WEBVTT
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we have been discussing about n m r quantum
computing we have gone through the basics
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of how n m r operates is basics spectroscopic
ideas how it has become such a popular device
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of a spectrometer to start with now will be
looking at it in terms of quantum computing
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how the quantum or the qubit representation
comes and we discuss from there so the qubit
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representation for the n m r quantum computing
actually in general the first few may not
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be exactly for just n m r it can be for any
case is that a single qubit has the computational
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basis which is given as the zero one kind
of a representation and that takes on to the
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states so the computational basis is based
on the states zero and one the wave function
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is represented as a combination of the basis
sets zero and one so the alpha and beta represents
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the corresponding amounts so alpha squared
beta squared are the corresponding amounts
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of alpha sorry zero and one state that is
their basis states which are there in the
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final state
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so in terms of how it is represented we
have already discussed this is how it looks
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likes that its a block spheres in which we
have the state basis sets with certain
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proportional representation probabilities
of alpha and beta square which are being used
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so at any point of time it will be alpha squared
plus beta squared is equal to one that is
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the total amount of amount which is there
which is what happens when you make a measurement
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it will give raise to either zero or one
with the probability which will be represented
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by their individual amounts and so that is
always is the final constrain that exists
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when you are looking at the qubit representative
so a single qubit will be represented by the
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individual basis sets which will be if constrained
by these parameters all possibilities exist
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however whenever the measurements are made
that as a probability of one verses the other
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in the in the way that the constraint has
been designed
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now for two qubit representation we have similarly
the tens a product of these individual
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states which is represented by zero zero if
it is any one of the other then it is represented
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in this passion so this is one kind this is
one its of the other kind and depending on
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the fact that we can distinguish between
one this with respect to this will have the
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different four different possibilities
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so these are the four different possibilities
and they have their own probabilities of
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their existence alpha square beta square mode
of alpha mode of beta mode of gamma mode of
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delta and some of them would always be equal
to one because the total probability as to
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be one so in general for n qubit quantum computer
we have two digit power n states basis states
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and they can be represented therefore by
the wave function which would be having all
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possibilities of their independent basis
sets with the alpha i as their particular
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contribution so the probability of each of
them being present is alpha i squared some
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over all of them
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so this is how the qubit representation goes
we can in case of the nuclear spin hamiltonian
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which is what we will be using in case of
n m r for a single spin the hamiltonian
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is given by the interaction that we just talked
about the energy gap h cross gamma b zero
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i of z which h is equal to minus h cross omega
naught i of z equal to minus h cross omega
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naught over two minus h cross omega naught
over two plus h cross omega naught over
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two and this is the hamiltonian that we work
with for multiple spins without coupling can
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be simply written as a continuous form
of the same way where n this is the delta
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i is the interaction is the shielding due
to each of them so this is the shielding as
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we had discussed due to the other electrons
present and we can have the final hamiltonian
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in the same format
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there are as we mentioned when the field intensities
are higher these coupling terms which are
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due to the spins pin of each of them which
comes in separately which is the h j so in
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most cases the total hamiltonian is thought
of as the spin coupling term independent
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h naught with the coupling term spins pin
coupling term which is h of j this is how
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the total hamiltonian is pick looked at when
you are looking at the nucleus spin hamiltonian
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if you go ahead and do calculations
with this and look at the reason and conditions
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and solve how the populations change for
each of these states the excite state with
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his govern state in find that they basically
oscillate between the ground and excided states
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and they go through these rabi oscillations
where the area under the applied filed is
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the one which determines how it is going to
go into the excited state or not and
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its going to go back and four between the
ground and excited state thats because
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they are in the two states are coupled such
that going into one case with the right amount
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of energy would also have mean that when at
the energy exist for a long enough time it
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will also be able to bring it down and so
on and so forth so it it can flop and that
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see rabi flopping principle excited state
energy and depending on how it is applied
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all these kinds of parameters can be looked
at this is a particular case where the shape
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of the field is hyperbolic secant this is
suppose to be one of the exact solution
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the gaussian is another way of looking at
the field where a rectangular uh applied field
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which is also done mostly the assumptions
are rectangular because the turn on and turn
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off nothing can be perfectly rectangular so
one some of these other cases also been looked
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at in terms of how this works the steps involved
in the n m r quantum computing we have an
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initialization step as because the before
the computational starts the qubits must be
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initialize to well defined state so that is
our initialization step the information
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is then process by applying unitary transformations
so ones the initialization is properly done
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we are having our quantum register and ones
we have our quantum register which has the
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information in that set that we want to
start our computing n we apply the unitary
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transformations one after the other so the
first step would take it to the application
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of one unitary transform into the other and
so on and so forth and the end of the computation
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the result is processed and in the read out
process processed in the read out process
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so this is our final read out process which
is been looked at as the final output
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the initialization process for quantum
algorithms generally assume or demand that
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the qubits can be prepared in a pure state
usually in the ground state nuclear spins
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are in thermal equilibrium at room temperature
and are subject to a reasonable static magnetic
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field in a highly mix state and is given
the density matrix and so this is a situation
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where generating a perfect pure state which
is basically the perfect ground state is a
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very hard job and so it require creation of
effective pure state with a density matrix
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of the form of something like that where
it is not exactly all of them are in the
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ground state but there are but its done in
such a ways so that they effectively are able
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to generate the same kind of result so
in reality all the all this spins are in the
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same direction is what a pure state is suppose
to be essentially you want everything to
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be alpha but so that is that is sort of
like what you would be wanting so if you are
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talking about term two spin case you would
like all your four states to be at the zero
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zero condition the pseudo pure state is the
one we create a state will which all levels
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except one have equal population so basically
this is the condition where we can actually
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have a situation where everything if they
can be balanced out then then all the levels
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except the one have equal populations then
it mimics a pure state because thats the state
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which essentially has has the property
which is not going to be the same as the others
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and therefore that pseudo pure state was pretty
much the same way as a pure state would a
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occurred because the others other populations
are balanced out in terms of having equal
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populations because all energy levels except
one have zero population such a state is very
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difficult to produce in case of room temperatures
for n m r condition so in the pure state
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technically demands there trays of rho is
equivalent to trays of rho squared which is
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equal to one for a diagonal density matrix
this condition require that all energy levels
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expect one have zero population such a state
to is difficult to create n m r so what is
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done is that um under high temperature approximation
pseudo pure states are taken where we create
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a state in which all levels except one have
equal population such a state mimics pure
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state and so for example alpha can be a
very high number in this case for instance
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but as long as its not going to create a issue
its going to work out so so for a two qubit
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a system in equilibrium we can have this situation
where that pseudo pure state would would
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therefore the equilibrium condition is like
this where every possible case is existing
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by the pseudo pure state is the one which
will work for us where is four zero
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zero zero its going to work out for us how
to create this is can be done by his special
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averaging temporal averaging logical labeling
or specially average logical labeling principle
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so method for preparing effective pure state
therefore can be done by logical labeling
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which consist of applying a pulse sequence
that rearranges the thermo populations such
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that a subset of this spins is in an effective
pure state something like this for instance
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there is one some molecule molecule penta
fluoride beta dienyl cyclopentadienyl dicarbonyl
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complex in thermal equilibrium the spectrum
after preparing in effective pure state is
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looks like this this is courtesy of some
the work which been carried out in a i i c
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bangalore anil kumars group
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so again as another way which is temporal
averaging consists of adding up spectra a
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multiple experiments each experiments starts
with a different state preparation pulse sequence
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consist of c naught hence not operations and
then these special averaging case which uses
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a pulse sequence containing magnetic field
ingredients to equalize all populations now
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these are all practical principles for
us when we want to learn how to do this in
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terms of a real applications or just for the
case of undemanding how n m r quantum computing
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does it is just to understand that its not
a case which is very simple but it turns out
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that there are ways of making sure that you
can get to a result which would be sort of
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like what you would be expecting or where
you would be starting of as a simple case
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now unitary transformations are the once
which are applied using quantum logic gates
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n m r technique provides a universal set of
hamiltonian that can be used to implement
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any unitary evolution including quantum logic
gates so the building of quantum logic gates
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is very similar to designing conventional
n m r pulse sequence and therefore it is as
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something which has been possible to be done
as i mention before in the principle of the
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n m r was that you would be essentially applying
r f pulses to do the operations where you
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do spectroscopy with it in this particular
case instead of doing that as a spectroscopic
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tool you would be essentially designing
logic gates which are the n m r which are
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in tune with designing the n m r pulse sequences
that is use first spectroscopy in in the
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conventional n m r case their simplest gates
are rotations about the axis and x y plane
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single qubit gates and these can be implemented
resonant r f pluses other gates are obtain
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by jointly using these basic gates
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now so for example the unitary operations
for single qubit gates can be as simple
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as the x gate the y gate z gate or the s gate
uh all of which are the simple single quibit
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gates that we have already known the two qubit
gates are of the ones which have the hadamard
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which had a result of ninety degree pulse
in the y dimension and a hundred and eighty
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degree pulse in the x dimension
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now just for clarity let me actually tell
you that these pulses these ninety and
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these are essentially measured with respect
to the area under this and this is the
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these are in these are on for a certain
time on off and this is the omega which is
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being applied so so omega t the area under
this pulse omega t the area is a dimensionless
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number
and this dimensionless quantity a is being
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represented by ninety so basically its pi
by two pi or some of this depending on the
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product of these two numbers so thats how
the hadamard gate turns out to be a sequence
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of ninety degree along the y axis and hundred
and eighty degree along the x axis so there
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are two axis that can be utilize so the
the field is propagating along the z dimension
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as we have discussed this is the z dimension
applied filed b or h wherever way you look
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at it and the
the x and y are the ones along which the r
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f is being applied so the coils could be used
along these dimensions and thats how these
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coils are been applied for the r f pulses
of ninety pi hundred and eighty pi and so
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on so forth c naught is of a different
kind of gate which is being discussed here
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so the quantum circuits can have control not
which can go along and provide all the different
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directions of applying the fields there
can be controlled phase gates which are just
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rotation gates and depending on how these
where pulse sequences have been used it can
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generate these kinds of gates so in control
not in n m r essentially works in the principle
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that you are applying these fields
applying these so that they can undergo the
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spin flips and if they are going spin flips
then you see the z pulses something like
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that so here for instance if this spin
b is up then it under goes this possibility
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of putting one through the other and then
there is is delay which is very important
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because his delay essentially allows the system
to relax if you are in the z dimension
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if you are put it back in the x dimension
it can fan out and relax and that delay is
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the one which takes care of the fact that
it will be actually utilizing that as a part
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of its pulse cycling so here it is a y
ninety degree pulse followed by a delay to
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give the half coupling j a b coupling and
then its again an x a ninety degree pulse
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if the spin b is down then you provide
a different way of doing this possibility
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so so this is sort of like if you have
his two spin system then you have a flit
19:39.610 --> 19:46.860
flip a if b is down and y vice verse so this
is how generally a m r f pulse sequence would
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look like for instance when a real situation
for a carbon thirteen so if you notice
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the carbon thirteen is one which as the spin
proton as this spin and proved in nineteen
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as this spin all these three are spin which
are going to interact in terms of this spin
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cases we have three different cases and based
on these three it is being huts to to the
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r f pulse sequences to do this kind of sense
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so these are basically certain examples
coming from research which we are not going
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to discuss in detail we are just showing it
to you so that you understand that its possible
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we will perhaps not be able to get into the
need not be able to get time to get into
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the details of these these are just to show
that these things in reality workout and this
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is how they kind of look like where you are
providing the pulses and how they are coming
20:39.150 --> 20:43.980
in different windows and timescales and how
did interact and how you get these things
20:43.980 --> 20:48.950
to get to the simulation of what you are looking
for so this is the case to show you can generate
20:48.950 --> 20:54.480
x y hamiltonian and so on and so forth for
a molecule of this kind
20:54.480 --> 21:00.510
so the finally the n m r computer also
works with the idea that you are working what
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a ensemble of spin which is why the measurement
principle is very very important it produce
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an observable microscopic signal which can
be picked up by a set of coils positioned
21:10.910 --> 21:18.950
in the x y plane the signal measures the change
in the rate of the magnetic field created
21:18.950 --> 21:23.990
by a large number of spins in the sample rotating
around this z axis called the free induction
21:23.990 --> 21:29.070
decay as we had mentioned earlier and this
free induction decay is then fourier transform
21:29.070 --> 21:33.740
to get to the result the magnetization detected
by the coil is proportional to the trays of
21:33.740 --> 21:39.860
the product of the density matrix with the
sigma plus which is basically sigma x plus
21:39.860 --> 21:48.670
i sigma y so this enables the in detection
of the magnetization which is being red by
21:48.670 --> 21:55.730
the coil as it is a proportional to the trays
and in terms of measurement this schematic
21:55.730 --> 21:59.810
of the diagram look like this you have this
sample tube which has the liquid sample in
21:59.810 --> 22:05.660
it which is being subjected to the static
magnetic field so b zero has been showed here
22:05.660 --> 22:10.531
and these are the r f coils which are along
their dimension which gives raise to the x
22:10.531 --> 22:16.710
y plane and the capacitor filters the signal
out there is a directional couple and with
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which we can actually have the signal being
controlled and provided through the
22:23.240 --> 22:31.630
r f oscillator by the help of a computer so
this is how a typical n m r spectrometer this
22:31.630 --> 22:38.150
is three hundred mega hertz spectrometer
looks like there are many different versions
22:38.150 --> 22:44.140
of it now a days there are eight hundred mega
hertz n m r spectrometers which are very powerful
22:44.140 --> 22:50.380
compare to these and it can be a lot more
interesting work with it molecule selection
22:50.380 --> 22:59.600
for n m r quantum computing have to be also
done very carefully desired properties are
22:59.600 --> 23:05.750
for the case of spin half systems which
are say for example proton carbon thirteen
23:05.750 --> 23:12.360
fluid in fifteen nitrogen fifteen they can
be they need to have long relaxation times
23:12.360 --> 23:20.681
it could be useful to get hatronuclear
or large chemical shifts systems so that
23:20.681 --> 23:26.220
you can also use require to make spins
of the same type addressable you need to
23:26.220 --> 23:30.330
have good j coupling networks so that you
know you can actually work with them and make
23:30.330 --> 23:36.050
them happen as and you want them to be stable
available and soluble in the system that we
23:36.050 --> 23:37.200
are looking at
23:37.200 --> 23:43.250
so it was used it has being shown with
the n m r quantum computing grover search
23:43.250 --> 23:49.080
algorithm has been shown quantum fourier transform
has been shown shors algorithm has been
23:49.080 --> 23:52.990
shown deutsch jozsa algorithm was also been
shown order finding error correction codes
23:52.990 --> 24:00.060
and dense coding has been shown the typical
setup essentially if you look at it a little
24:00.060 --> 24:09.840
bit more close has this kind of setup which
has these electric filed as i just showed
24:09.840 --> 24:15.670
here so this is the magnetic pole pieces
in which the is a permanent magnate and then
24:15.670 --> 24:19.170
there is this super conducting magnet which
is also used along with this to make sure
24:19.170 --> 24:26.530
that this one works properly this is internal
externally applying an oscillating magnetic
24:26.530 --> 24:30.560
field to the spin this spin will gradually
move to the state from down or up and vice
24:30.560 --> 24:34.350
verse and this is how this gates have been
done
24:34.350 --> 24:39.640
so ultra high sensitivity of n m rs have
also been developed and people have working
24:39.640 --> 24:47.830
on this area also to try to see this can make
things work better for n m r the r f and
24:47.830 --> 24:52.140
the magnetic field availability is the most
important thing in this cases the signal to
24:52.140 --> 24:57.660
noise ratio is defendant on the amount of
magnetic field that can be applied also so
24:57.660 --> 25:03.610
they have the magnetic field the better and
so thats one of the sick ideas here so here
25:03.610 --> 25:08.500
is a simple cartoon of how things works that
is the so in this particular case for instance
25:08.500 --> 25:14.310
the carbon and the proton in the chloroform
as i had discussed earlier have one of the
25:14.310 --> 25:20.990
proper molecules which use where used for
two qubit system is one of the first molecules
25:20.990 --> 25:28.950
which is use for quondam computing to show
hadamard transform happen for a for by
25:28.950 --> 25:35.031
using this chloroform molecule where the carbon
and the proton was used and the radio frequency
25:35.031 --> 25:41.600
pulse was used to address the hydrogen
nuclei and causes it to rotate to form
25:41.600 --> 25:46.350
a zero state to a so position state interactions
through in through the chemical bond would
25:46.350 --> 25:50.910
allow multiple qubit logic to be perform so
this was one of the first cases which we have
25:50.910 --> 25:56.130
shown i have already mention that carbon twelve
has no spin carbon thirteen is the one which
25:56.130 --> 26:00.200
we would like so in the case of
26:00.200 --> 26:06.740
for example in and the chloroform that i just
showed the carbon thirteen and the proton
26:06.740 --> 26:13.520
shows this spin half case and these chlorines
dont have anything its its properly designed
26:13.520 --> 26:18.840
radio frequency pulse can rotate the carbon
spin downwards to the horizontal plane and
26:18.840 --> 26:23.590
so on so forth the geometry of the molecule
is constrain because of the way the structure
26:23.590 --> 26:32.090
of the molecule is the protons placed within
the fix magnetic field can induce the change
26:32.090 --> 26:36.000
of direction by magnetic filed the oscillates
at radio frequencies as you have been saying
26:36.000 --> 26:41.450
only a few million to be second such radio
frequencies with rotate the nuclear spin about
26:41.450 --> 26:45.790
the resurrection of the oscillating field
which is typically to try at right angle so
26:45.790 --> 26:50.080
the fix filed so thats how we had shown along
the right angles hundred and eighty degree
26:50.080 --> 26:55.300
pulse ninety degree pulse it depends on how
you are providing them at each point and they
26:55.300 --> 27:00.710
flip or they remain in the same place if they
flip then they actually precise wherever they
27:00.710 --> 27:05.010
are they will precise and thats what happens
when you if the maximum precision is at ninety
27:05.010 --> 27:09.610
degrees press if the oscillating radio frequencies
loss just long enough to rotate spin per hundred
27:09.610 --> 27:14.760
and eighty the excess magnetic field previously
aligned in parallel with the fix field will
27:14.760 --> 27:20.000
now point in the opposite anti parallel direction
a pulse of half that duration will leave the
27:20.000 --> 27:23.990
particles in a equal probability of being
align parallel or anti parallel and thats
27:23.990 --> 27:28.500
the idea behind hundred and eighty pulse verses
ninety degree pulse and that was the area
27:28.500 --> 27:31.670
under the pulse principle that i will in showing
before
27:31.670 --> 27:38.340
so i mean in generally whatever i have
shown until now is sort of shown here pictorially
27:38.340 --> 27:43.160
which says that after certain short times
is this trying to explain whatever i said
27:43.160 --> 27:51.540
in pictorial form the carbon will point either
in one direction or exactly the opposite depending
27:51.540 --> 27:57.220
on whether the spin of the neighboring
hydrogen was up or down at that instant we
27:57.220 --> 28:02.640
apply another radio frequency pulse to rotate
the carbon nuclear another ninety degrees
28:02.640 --> 28:08.360
that many over than flips the carbon nuclei
into the down position if the adjacent hydrogen
28:08.360 --> 28:14.790
was up or into the up position if the hydrogen
was down a basic limitation of the chloroform
28:14.790 --> 28:20.760
computer this you have to you have been discussing
lately is clearly in its small number of qubits
28:20.760 --> 28:26.110
thats what happens in most of the n m r principles
unfortunately the number qubits could be expanded
28:26.110 --> 28:29.930
but n could not be larger than the number
of atoms in the molecule employed because
28:29.930 --> 28:35.080
this is a molecule only computer thats the
point of how it works
28:35.080 --> 28:42.460
so this is how these sequences they have been
showing in different ways either by showing
28:42.460 --> 28:50.310
through animation or by showing the different
stages this is how it goes and the the elaboration
28:50.310 --> 28:57.470
of quantum computing by n m r one of the largest
one has been the seven qubit computer in
28:57.470 --> 29:04.660
i b m where they use a seven qubit molecule
alanine an aluminum acid to factorize the
29:04.660 --> 29:10.590
number fifteen which showed algorithm and
this was done by h one why you are tell in
29:10.590 --> 29:18.260
i b m and this he was able to show this with
the help of nuclear of five fluorine and two
29:18.260 --> 29:25.550
carbon atoms interacting with each other to
provide this process of the shors algorithm
29:25.550 --> 29:29.180
where he was able to factorize the number
fifteen we have done the factorization of
29:29.180 --> 29:33.740
number fifteen separately in this course
to show the steps and so it will become clear
29:33.740 --> 29:40.380
that how he applied this with respect to n
m r quantum computing by using this molecule
29:40.380 --> 29:44.910
typical difficulties in n m r quantum computing
lies in the fact that the number of qubits
29:44.910 --> 29:52.490
is difficult in scaling standard q c is
based on pure states in n m r single spins
29:52.490 --> 29:59.830
are too weak to measure so we must consider
ensembles thats one of the biggest thing here
29:59.830 --> 30:06.110
which is different from the regular standard
quantum computing principles in a q c measurement
30:06.110 --> 30:10.930
are usually projective in n m r we get the
average over all molecular values tendency
30:10.930 --> 30:15.730
for spins to align with field is weak even
at equilibrium most spins are at random thats
30:15.730 --> 30:20.350
one of the difficulties here how way this
is overcome by the method of effective
30:20.350 --> 30:25.460
pure states and thats the reason why i mentioned
little bit one their effective pure
30:25.460 --> 30:29.180
state pseudo pure states which have been done
some of these sections will again be dealt
30:29.180 --> 30:35.820
with a little bit when we do more on the
theory side of these principles where we
30:35.820 --> 30:42.150
do density matrices pure state pseudo
pure states how they interact how they operate
30:42.150 --> 30:47.340
and those kinds of things this was more to
do with implementation aspects and with
30:47.340 --> 30:54.120
that i would like to thank you for todays
class we will take on further aspects
30:54.120 --> 31:10.440
of quondam computing in the next session
31:10.440 --> 31:19.800
thank you