in the figure right here we plotted
the cost function.
Now until now we considered the ideal
efficiency measurement. Then uh we were
also interested to see how the
classification accuracy changes with
respect to the um number of finite
number of projective measurements and we
saw that around 2,000 measurements it
already reaches the the accuracy already
reaches the ideal case. So even if we
have a small statistical error in our
measurements still uh this methodology works.
works.
Now let's move to the second work um
which is 2 cubit quantum sensor for
detection of noise correlations by
machine learning. Here we u proposed a
design of 2 cubit sensor uh to detect
spatial and temporal correlation of
noise by expanding our ideas which is
which is used in the previous work.
And here we have two cubits with
energies epsylon 1 and epsylon 2
respectively. And they are coupled with
each other by xxising interaction. And
they are also driven by u two local
fields w1 and w2 and are also subjected
to local defasing noise.
So here then the question arises is it
possible to apply stup like protocol
also in this case. Um so let's first
look at the structure of the system. So
for simplicity we considered um that the
two cubits are identical so they have
the same energies and we can diagonalize
the system Hamiltonian into two
invariant subspaces. One is with even
parity, another one is with odd parity
and the corresponding values and states
are given in the table here. The even
parity states are um one 0 and one and
the odd parity states are three and two.
And uh now if we drive the two cubits
symmetrically so um we have so we have
w1 is equal to w2. Then the state three
is uncoupled with the rest and we are
left with three levels 0 2 and one which
is exactly what we need for the
application of syrup. Furthermore,
if we apply a two-tone pulse um where
the frequency small omega s is in
resonance with the transition 2 to1 and
small omega p is in resonance with the
transition 0 to2 and we perform rotating
web approximations um by neglecting the
fast oscillating terms and the terms
oscillating with uh g are neglected in
strong ultra strong coupling regime.
Then we have the ladder configuration
and we have exactly the steerup
Hamiltonian. Okay. But this is um just
uh this is the ideal case just for the
illustrative purpose. But anyway for our
simulations and analysis we use the full
system without any approximation and in
fact we get more information from the
non idealities.
um and uh in fact the presence of noise
disturbs everything. So it induces
fluctuations in the energy levels 0 and
one and also causes coupling between
state two two and three and couplings
between the states zero and one. Uh in
the figure right here we plotted the
populations for the four levels and uh
we see that even if there is a small
amount of noise present the state three
gets populated and this is actually
useful for uh detecting the correlations
uh between the noise and again like the
previous work uh we use the similar
noise uh classes but uh here the the
noise are temporal and specially uh
correlated and also we here we included
the uncorrelated marovian noise. So in
total we have six noise classes and this
is the result. So again we have the
three messages. The neural network is
able to distinguish very well the
non-marovian and marovian noise. The
second message is it is also able to
distinguish the special correlations
within the non-marovian class. And the
third is it it is able to distinguish al
also also the special correlations of
the uh marovian class of course with
certain amount of errors and overall we
can get an classification accuracy of um 86%.
86%.
Okay. So in summary we saw that it is
possible to combine quantum control and
machine learning to detect noise
correlations in multi-level quantum
systems. We exploit uh the sensitivity
of startup or situp protocol for
detecting noise correlations in
three-level system. It is it is possible
to distinguish very well the
non-marovian and marovian uh noise and
it is also possible to uh distinguish
the energy correlations in the
non-marovian class almost with 100%
accuracy but it here it fails to
distinguish the energy correlations in
the marovian uh class. Instead for the
two two cubits uh sensor um the neural
network is able to distinguish also the
space space correlations in marovian
noise uh class with around 80% accuracy.
Now let's uh move to the third um and
final uh work which is m machine
learning approach for detection of noise
gausianity in quantum systems. Um so
until now we did uh noise classification
based on their correlations um
correlations and so the next question
that comes that what is the statistical
signature of the noise sources is it
gausian like it is often assumed in open
quantum system problems or it is non-gausian.
non-gausian.
So a stoastic process is said to be
gausian if it can be fully characterized
by its mean and two time autocorrelation
function. H in this case all the higher
order cumulants vanish. And one of the
examples of this kind of noise is
electromagnetic noise. Instead for the
nonausian noise this is not the case.
The higher order cumulants are non zero
and they can also have relevant effects
in the deep coherence. And uh some
examples of this kind of noise are
impurities and 1 / f noise in
superconducting and semi semiconducting
devices. So note that um noise with the
same power spectra can be gausian or
non- gausian. For example, random
telegraph noise and on stanul and back
noise both of them have um have same two
time autocorrelation function thus the
same spectra but RTN is non-gausian
while O is gausian so they have
different effects on the system for the
RTN the important parameter is the
switching rate while for the stanul and
vectors the important parameter is the
um inverse of correlation time. Here we
plotted uh one realization of both type
of noise and in this work our goal is to
use a single cubit as a probe together
with machine learning to detect
gausianity and characterize these uh noises.
noises.
So here we considered um cubit which is
coupled with n number of bstable
fluctuators each mimicking random
telegraph noise. uh the cubit
Hamiltonian is given by H cis where
epsylon is its energy and delta is the
coupling and the h noise is given by the
random telegraph noise multiplied by
their coupling sum over all random
telegraph noise is coupling by
multiplied by their coupling strength
and here we assume that um the random
variable xt switches between two values
plus and minus one and the switching
rates between these two values are gamma
/ two.
Um now for simplicity we considered
purity phasing case. So we set delta is
equal to zero. And for um n um identical
fluctuators. So if uh the couplings are
the couplings of the fluctuators are the
same and their uh switching rates are
also the same. Then the an analytical
expression for the coherence of the
cubit evolving under n identical rtn
sources is given by this expression here.
here.
And now something interesting happens.
If we increase the number of fluctuators
n and simultaneously decrease the
parameter g which is the ratio between
the coupling and the switching rate such
that ng² is equal to constant then the
cumul cumulative effect of all the rtn
sources becomes equivalent to on stanul
and back noise which is a causian
process and the expression the the
expression for the coherence is reduced
to a form like this. And this is the
exact uh similar form if we consider
that the cubit evolves um under an
ownback noise with mean zero and
exponential correlation time.
Um so here we analyzed two scenarios.
First that the environment is composed
of finite number of identical earth
fluctuators. Here n equal to once means
that the that the noise is strongly non non-gausian
non-gausian
and as we increase the number n it
becomes more and more gausian. So here n
acts as a me measure of gausianity. So
the machine learning task is to infer
two parameters the number n and the
switching rate gamma by observing the
temporal evolution of the system. And in
the second scenario uh we considered a
hybrid uh environment where the cubit uh
where one fluctuator is strongly coupled
with the cubit while the rest is uh
weakly coupled and the cumulative effect
of the weakly coupled ones uh become
equivalent to an onstand backto noise
and in this case actually the
environment has both gaussian and
non-gausian components. So in this case
the machine learning task is to infer
four parameters. The coupling strength
of the strong fluctuator VRTN its
switching rate gamma and the coupling of
the equivalent uh equivalent coupling of
the weak fluctuators V and its inverse
correlation time 1 / C.
And these are the uh results actually.
Here we measured the average of sigma x
at discrete times during the evolution
of the system. And the neural network
here predicts the two parameters n and
gamma. And in the pictures uh right here
we in the y-axis we plotted the
parameters that we want to estimate n
and gamma uh and in the x-axis we
plotted the number of test sample. The
black dots here represent the target
values while the orange dot is the are
the predicted values. And we can see
that the neural network is able to
predict uh the the value of n until 9
around 9. But after that it fails to
estimate the value of n because in this
case the evolution of the cubit becomes
very similar to the gausian uh evolution
and the neural network is not able to
distingu distinguish it anymore and
instead for the gamma uh the switching
rate it uh estimate uh it the neural
network is able to estimate it quite
well. For the second scenario instead we
try to estimate the four parameters as I
mentioned and we see that the most
difficult parameter to estimate is uh
the gamma the switching rate instead for
the other uh parameters we see that it
is able to estimate good good enough. So
um in summary here we saw that we
developed a datadriven framework for
characterizing gausian and non-gausian
noise using a single cubit as a probe.
We saw that uh using supervised learning
it is possible to infer key noise
parameters and we saw that the potential
of um machine learning as a powerful
tool for noise spectroscopy and for uh
outlook we are trying to apply um
unsupervised learning for the first two
um works and for the last work we are
also doing the similar analysis on 1 / f
+ one shops
and these are the publications that I
had during my PhD. The last two works
are not included in my thesis. Uh thank
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