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 ah the n m r quantum computing
actually in general the first ah 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 ah 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 ah 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 ah
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 ah amount which is there
which is what happens when you make a measurement
it will give raise to either ah 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 ah 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 ah 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 ah 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 ah
the wave function which would be having all
possibilities of their ah independent basis
sets with the alpha i as their ah 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
ah we can in case of the nuclear ah spin hamiltonian
which is what we will be using in case of
n m r ah 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 ah 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 ah continuous form
of the same way where n this is the delta
i is the interaction ah 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 ah hamiltonian
in the same format
there are as we mentioned when the field intensities
are higher ah 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 ah 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
ah if you go ahead and do ah calculations
with this and look at the reason and conditions
and solve how the populations ah 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
ah 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 ah into the excited state or not ah and
its going to go back and four between the
ground and excited state ah 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
ah 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 ah
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 ah in that set that we want to
start our computing n we apply the ah 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 ah output
the initialization process ah for quantum
algorithms generally assume or demand that
the qubits can be prepared in a pure state
usually in the ground state ah nuclear spins
are in thermal equilibrium at room temperature
and are subject to a reasonable static magnetic
field ah in a highly mix state and is given
the density matrix and so this is a situation
where generating ah 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 ah of something like that where
it is not exactly all of them are in the ah
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 ah
in reality all the all this spins are in the
same direction is what a pure state is suppose
to be ah essentially you want everything to
be alpha but ah 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 ah then all the levels
except the one have equal populations then
it mimics a pure state because thats the state
which essentially has ah 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 ah 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 ah 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 ah we create
a state in which all levels except one have
equal population such a state mimics pure
state and so for example ah 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 ah so for a two qubit
a system in equilibrium we can have this situation
where that pseudo pure state ah would ah would
therefore the equilibrium condition is like
this where ah every possible case is existing
by the pseudo pure state is the one which
will ah work for us where ah is four zero
zero zero its going to work out for us how
to create ah 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 ah 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 ah 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 ah 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 ah 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 ah
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 ah
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 ah 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 ah so for example the unitary operations
ah 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 ah 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 ah 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 ah these ninety and
these are essentially measured with respect
to the area under this and this is ah the
these are in ah 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
ah product of these two numbers so thats how
the hadamard gate turns out to be a sequence
of ah ninety degree along the y axis and hundred
and eighty degree along the x axis so there
are two axis that can be utilize ah so the
the field is propagating along the z dimension
as we have discussed this is the z dimension
applied filed ah 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 ah r f pulses
of ninety pi hundred and eighty pi and so
on so forth c naught ah is of a different
kind of ah 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 ah there
can be controlled phase gates which are just
ah 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 ah you are ah applying these fields ah
applying these so that they can undergo the
spin flips and if they are going spin flips
then you see the z ah pulses something like
that so here for instance ah 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 ah 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 ah 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
ah if the spin b is down then you provide
a different way of doing this ah possibility
so ah so this is sort of like ah if you have
his ah 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
ah 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 ah 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 ah to the
r f pulse sequences to do this kind of sense
so these are basically certain ah 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
ah 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 ah 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 ah 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 ah provided ah through the
r f oscillator by the help of a computer so
this is how a typical n m r spectrometer this
is ah three hundred mega hertz spectrometer
looks like ah 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
ah interesting work with it ah molecule selection
for n m r quantum computing have to be also
done very carefully desired properties are
for the ah 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
ah it could be useful to get hatronuclear
or large chemical shifts systems ah so that
ah you can also use require to make spins
of the same type addressable ah 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 ah it has being shown with
the n m r quantum computing grover search
algorithm has been shown quantum fourier transform
has been shown ah shors algorithm has been
shown deutsch jozsa algorithm was also been
shown order finding error correction codes
and dense coding has been shown ah the typical
setup essentially ah if you look at it a little
bit more close ah has this kind of setup which
has these ah electric filed as i just showed
here so this is the magnetic ah 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 ah 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 ah 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 ah 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 ah to show
hadamard transform happen ah for a for by
using this chloroform molecule where the carbon
and the proton was used and the ah radio frequency
pulse was used ah to address the hydrogen
nuclei ah 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 ah 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 ah 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 ah 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 ah
shown until now is sort of ah shown here pictorially
ah 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 ah 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 ah different ways either by showing
ah 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 ah 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 ah 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 ah
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
ah 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 ah 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 ah
pure states and thats the reason why i mentioned
little bit one their ah effective ah pure
state pseudo pure states which have been done
some of these sections will again be dealt
with a little bit when we do ah more on the
theory side of these ah principles where we
do ah ah 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 ah with
that i would like to ah thank you for todays
class we will ah take on ah further aspects
of quondam computing in the next session
thank you
