we have been discussing about n m r quantum computing we have gone through the basics
of how n m r operates is basics spectroscopic ideas how it has become such a popular device
of a spectrometer to start with now will be looking at it in terms of quantum computing
how the quantum or the qubit representation comes and we discuss from there so the qubit
representation for the n m r quantum computing actually in general the first few may not
be exactly for just n m r it can be for any case is that a single qubit has the computational
basis which is given as the zero one kind of a representation and that takes on to the
states so the computational basis is based on the states zero and one the wave function
is represented as a combination of the basis sets zero and one so the alpha and beta represents
the corresponding amounts so alpha squared beta squared are the corresponding amounts
of alpha sorry zero and one state that is their basis states which are there in the
final state
so in terms of how it is represented we have already discussed this is how it looks
likes that its a block spheres in which we have the state basis sets with certain
proportional representation probabilities of alpha and beta square which are being used
so at any point of time it will be alpha squared plus beta squared is equal to one that is
the total amount of amount which is there which is what happens when you make a measurement
it will give raise to either zero or one with the probability which will be represented
by their individual amounts and so that is always is the final constrain that exists
when you are looking at the qubit representative so a single qubit will be represented by the
individual basis sets which will be if constrained by these parameters all possibilities exist
however whenever the measurements are made that as a probability of one verses the other
in the in the way that the constraint has been designed
now for two qubit representation we have similarly the tens a product of these individual
states which is represented by zero zero if it is any one of the other then it is represented
in this passion so this is one kind this is one its of the other kind and depending on
the fact that we can distinguish between one this with respect to this will have the
different four different possibilities
so these are the four different possibilities and they have their own probabilities of
their existence alpha square beta square mode of alpha mode of beta mode of gamma mode of
delta and some of them would always be equal to one because the total probability as to
be one so in general for n qubit quantum computer we have two digit power n states basis states
and they can be represented therefore by the wave function which would be having all
possibilities of their independent basis sets with the alpha i as their particular
contribution so the probability of each of them being present is alpha i squared some
over all of them
so this is how the qubit representation goes we can in case of the nuclear spin hamiltonian
which is what we will be using in case of n m r for a single spin the hamiltonian
is given by the interaction that we just talked about the energy gap h cross gamma b zero
i of z which h is equal to minus h cross omega naught i of z equal to minus h cross omega
naught over two minus h cross omega naught over two plus h cross omega naught over
two and this is the hamiltonian that we work with for multiple spins without coupling can
be simply written as a continuous form of the same way where n this is the delta
i is the interaction is the shielding due to each of them so this is the shielding as
we had discussed due to the other electrons present and we can have the final hamiltonian
in the same format
there are as we mentioned when the field intensities are higher these coupling terms which are
due to the spins pin of each of them which comes in separately which is the h j so in
most cases the total hamiltonian is thought of as the spin coupling term independent
h naught with the coupling term spins pin coupling term which is h of j this is how
the total hamiltonian is pick looked at when you are looking at the nucleus spin hamiltonian
if you go ahead and do calculations with this and look at the reason and conditions
and solve how the populations change for each of these states the excite state with
his govern state in find that they basically oscillate between the ground and excided states
and they go through these rabi oscillations where the area under the applied filed is
the one which determines how it is going to go into the excited state or not and
its going to go back and four between the ground and excited state thats because
they are in the two states are coupled such that going into one case with the right amount
of energy would also have mean that when at the energy exist for a long enough time it
will also be able to bring it down and so on and so forth so it it can flop and that
see rabi flopping principle excited state energy and depending on how it is applied
all these kinds of parameters can be looked at this is a particular case where the shape
of the field is hyperbolic secant this is suppose to be one of the exact solution
the gaussian is another way of looking at the field where a rectangular uh applied field
which is also done mostly the assumptions are rectangular because the turn on and turn
off nothing can be perfectly rectangular so one some of these other cases also been looked
at in terms of how this works the steps involved in the n m r quantum computing we have an
initialization step as because the before the computational starts the qubits must be
initialize to well defined state so that is our initialization step the information
is then process by applying unitary transformations so ones the initialization is properly done
we are having our quantum register and ones we have our quantum register which has the
information in that set that we want to start our computing n we apply the unitary
transformations one after the other so the first step would take it to the application
of one unitary transform into the other and so on and so forth and the end of the computation
the result is processed and in the read out process processed in the read out process
so this is our final read out process which is been looked at as the final output
the initialization process for quantum algorithms generally assume or demand that
the qubits can be prepared in a pure state usually in the ground state nuclear spins
are in thermal equilibrium at room temperature and are subject to a reasonable static magnetic
field in a highly mix state and is given the density matrix and so this is a situation
where generating a perfect pure state which is basically the perfect ground state is a
very hard job and so it require creation of effective pure state with a density matrix
of the form of something like that where it is not exactly all of them are in the
ground state but there are but its done in such a ways so that they effectively are able
to generate the same kind of result so in reality all the all this spins are in the
same direction is what a pure state is suppose to be essentially you want everything to
be alpha but so that is that is sort of like what you would be wanting so if you are
talking about term two spin case you would like all your four states to be at the zero
zero condition the pseudo pure state is the one we create a state will which all levels
except one have equal population so basically this is the condition where we can actually
have a situation where everything if they can be balanced out then then all the levels
except the one have equal populations then it mimics a pure state because thats the state
which essentially has has the property which is not going to be the same as the others
and therefore that pseudo pure state was pretty much the same way as a pure state would a
occurred because the others other populations are balanced out in terms of having equal
populations because all energy levels except one have zero population such a state is very
difficult to produce in case of room temperatures for n m r condition so in the pure state
technically demands there trays of rho is equivalent to trays of rho squared which is
equal to one for a diagonal density matrix this condition require that all energy levels
expect one have zero population such a state to is difficult to create n m r so what is
done is that um under high temperature approximation pseudo pure states are taken where we create
a state in which all levels except one have equal population such a state mimics pure
state and so for example alpha can be a very high number in this case for instance
but as long as its not going to create a issue its going to work out so so for a two qubit
a system in equilibrium we can have this situation where that pseudo pure state would would
therefore the equilibrium condition is like this where every possible case is existing
by the pseudo pure state is the one which will work for us where is four zero
zero zero its going to work out for us how to create this is can be done by his special
averaging temporal averaging logical labeling or specially average logical labeling principle
so method for preparing effective pure state therefore can be done by logical labeling
which consist of applying a pulse sequence that rearranges the thermo populations such
that a subset of this spins is in an effective pure state something like this for instance
there is one some molecule molecule penta fluoride beta dienyl cyclopentadienyl dicarbonyl
complex in thermal equilibrium the spectrum after preparing in effective pure state is
looks like this this is courtesy of some the work which been carried out in a i i c
bangalore anil kumars group
so again as another way which is temporal averaging consists of adding up spectra a
multiple experiments each experiments starts with a different state preparation pulse sequence
consist of c naught hence not operations and then these special averaging case which uses
a pulse sequence containing magnetic field ingredients to equalize all populations now
these are all practical principles for us when we want to learn how to do this in
terms of a real applications or just for the case of undemanding how n m r quantum computing
does it is just to understand that its not a case which is very simple but it turns out
that there are ways of making sure that you can get to a result which would be sort of
like what you would be expecting or where you would be starting of as a simple case
now unitary transformations are the once which are applied using quantum logic gates
n m r technique provides a universal set of hamiltonian that can be used to implement
any unitary evolution including quantum logic gates so the building of quantum logic gates
is very similar to designing conventional n m r pulse sequence and therefore it is as
something which has been possible to be done as i mention before in the principle of the
n m r was that you would be essentially applying r f pulses to do the operations where you
do spectroscopy with it in this particular case instead of doing that as a spectroscopic
tool you would be essentially designing logic gates which are the n m r which are
in tune with designing the n m r pulse sequences that is use first spectroscopy in in the
conventional n m r case their simplest gates are rotations about the axis and x y plane
single qubit gates and these can be implemented resonant r f pluses other gates are obtain
by jointly using these basic gates
now so for example the unitary operations for single qubit gates can be as simple
as the x gate the y gate z gate or the s gate uh all of which are the simple single quibit
gates that we have already known the two qubit gates are of the ones which have the hadamard
which had a result of ninety degree pulse in the y dimension and a hundred and eighty
degree pulse in the x dimension
now just for clarity let me actually tell you that these pulses these ninety and
these are essentially measured with respect to the area under this and this is the
these are in these are on for a certain time on off and this is the omega which is
being applied so so omega t the area under this pulse omega t the area is a dimensionless
number and this dimensionless quantity a is being
represented by ninety so basically its pi by two pi or some of this depending on the
product of these two numbers so thats how the hadamard gate turns out to be a sequence
of ninety degree along the y axis and hundred and eighty degree along the x axis so there
are two axis that can be utilize so the the field is propagating along the z dimension
as we have discussed this is the z dimension applied filed b or h wherever way you look
at it and the the x and y are the ones along which the r
f is being applied so the coils could be used along these dimensions and thats how these
coils are been applied for the r f pulses of ninety pi hundred and eighty pi and so
on so forth c naught is of a different kind of gate which is being discussed here
so the quantum circuits can have control not which can go along and provide all the different
directions of applying the fields there can be controlled phase gates which are just
rotation gates and depending on how these where pulse sequences have been used it can
generate these kinds of gates so in control not in n m r essentially works in the principle
that you are applying these fields applying these so that they can undergo the
spin flips and if they are going spin flips then you see the z pulses something like
that so here for instance if this spin b is up then it under goes this possibility
of putting one through the other and then there is is delay which is very important
because his delay essentially allows the system to relax if you are in the z dimension
if you are put it back in the x dimension it can fan out and relax and that delay is
the one which takes care of the fact that it will be actually utilizing that as a part
of its pulse cycling so here it is a y ninety degree pulse followed by a delay to
give the half coupling j a b coupling and then its again an x a ninety degree pulse
if the spin b is down then you provide a different way of doing this possibility
so so this is sort of like if you have his two spin system then you have a flit
flip a if b is down and y vice verse so this is how generally a m r f pulse sequence would
look like for instance when a real situation for a carbon thirteen so if you notice
the carbon thirteen is one which as the spin proton as this spin and proved in nineteen
as this spin all these three are spin which are going to interact in terms of this spin
cases we have three different cases and based on these three it is being huts to to the
r f pulse sequences to do this kind of sense
so these are basically certain examples coming from research which we are not going
to discuss in detail we are just showing it to you so that you understand that its possible
we will perhaps not be able to get into the need not be able to get time to get into
the details of these these are just to show that these things in reality workout and this
is how they kind of look like where you are providing the pulses and how they are coming
in different windows and timescales and how did interact and how you get these things
to get to the simulation of what you are looking for so this is the case to show you can generate
x y hamiltonian and so on and so forth for a molecule of this kind
so the finally the n m r computer also works with the idea that you are working what
a ensemble of spin which is why the measurement principle is very very important it produce
an observable microscopic signal which can be picked up by a set of coils positioned
in the x y plane the signal measures the change in the rate of the magnetic field created
by a large number of spins in the sample rotating around this z axis called the free induction
decay as we had mentioned earlier and this free induction decay is then fourier transform
to get to the result the magnetization detected by the coil is proportional to the trays of
the product of the density matrix with the sigma plus which is basically sigma x plus
i sigma y so this enables the in detection of the magnetization which is being red by
the coil as it is a proportional to the trays and in terms of measurement this schematic
of the diagram look like this you have this sample tube which has the liquid sample in
it which is being subjected to the static magnetic field so b zero has been showed here
and these are the r f coils which are along their dimension which gives raise to the x
y plane and the capacitor filters the signal out there is a directional couple and with
which we can actually have the signal being controlled and provided through the
r f oscillator by the help of a computer so this is how a typical n m r spectrometer this
is three hundred mega hertz spectrometer looks like there are many different versions
of it now a days there are eight hundred mega hertz n m r spectrometers which are very powerful
compare to these and it can be a lot more interesting work with it molecule selection
for n m r quantum computing have to be also done very carefully desired properties are
for the case of spin half systems which are say for example proton carbon thirteen
fluid in fifteen nitrogen fifteen they can be they need to have long relaxation times
it could be useful to get hatronuclear or large chemical shifts systems so that
you can also use require to make spins of the same type addressable you need to
have good j coupling networks so that you know you can actually work with them and make
them happen as and you want them to be stable available and soluble in the system that we
are looking at
so it was used it has being shown with the n m r quantum computing grover search
algorithm has been shown quantum fourier transform has been shown shors algorithm has been
shown deutsch jozsa algorithm was also been shown order finding error correction codes
and dense coding has been shown the typical setup essentially if you look at it a little
bit more close has this kind of setup which has these electric filed as i just showed
here so this is the magnetic pole pieces in which the is a permanent magnate and then
there is this super conducting magnet which is also used along with this to make sure
that this one works properly this is internal externally applying an oscillating magnetic
field to the spin this spin will gradually move to the state from down or up and vice
verse and this is how this gates have been done
so ultra high sensitivity of n m rs have also been developed and people have working
on this area also to try to see this can make things work better for n m r the r f and
the magnetic field availability is the most important thing in this cases the signal to
noise ratio is defendant on the amount of magnetic field that can be applied also so
they have the magnetic field the better and so thats one of the sick ideas here so here
is a simple cartoon of how things works that is the so in this particular case for instance
the carbon and the proton in the chloroform as i had discussed earlier have one of the
proper molecules which use where used for two qubit system is one of the first molecules
which is use for quondam computing to show hadamard transform happen for a for by
using this chloroform molecule where the carbon and the proton was used and the radio frequency
pulse was used to address the hydrogen nuclei and causes it to rotate to form
a zero state to a so position state interactions through in through the chemical bond would
allow multiple qubit logic to be perform so this was one of the first cases which we have
shown i have already mention that carbon twelve has no spin carbon thirteen is the one which
we would like so in the case of
for example in and the chloroform that i just showed the carbon thirteen and the proton
shows this spin half case and these chlorines dont have anything its its properly designed
radio frequency pulse can rotate the carbon spin downwards to the horizontal plane and
so on so forth the geometry of the molecule is constrain because of the way the structure
of the molecule is the protons placed within the fix magnetic field can induce the change
of direction by magnetic filed the oscillates at radio frequencies as you have been saying
only a few million to be second such radio frequencies with rotate the nuclear spin about
the resurrection of the oscillating field which is typically to try at right angle so
the fix filed so thats how we had shown along the right angles hundred and eighty degree
pulse ninety degree pulse it depends on how you are providing them at each point and they
flip or they remain in the same place if they flip then they actually precise wherever they
are they will precise and thats what happens when you if the maximum precision is at ninety
degrees press if the oscillating radio frequencies loss just long enough to rotate spin per hundred
and eighty the excess magnetic field previously aligned in parallel with the fix field will
now point in the opposite anti parallel direction a pulse of half that duration will leave the
particles in a equal probability of being align parallel or anti parallel and thats
the idea behind hundred and eighty pulse verses ninety degree pulse and that was the area
under the pulse principle that i will in showing before
so i mean in generally whatever i have shown until now is sort of shown here pictorially
which says that after certain short times is this trying to explain whatever i said
in pictorial form the carbon will point either in one direction or exactly the opposite depending
on whether the spin of the neighboring hydrogen was up or down at that instant we
apply another radio frequency pulse to rotate the carbon nuclear another ninety degrees
that many over than flips the carbon nuclei into the down position if the adjacent hydrogen
was up or into the up position if the hydrogen was down a basic limitation of the chloroform
computer this you have to you have been discussing lately is clearly in its small number of qubits
thats what happens in most of the n m r principles unfortunately the number qubits could be expanded
but n could not be larger than the number of atoms in the molecule employed because
this is a molecule only computer thats the point of how it works
so this is how these sequences they have been showing in different ways either by showing
through animation or by showing the different stages this is how it goes and the the elaboration
of quantum computing by n m r one of the largest one has been the seven qubit computer in
i b m where they use a seven qubit molecule alanine an aluminum acid to factorize the
number fifteen which showed algorithm and this was done by h one why you are tell in
i b m and this he was able to show this with the help of nuclear of five fluorine and two
carbon atoms interacting with each other to provide this process of the shors algorithm
where he was able to factorize the number fifteen we have done the factorization of
number fifteen separately in this course to show the steps and so it will become clear
that how he applied this with respect to n m r quantum computing by using this molecule
typical difficulties in n m r quantum computing lies in the fact that the number of qubits
is difficult in scaling standard q c is based on pure states in n m r single spins
are too weak to measure so we must consider ensembles thats one of the biggest thing here
which is different from the regular standard quantum computing principles in a q c measurement
are usually projective in n m r we get the average over all molecular values tendency
for spins to align with field is weak even at equilibrium most spins are at random thats
one of the difficulties here how way this is overcome by the method of effective
pure states and thats the reason why i mentioned little bit one their effective pure
state pseudo pure states which have been done some of these sections will again be dealt
with a little bit when we do more on the theory side of these principles where we
do density matrices pure state pseudo pure states how they interact how they operate
and those kinds of things this was more to do with implementation aspects and with
that i would like to thank you for todays class we will take on further aspects
of quondam computing in the next session
thank you