we have been looking at various different
aspects of implementation of quantum computing
as the final series of lectures of this course
and we have looked at many different implementations
one of the things which we have not looked
at in detail which i thought i will cover
in this lecture is the optical aspect of the
implementation ah we have covered the n m
r aspects we have covered the iron trap aspects
we have covered the commercial aspects but
the aspect involving more of the optical scheme
is the one which we will be doing here so
in some of it we have already seen earlier
and here we are going to focus on the optical
approaches to quantum computing
in some sense photons could be ideal for making
qubits so there are two essential points to
remember one is the spin and the other one
is the photon spin is very important because
those are the ones which can be changed by
different principles which does not effect
many other interactions as well as photons
are also very important because they can also
be put together in such a way so that they
do not interact with each other also they
can be transmitted so here we say in particular
the fact that photons dont interact with one
another under normal circumstances gives an
advantage a superposition state of say a photon
spin could be immune to decoherence by stray
electromagnetic fields that suggest a need
for fireless error correction than for a quantum
computer based on matter qubits but the photons
lack of mutual interaction is also a big problem
for creating multiple qubit logic gates this
like the classical equivalents are nonlinear
devices requiring qubit careers to interact
with one another such interactions required
nonlinear interactions that are hard with
photon so these are the pluses and minuses
with photons been used for quantum computer
they have been successful approaches and here
is one of the ones which we discussed in the
class which is the linear optical approach
where in linear devices in optical framework
can be used in effect to carry out nonlinear
operations and this can be possible because
photons are bosons when two photons enter
fifty fifty reflecting beans splitter from
opposite sides at the same time they will
always leave the device along the same path
and this sticking together constitutes kind
of interaction so this was one of the advantages
which was utilized and this work was first
shown in the year two thousand by quite anothers
in terms of showing that it can be applicable
to grover search algorithm in another linear
approach discussed the laser cavity was used
as an implementation of the grovers search
algorithm where the read out was accomplished
by measuring a mode with a photo director
which destructively determined whether one
or more photons represent in a given mode
and so this was shown in two thousand two
by using classical fourier optics so there
are several ways exist to get around the lack
of interaction photons for example using and
one of the ways is over and above these particular
approaches is to use matter as an intermediary
between two sets of photons which work so
in a few of the slides which follow we would
be looking at such other approaches of optical
quantum computing the power of the quantum
in this sense can be looked at as three levels
of computational ability starting from its
weakest to strongest one is the simple classical
bit based computing which is the todays digital
machines which also uses optical technology
quite a lot in terms of storage transfer and
in some cases even computing the other one
is the classical light wave computing which
uses limited aspects of quantum computing
for example its wave nature in terms of super
position and that was often used in terms
of using say the laser cavity for demonstrating
grovers algorithm or linear optical devices
to implement superposition principles in terms
of classical light wave computing the most
intricate one involves the quantum computing
that take advantage of the entanglement of
quantum states as well as their wave nature
to speed up the processing exponentially for
certain problems even otherwise if it is not
exponentially speeding it up the resources
used for whenever the entangle process is
available in terms of quantum states the resource
handling becomes much simpler and much less
requirement is there on the resources same
problem done in a classical light wave computing
versus in the quantum computing sense would
require much less resources in the quantum
computing sense
so computations based on quantum interference
is the part which we discussed at the beginning
where the information processing was done
without entanglement and the information storage
and retrieval was possible through quantum
face but it did not involve any entanglement
so in this particular case all though matter
was used to have an interaction with the light
the interaction only resulted in creating
supervision of states rather than entanglement
and it only ended up producing similar aspects
as the wave property have had so in this case
all though quantum phase was involved it only
had states which could be superimposed so
this particular work did not use entanglement
and was possible to show grovers algorithm
but was resource heavy in particular they
use the cesium atoms rydberg state to write
the data register and then read it out within
the decoherence time scale by using a shape
laser pulse to amplify and detect the electric
field induced ionization
so there advantage was that they were able
to distinctively show at single quantum systems
possessing no entanglement can implement search
algorithms in the same spirit as the optical
waves without any entanglement could show
grovers algorithm also this work was later
on extended by strout theoretically to prove
that this comes at a price of reducing the
effective element size however if such n register
qubits which are known as qubit size as of
now can entangle effectively within themselves
to provide a high level of potential parallelism
then the difficulty of this problem can again
be reduced how its a much more difficult problem
experimentally to be addressed and that is
yet to be done in general the data registers
in these cases were loaded through optical
techniques of exciting the system to a rydberg
state through a two photon excitation process
by using a shape pulse two essentially write
down the information in terms of phase encoding
of the excise state and then using a decoding
pulse which could be an optical pulsar terahertz
half cycle pulses to essentially ionize the
correct states
to get the information out so it was essentially
reading the rydberg wave packet by the data
register was stored instead of using a matter
to interact with shape pulses walmsley show
that the principle of pulse shaping can itself
be utilized as in the previous cases of no
entanglement with light waves to also do the
optical grover search principle application
and in their approach they were able to put
in the data register using a pulse shaper
an acousto optic modulator where in the input
data were able to be read once again by a
part of the original beam of light to find
the correct answer and it was possible as
the diffraction ratings spread the pulse in
two component spectrum bands of which corresponded
to the fifty database elements the modulator
shifts the phase of one band that is the target
database element ordinary wave interference
cancels the and shifted brands and the spectrometer
reads of the remaining light as the target
element
so its all classical and yet it was possible
to do this entire processing by using the
way property because in the fourier domain
this was done so its application of the optical
fourier transform principles to encode and
then decode the data however this is all classical
as i just mention and so in terms of using
the light wave its just acting as a wave which
is using the superposition of the states of
the information content in the system and
much more interesting problem to be looking
at in terms of these interactions is the problem
of decoherence which as we discussed in the
earlier case of interactions for iron traps
and n m r and other principles in optical
case also the laser molecular interaction
has the difficulty of decoherence which could
be off different kinds one which is intermolecular
which is due to diffusibility and mobility
and their times is depend on environmental
conditions if it is a liquid it has one time
scale if it is a gas it has another times
scale and if it is a solid so depending on
the states of matter this can have different
time scales proximity and the motion of the
particles everything put together the other
intrinsic decoherence comes from the intermolecular
aspect which is intrinsic to molecular states
these time skills typically vary from nanoseconds
or below depending on whether electronic vibration
or rotational states are involved in the process
of the laws of the coherence it can be looked
at in this particular format where the states
of the molecule are getting coupled to the
other states other nearby states which could
be affected due to the environment as well
to take away the coherence of this system
from the quantum system
so this happens to be a major aspect of problem
for chemistry also because ever since the
early days of quantum mechanics has been an
implicit dream of controlling atomic or molecular
dynamics which lead to chemical changes it
was pursued with even greater figure with
the discovery of the laser however such quantum
mechanical control has remained as elusive
and evading and is often considered as a dream
the major reason of bits elusive nature is
the energy and coherence randomization due
to the typically strong coupling amongst the
molecular degrees of freedom such as intermolecular
vibration or relaxation or i v r the equivalence
of the two pictures which are provided here
essentially mean that the oscillator strength
from a single excitation is distributed amongst
many eigenstates either way hows ever this
picture is being looked at the effect of decoherence
is the same that the energy localization cannot
be achieved
so this has been seen in many different cases
in particular for example the work which led
to the first observation of how molecular
dynamics work in reality in a chemical reaction
which led to the nobel prize of zewall also
has features of how the intermolecular vibration
or relaxation process keeps on having this
feature and this is similar to the t two oscillating
feature that we discussed in the earlier case
where the system essentially the quantum state
got entangle with the environment and here
also it undergoes oscillations because there
are few states into which the energy can flow
in and then they can have a few iterations
of coming back and forth and as a result you
see the oscillatory behaviour if the coupling
is strong so that is essentially goes away
into these other states and doesnt have a
chance of coming back then there is a persistent
decay as can be seen in these cases the lifetime
of the excited state which is otherwise labeled
as the t one state almost always have a exponential
decay because that is related to the time
scale on which the state can exist so based
on this work because possible later to extract
the hamiltonian associated with such kind
of transitions at least for molecules where
the states were recognized it was possible
to do this so in this particular case of anthracene
study which was done initially by zewall and
peter felker was later on possible to have
the hamiltonian deciphered for this system
and so this could be this became a good model
system for understanding and using the decoherence
principle better so it was therefore possible
in later years to show that the principle
of decoherence which exist for these kinds
of states can then be also controlled by using
the pulses which can be shaped in certain
way or the other we had eluded to this in
an earlier lecture where we mention that the
different ways of controlling the coherence
one of the approaches is to actually change
or modulate the pulses and here is an example
this is typically also done in the case of
n m r where the decoherence or states properties
are also modulated by use of pulse shaping
in terms of radio frequency pulses which are
used for modulating the spins in n m r
in case of optics it gets much more complicated
because the time skills involved are much
faster it has the benefit of the faster time
scales which means that the gates can be operated
and worked at a much faster rate however it
has to be carefully applied because for example
here there are two different shapes of pulse
which is being discussed here and this is
based on the principle of adiabatic interactions
of the system with the applied field and if
the field applied has its frequency changing
linearly and if the change in linear frequency
sweep is slow enough and there will be a period
over which as is indicated by the horizontal
arrow here and there will be a condition where
the two states will remain in coherence and
the one and two states and it will not be
possible for the other states three four five
to be able to extract energy from these two
states
so under that condition when the two states
are not going to talk to the other states
then it is possible to have the energy localized
only between the ground excised state or other
words they can be incoherence and the decoherence
arising from the decay into the states three
four and five does not occur however when
the frequency sweep goes more further away
from resonance then the system again follows
the applied field and it goes into the excised
state so this is the principle of adiabatic
passage with where in this can be seen at
certain frequencies are at certain sweeps
of the frequency a better approach would be
to essentially bring the field in such a way
so that it is the frequency is swept to resonance
and is kept there and that is typically known
as adiabatic half passage and this is a typical
principle which is applied in n m r to bring
systems in to resonance and keep it there
and in doing so adiabatically it is possible
to ensure that the system essentially comes
to coherence and stays there unfortunately
in optics such pulse shapes are very difficult
to do because such a pulse shape would require
change of frequencies occurring over very
rapid
time scales and so these are quite challenging
unless certain property of the system itself
enables search frequency components to be
generated so these are the reasons why this
technology is often difficult in optics however
if possible to be achieved it enables complete
coherence between the ground in excise states
in such a way so that the the other decoherent
states have no role to play in such interactions
so that is one of the important aspects of
decoherance which can be looked at and the
theory on this was possible to be developed
only based on the fact that the hamiltonian
existed from earlier experimental results
going forward on these the idea of pulse shaping
can still be further put to use by having
the pulses go through furthermore shaped conditions
either ramping up and then going down and
in this particular case this could be two
different pulses which are coming with linear
sweat frequencies and getting overlapped as
such that the interaction in essence brings
down the system back to its ground state
so its not that the system does not interact
with the applied field it does however the
interaction balances out in terms of bring
the system back down to its ground state in
terms of cycling it through and this particular
process of keeping the system although it
is being interacting with the applied field
is self induced transparency at resonance
and that is because at that point the light
will be able to pass through the system without
having a net interaction happening to the
system because it goes cycles through back
to the ground state so this particular idea
of producing an interaction where no population
transfer may occur has sometimes been termed
in literature as dark pulses because the pulse
interacts with the system and yet there is
no net change in the system as a result of
this interaction as if the pulse was non existence
so in some sense its a dark pulse on the other
hand when a system is interacting with the
pulse such that because of adiabatic sweeping
of the applied field the system follows the
applied field to go from ground to the excise
state by the end of the pulse we end up generating
robust chirped pulse inversion at resonance
and this is known as chirped pulse inversion
process due to adiabatic inversion and this
is another way of achieving hundred percent
inversion so complete population transfer
or inversion pulse are therefore possible
in case of pulses which have this character
where the frequency sweep is able to make
sure that the ground and the excise state
population are neatly transferred from the
ground to the excise state
so by using these kinds of pulses either which
is something which is a dark pulse which does
not create any change in population in spite
of its interaction with the system or an inversion
pulse which changes the population from the
ground to excited state on interaction with
the laser it is possible to come up with an
ensemble control not gate in a semi classical
sense as the inverting and the dark pulse
interaction creates a system to undergo changes
in this particular manner a quantum mechanical
ensemble be that can either be in the ground
state state zero or excited state one interacts
with the control pulse a which provides robust
chirp pulse inversion condition one and self
induced transparency set your dark pulse condition
zero and based on these conditions it can
undergo this so such interactions and such
gates have been shown to operate modeled based
on these principles so the overall problem
that we have been discussing is the idea of
molecular interactions due to intermolecular
redistributions and whenever molecules are
going to be used for quantum computing these
are important steps to be understood because
no matter how hard we try to excite a single
bond and even if a single bond vibrates within
a few femto second however the entire molecule
starts vibrating and that is because of this
principle that we just showed that the energy
is not possible to be localized and they leak
out and start vibrating as soon as time elapses
so based on this what we had discussed about
was the use of shape pulses and the principle
of shape pulses is something where we showed
already that it has been utilized in fourier
domain by either using modulators or mask
or programmable components to be able to create
the output wave front which is going to based
on the pulse shaping principles that unnecessary
so this is the overall idea of pulse shaping
and typically indirect approaches using fourier
transforms are necessary because the interaction
time scales for shorter and shorter pulses
become very difficult to be of much use when
the pulses become very short and therefore
its much easier to have this process happen
when it is converted into the fourier domain
and the interaction works without much of
a problem so this methodologies is essentially
is linear filtering scheme either in the time
domain or in the frequency domain in time
domain its convolution whereas in frequency
domain its a multiplication and vice versa
depending on how it is being utilized
so this has been a quite a useful technique
which has been applied for many cases in terms
of optics approaches and some of the technologies
that we discussed in terms of how modulations
or gates another thing can be built are based
on this methodology ideas in terms of coherence
of the two levels linearly polarized light
which is available from such laser pulses
can make the two states go through the interaction
were they are getting couple in the dipole
limit which can then be written in terms of
either the phase been expanded in terms of
the taylor series is or its derivative which
is the frequency and the entire process of
their interaction when it is being looked
at theoretically is by done through simulation
of the louville equation which is a time dependent
form of evolution for the ensemble rather
than a single wave function and we have done
a lot of density matrices and its interactions
throughout this course and we have also mentioned
the time dependent evolutions of these density
matrices which are nothing but the louville
equation and the result of these essentially
lead to the numeric integration of how the
equation or how the system evolves with time
as the interaction goes on
the time dependent phase change is the frequency
change which occurs within the pulse and this
appears as additional resonance offset for
the frequency modulated hamiltonian which
is what is been used when these are been continuously
interacting so these are the different parameters
which are used and the rabi is the exact resonance
energy gap the difference in energy from the
applied field to the energy gap of the two
states is the omega r is the resonance energy
gap and the difference of this energy gap
to the applied energy field 
is the one which gives rise to the detuning
delta as it been shown here now if it is not
a one single photon interaction but if it
is a multiple photon interaction then it can
be in terms of n photon interaction and therefore
the overall detuning can be written for the
most general case as omega r minus n omega
which could then be related to the energy
of the applied field and the dipole moment
and all these properties which can be effective
values in terms of multi photon conditions
and they can be made to work in similar ways
as the single photon problem has been developed
over the years there are some cases where
these simple logics will not work but in most
cases when a single photon property does not
exist and only multi photon properties are
going to happen then this simplification is
possible and that is often true when l intense
laser pulse interacts with a simple system
so here are the basic notations of the states
or the system that we have used in this frame
and rotational frequency is the frequency
over which the system is rotating its similar
to the principal of how we are standing on
the earth and still not facing the fact that
the earth is rotating all the time so thats
the modulation frame that we are in where
the effect of the overall rotation is not
going to be of much effect on the system which
is being looked at
so these are the different ways of setting
up the problem so that this can be modeled
to get to the results that we have been discussing
in terms of the applied electric field in
with the system interacting it has to be noted
though similar principles work also for magnetic
fields and so similar ideas of this properties
can be extended to magnetic interactions also
so here is the cartoon of the block sphere
that i was trying to draw in the other particular
case and we have also mentioned before in
one of the lectures that when we have mix
states instead of pure states then these points
that we have they do not actually traverse
only on the surface they are not coming on
the way to the surface but those are for cases
when they get mixed but in for pure states
the representation is only on the surface
of the states that we are looking at sphere
so in terms of molecules and quantum computing
adiabatic manipulations are aimed at i v r
control which have been shown to generate
hadamard gates which is how it is being shown
here in this case where two level system interaction
is going to have them become equally probable
and they can be put to work together a pseudo
two level system can be generated from any
i v r multilevel system because we are able
to turn off the interactions with other states
which are all bunched here can be three four
five for instance depending on the number
of states that have been used the street could
be either in the ground state zero excited
state on interaction and this is what we just
described in terms of the control not that
we talked about when we interacted with the
state in terms of a simple interaction with
a laser pulse the system keeps on oscillating
between ground and excited state however when
we have the adiabatic principle applied and
we apply chirp pulse its possible to have
only one interaction or the other
so here is for example adiabatic gets with
chirp pulse pseudo two level system which
are generated based on the idea that the other
states and not interacting with the system
so these are basically the points which we
have looked at it is then possible to extend
such an optical scheme into a case where we
can apply the grovers algorithm so in general
a grovers algorithm essentially utilizes a
superposition of qubits which is being iteratively
manipulated to increase the probability amplitudes
of the index state which match search criteria
and the search criteria is provided through
in oracle and then finally the qubits are
measure to retrieve one of the matching indices
probabilistically and so this is being followed
multiple times for the matching indices to
work so it is basically a quantum computing
unstructure so we need a unstructured search
algorithm so the grovers algorithm is a unstructured
search algorithm in the quantum computing
sense and its one of the major quantum computing
algorithms the other one as we know is the
source algorithm it works on the principle
of superposition typically classical algorithms
can optimally work in order time n by two
to be precise minimum and grovers non distributed
principal works in order root n time and one
of the things which is being now will look
at is whether it is possible to distributed
quantum computing
as until now what we have been discussing
is the principle of only using a single quantum
computer why would anybody think of distributed
quantum computing thats because many of the
issues of implementing quantum computing that
we have done throughout this course have come
to the point of saying that scaling of the
quantum computer is always a problem so one
of the ways to get rid of this problem would
be to be able to have a principle of distributed
quantum computing so that the smaller nodes
of available quantum computers can be put
to use to develop a larger quantum computer
now the easy or the hard part of the problem
is that it is hard to recognize a solution
when we are going to go ahead with the distributed
quantum computing typically the oracle which
is used in grovers algorithm evaluates by
index and converts each item to a usable representation
and this can be utilized to ensure that when
the system is getting into the distributed
quantum computing mode this particular oracle
evaluation is put to a proper use and the
overhead of this process can be put back into
the principal to see if we can have an advantage
and if one goes by this approach it is found
that given an overhead of n log base two n
in addition to the single solution query approach
it is possible to have a distributed quantum
computing approach to grovers algorithm
now this is a very important step because
as this is developed it essentially brings
back the scalability issue of quantum computer
into practice and so an optical approach of
doing scalable quantum computing in this sense
is very effective the reason why this would
work most fruitfully with optical computing
approaches is because the optical qubits are
the best in terms of distributing because
they are the bits they are the qubits which
can be made to be distributed across different
nodes in terms of other kinds of computing
environments like n m r or other principles
it may be very difficult to do that but in
case of optics this is possible so that way
it is an attractive approach so the description
of the algorithm is similar to what it is
been always in it has the initialization step
which is essentially a what is gotten after
ahead amount gate in terms of all the states
been put as equally state then there is an
oracle which sets the stage for the entire
problem to work and that part is a critical
part in terms of generating the preparing
the oracle
and that we have done several times is been
provided here for completeness in this second
step here and the important part is to recognize
that we have a solution term index which we
are going to use as we go ahead the oracle
is an applied to the initial states that are
prepared and a the better way of this entire
grovers algorithm process involves the q f
t application process where its a unitary
transformation which is a discrete fourier
transform the amplitudes of the quantum state
which is being done in the subsequent step
and finally we can go through a loop where
the oracle and the q f t keeps on applying
on each other before doing that however would
be reversing the sign of the quantum state
in question and then keep on applying the
inverse q f t and this particular loop approach
it is possible to get to the marked signal
to be amplified and thats the basic point
here in terms of doing the grovers search
in order to distribute the grovers algorithm
which was first proposed as late as two thousand
fifteen lecture notes of computer science
it was possible to show that it distributes
the computing load on a set of quantum computers
and the critical part on this was a single
query solution has an overhead of n log base
two n steps to process and prepare the query
and this allows for virtually large pool of
available qubits by networking and data sharing
so here is the principal where eta identical
subsets have been setup and the interaction
of these subsets which are quantum computing
nodes are being connected through optical
fiber links because the information content
or the points of interaction with them can
be done through optical connects and so this
is how this principle have been looked at
through various different aspects of the nodes
being put to work simultaneously and in this
case also the principle is the same it has
an initiation process where the states are
all in equilibrium equally placed and each
state vector is in n to the power eta terms
because they have been put in this each of
these individual qubits states individual
quantum computers q c one q c two for example
the second step involves preparation the oracle
which again remains the same as before in
terms of the regular grovers algorithm application
of the oracle is now going to be on most the
oracle will now be applied to each of these
different quantum states and the state vectors
would then be prepared as a result of the
application and then the quantum fourier transform
which is the unitary transform of discrete
fourier transform the amplitude of the quantum
state will again be operating on them
each of these states will then undergo the
interactions as we have discussed before except
that the reversal of the states that we now
do in terms of these as we have discussed
is going to work for all except the zeroth
state and we apply the inverse of q f t and
repeat the loop from step three by the oracle
application process starts so this loop continues
and what we generate after n cycles is a case
where each of these have been undergoing independently
the process the general case is that would
be an order of root k and the advantage is
that when there are small sizes then that
is when your eta of this order of n then it
goes back to the original condition of log
base n and the upper limit of applicability
is when n is less than half of root n else
all the subsystems are in the qualified state
so overall we find that we have a limit over
which this particular process is possible
to give rise to the same order as per the
grovers overall club order of applicability
there are a couple of kvs to worry which is
the classical limit is what the answer comes
out for smaller sizes and so when the size
gets really small then there is no advantage
of going into this distributed mode because
it finally goes back to the classical search
condition is no advantage in some sense by
applying this
however a point to note is distribution for
the small size is impractical anyway so nobody
would want to do a distribution for a very
small size of a search algorithm the reason
why it is going to be applicable is the case
when the size of the problem is large enough
that you would like to distribute the problem
so the point to note here is that unless the
problem size is large there is no advantage
or there is no point of given distributing
this principle of quantum computing or in
this particular case of doing grovers algorithm
in this particular manner however when the
data size is large and it is not possible
to be done by using the traditional grover
search algorithm because of the size of the
problem it does make sense to distribute the
problem as there is an advantage while going
into the system
and last but not the least i thought i will
mention the molecular hadamard gate so many
times i should also show it here and this
is one of the things where in the use of these
many many states which are these i v r states
that we have been discussing it is possible
to in fact create a situation where all the
states would have equal superposition between
the quantum states and this could be possible
if we have a bunch of states which are coupled
to the ground state and once this couplings
work then we call them bright states and if
we have so in typical approaches we have used
only one bright state with respect to the
ground state and the rest of them are all
states which are going to take away the energy
from them so that we do not want them to be
coupled and so we call them dark states as
we can decouple them
however if there are states which are directly
coupled dipole couple to the ground state
so that they all get excited when the energy
is available and by applying of principle
of adiabatic sweeping of the laser we can
keep the unwanted states to draw energy away
from these optically couple states will be
having unequal superposition of bride states
and such multiple coherent states if can be
generated then they are nowadays called qudits
which are quantum objects where the number
of possible states or levels is greater than
two the numbers are denoted by the letter
d and this was first given by stroud as i
mentioned in this particular work more than
two states called qudits so that was the final
said that i wanted to mention and with that
i would like to end todays lecture on the
optical aspects of quantum computing that
in this particular end state we have managed
to show a little bit more than what we have
actually discussed earlier in the sense that
there is a possibility in research at least
or in principles that has been developed as
of now that there is indicated that distribution
of quantum notes or architecture may perhaps
be possible although this is too preliminary
to make those statements but if they are in
the scalability would become important and
with that i would like to thank you and see
you next time
