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 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 ead 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