WEBVTT
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the cnot maps the inputs in such a way that
one of them remains the same only if that
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the control bit or qubit and this is valid
only for pure states so these mappings where
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the particular qubit is not changing is only
valid for pure states however this can serve
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as a non demolition measurement gate because
of the control bit which can preserve the
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measurement process in this so these control
not gates are very useful and has been used
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for many purposes we will look into their
operation very soon here is one approach of
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implementing control not gate via linear optics
so in this approach as we have seen before
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in terms of linear optics in this particular
case a beam splitter is being used and the
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detectors at a and b would be measuring how
the outputs are so two qubits in this case
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coming from the source light source can be
encoded in one photon one in terms of the
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momentum or direction and the other in terms
of polarization of the photon
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the polarization controls the change in momentum
of the photon also however this cannot be
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scaled up directly but this demonstrates an
implementation of a two qubit gate the scaling
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of this is difficult because if you want to
increase the number of photons in this and
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simultaneously have more qubits encoded it
doesn't scale that easily because the photons
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cannot be treated in this particular format
of two qubits individually and so that's the
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difficulty however this is an important demonstration
of the use of linear optics in control not
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gate a three input gate is also easily possible
where instead of having two controls three
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inputs can be having in which two of them
can act as control and the other one does
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the bit flip so a b are the control bits and
c is the one which undergoes the change and
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this is more often also known as that a toffoli
gate so it is either known as cc not gate
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or the toffoli gate and a typical matrix for
such a gate is given by this three qubits
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a generalized control gate that can control
some one qubit unitary operation u are useful
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and that can be looked at in this format where
every time you have an operation going we
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can label them in terms of the operation
so for example just a unitary operation of
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the kind in circuit times would be looking
like this which we represent by u once we
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use it with respect to a single control then
it is a control on top of the operation so
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that's an example of the control not so
our unitary operation was essentially the
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not operation that we showed as unitary and
that's the one which is being used in all
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these cases here with some control if we use
two controls then it becomes control control
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unitary and this can be scaled as we have
seen to further kind of processes where more
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and more control bits can be used
however to have a universal gate set which
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will implement any unitary operation on n
qubits exactly would require an infinite number
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of gate types and so the principle that we
showed for a single qubit case where we were
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able to use only two gates as a complete
set the hadamard and the phase rotation is
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not as simple as we go to higher number of
qubits and so the complete set gets harder
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and harder to be defined the infinite set
of all two input gets is universal for instance
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so any n qubit unitary operation can be implemented
using so and so many gates and this is
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sort of taken from some work which was done
back in nineteen ninety four by reck et al
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where they were able to show how many operations
unitary cases and orders of the gates that
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are required for this so it turns out as i
mentioned that its not quite possible to come
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up with finite number in these cases and
so c not and the infinite set of one qubit
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gates is universal
so c not is with two qubit universal gate
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and but then there are the other infinite
set of all one qubit gates that are universal
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so that's why it is difficult to keep on defining
finite set of gates which will make it
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universal in this kind of an approach so in
order to have discrete universal gate sets
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the error on implementing say particular unitary
operator u by another v can be defined in
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this kind of a functional form and if we can
have u gates that can be implemented by k
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gates then we can simulate that many unitary
gates with a total error less than eta with
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a gate overhead that is polynomial in the
order which is log k over e
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now these kinds of work with their proofs
are parts of theoretical approaches to quantum
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computing which is beyond the scope to
some extent of this course we came up to here
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because we wanted to talk about the universality
of certain gates the number of gates are necessary
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the closed set and all those which are sort
of important in implementation purposes also
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however to be able to get into the exact nature
of how many or how to get to these definitions
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would become difficult so what we will do
is we will take it up to a point where will
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we have discussed as of now and we will just
come to note that a discrete set of gate types
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g is universal if this is a statement that
will keep which is a discrete set of gate
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types g is universal if we can approximate
any u or the unitary get to within an error
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which is eta slightly not too far away from
zero using a sequence of gates from the discrete
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gate types of g
so this is sort of a process which is utilized
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to ensure that among the infinite sets that
are possible the finite number of discrete
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sets are often used to within certain level
of precision so that the sequence of gates
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can be utilized with more efficiency so here
is an example of this particular approach
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so finally here is an example of this particular
approach of discrete universal gates set so
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for example four members standard gate set
in our particular approach as we have been
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discussing are the cnot gate the hadamard
gate the phrase gate and let say the rotation
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say here it's a pi over eight gate and all
of these are the discrete universal gate said
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that can be used as standard ones similarly
there are these cnot a hadamard phase and
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toffoli which can be another set of four
gate sets which are discrete universal gate
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such that can be utilized
so with this we just wanted to give you the
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idea of the different kinds of gates that
are implementable even with the simple linear
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optical approaches for quantum computing purposes
and i think we are now ready to look at certain
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circuits by using this approach so the quantum
circuits are important aspects that are necessary
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for the overall implementation of the processes
and in as far as definition goes these circuits
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are a sequence of quantum gates linked by
wires they are they are being put together
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in such a way that they can implement the
processes that we are interested in the circuit
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has fixed width corresponding to the number
of qubits which are being processed it's based
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on logic design both classical and quantum
which attempts to find the circuit structures
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for needed operation that are functionally
correct independent of physical technology
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and obviously for implementation purposes
they can require further aspects of low
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cost or the use of minimum number of qubits
or gates
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now unfortunately this is an area where a
lot more development is presently necessary
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as the quantum logic design is still not extremely
well developed there are quite a few adhoc
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designs that are known for many specific functions
and gates so here is an example taken from
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some work done back in nineteen ninety five
were a toffoli gate can be built from cnot
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control not gates where the a particular gates
implementation twice was essentially a unitary
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gate so here is for example a particular approach
which has been shown to implement certain
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specific functions and this is how some of
these wires and circuitries can be written
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so the final unitary operation is essentially
equivalent to the application of gates
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which turn out to be in this particular order
so for certain specific functions it's possible
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to write them out as has been done here it
is sometimes important to go through with
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them so here if we know how this is going
to go through and here is an example of what
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happens when we put in say the three qubits
in such a particular circuit diagram so once
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we put in these three bits through this equivalent
circuits which is an unitary operation
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what we will find is that the they will undergo
changes as per the sets which have been provided
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and finally we will end up producing a unitary
operation which results in giving rise to
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the same result
so it can produce different conditions depending
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on what the inputs are so if you have noticed
my control bit has changed between the last
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case that we looked at where my initial bit
was zero now we have changed it to one and
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we can immediately see the control not part
being operated on on the different points
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and can see the resultant and you can essentially
go ahead and simulate the other two remaining
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cases because i have just done it for zero
and one bit in the first case you will be
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having the same situation for the next case
also you will be finding that they essentially
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follow the same order so we can essentially
verify the unitary matrix of toffoli gate
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which can be looked at in the same way we
can calculate the unitary matrix u one of
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the first block from one side and that can
be done in this fashion where the unitary
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matrices of the control and the operation
is going to occur in this fashion we can
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again apply it the way we have done it in
the circuit diagram by using the matrices
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and we will get back the solution as we have
shown in the circuit is reason for doing this
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process is to essentially show you that the
circuits are undergoing the same changes as
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we are doing the operations and as a result
we will be getting the operations going the
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same way as we have expected and what we will
find is that the different inputs get permuted
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and that's why it's a tricky situation it's
not really remaining the exactly same as we
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say and it is important to evaluate the product
of all of them one after the other using the
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fact that we now have them going as identity
matrix and applying them one after the other
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is going to be unitary matrix and so this
can be looked at in this entire process the
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matrices are very sparse matrices all it matters
are those little points where they are going
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to interact and finally we end up producing
the particular sets where they are going to
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go undergo the changes to give rise to the
final result
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we can similarly calculate the different
u three matrices that will be necessary for
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this and which is a hermitian matrix so we
can transport and next calculate the complex
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conjugate we can denote the complex conjugates
by the bold symbols that's what is being done
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here and in all these cases these different
unitary matrices that we have been using are
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initially one after the other and that's why
they have been labelled as one two three in
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the subscript and that's how they have been
shown here so our fifth iteration once
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we go one after the other is going to be similar
to u one but has the x one and x two permitted
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because u one as in the in that other fashion
where we had a black dot closed dot in
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in the variable x two and the other one is
in the variable in the x one so this can be
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also checked in the definition which is here
so we had this situation here which was getting
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connected and so the this is calculated by
using this principle which we started off
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and so at every point whenever we are doing
the simulations or the calculations this is
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how they are evolving and we are looking at
the final result and the next step would be
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a unitary matrix of a swap gate and we can
use the five different products of the five
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eight by eight matrices which go all the way
from one to five using the fact again that
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their product is an identity matrix the v
and the v tagger whereas the their simple
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product is going to give rise to the unitary
operation and we can go ahead and find the
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solution here
so in many ways this whole process can
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be brought together by implementing each and
every step in the matrix whereas i we just
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pointed out the implementing of the half adder
on the other hand would mean that we have
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to implement a classical function which is
the sum of the two modulus and then carry
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on the product of the x one and x two x one
and x zero at the same time so these are our
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inputs and input qubits and these are our
outputs where we have the carry and the sum
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going in the last two bit and this is the
half adder that we are looking at
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so a generic design can therefore be implemented
by designing a matrix of this kind where all
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these elements would then be going through
the different principles as we have been
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discussing as of now and the specific reduced
design would then be comprised of a tiffoli
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which is a control control not and a cnot
to finally give rise to the carry forward
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one as well as the sum of the two which we
have used here on the control which is our
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x y so there are specific differences that
we have seen in terms of the classical versus
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the quantum bits as we discussed here the
classical bits were very basic in terms of
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off and on or just the specific numbers whereas
and they were mutually exclusive however in
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our particular case as we always know the
qubit has many many states which are a resultant
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of the initial gate zero and one and they
result in the superposition of states which
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are non continuous in nature and it gives
rise to entanglement and they can be described
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in reference to one another which are non
local properties which allows a set of you
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wish to be expressed a superposition of different
binary strings
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in terms of the qubit state being pure
what we have defined is that they are super
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position of their individual state with their
complex coefficients such that the square
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of the mod of those coefficients would give
rise to be equal to one some of this mod of
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this curve which means that they are normalized
and as such there would be eight possible
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states per cubic with this background let
us revisit the process or the principle that
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we are looking at which is the linear optical
quantum computing process where we have been
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essentially going over the details of these
quantum computing aspects with respect to
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the approaches of linear optical designs and
these are in some sense our photonic qubits
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which are our photons either in different
polarization states or as different momentum
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states they have their own advantages and
disadvantages as we have been discussing there
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are many other ways of using the optical principles
in to quantum computing but those are separate
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entities as of now we are just looking at
this particular approach we could also take
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advantage of photonic qubits with linear optics
in this particular process and the linear
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optical logic gates initially started off
with the theoretical idea but has been put
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into experimental realizations by using beam
splitters polarisers and wave plates there
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are also approaches which are clusters versus
one way quantum computing i don't know
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if we will be getting into these areas because
demonstration aspects of these are still very
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sparse but these are different ideas
once again linear optical one way quantum
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computing has been developed as a result of
this understanding which uses single photons
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linear optics as well as their measurement
and interface of photons and atomic ensembles
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have been used for quantum memory for polarization
of qubits and we will finally look at their
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summary and outlook as we go along in this
direction so revisiting this whole principle
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we can understand as to why we would like
to have this principle of optics coming into
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this picture of quantum computing and that
too linear and because that has the advantage
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where the information flow in the quantum
computing process can be carried on by the
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qubits which are subjects to design unitary
evolution which are being carried out in
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this particular case by optical approaches
so performing general transformation relies
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on the ability of the engineering arbitrary
interactions between the qubits the this task
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has been greatly simplified by following the
universal quantum computation theorem of lloyd
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and others which which have been worked on
in this area any unitary transform of an
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n qubit system can be implemented with single
qubit operations and quantum control not gates
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or equivalent to qubits gates that is what
we discussed in the beginning part of this
23:36.760 --> 23:44.520
lecture showing that it is important to realize
the university of these processes
23:44.520 --> 23:52.850
now the building block of these qubits are
our aspect of the superposition which not
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only its just a combination but also its a
coherence superposition such that the coefficients
24:00.390 --> 24:10.150
can be complex although they are mod squares
always add up to be equal to one they can
24:10.150 --> 24:15.990
also add together in terms of entanglement
in such a way such that the individual carries
24:15.990 --> 24:24.490
qubit carries no information at all but the
composite and together carries all the information
24:24.490 --> 24:28.480
the fact that the qubits can be incoherent
superposition and entangled states gives the
24:28.480 --> 24:33.790
extraordinary power to a quantum computer
that's what we know and that outperforms its
24:33.790 --> 24:37.500
classical counterparts
so whenever we are using any implementation
24:37.500 --> 24:42.890
approaches we have to see that this particular
approach or this particular advantage remains
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to our particular sense now there are many
different ways of looking at this from mathematical
24:50.660 --> 24:56.770
principles as well as several others there
is a subgroup which can be used which has
24:56.770 --> 25:02.600
the symmetry operational principles and it
can represent mostly as for example the two
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level quantum system has s u two symmetry
and that can represent a qubit there are
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many different ways of implementing qubits
for examples we have already talked about
25:16.130 --> 25:20.620
electron spins atoms with two relevant energy
levels these were all talked about in our
25:20.620 --> 25:27.480
introductions and we will be talking sometimes
later about superconducting josephson junctions
25:27.480 --> 25:33.770
and photon polarizations or special modes
but the more important part which we are dealing
25:33.770 --> 25:40.150
with right now are the photonic realization
of the qubits which is one of the most promising
25:40.150 --> 25:45.890
not only because they are easy you are important
but also because they are the ones which are
25:45.890 --> 25:52.770
important for quantum communication purposes
as well as for carrying forward quantum computing
25:52.770 --> 26:00.400
to multiple scalable levels that's what we
are after so in some sense having both
26:00.400 --> 26:05.570
polarizations encoding as we have been discussion
which depends on the horizontal and the vertical
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aspects of an horizontal vertical polarization
are one of these important aspects so the
26:13.390 --> 26:21.250
degrees of freedom in some sense given to
an individual photon is important so here
26:21.250 --> 26:28.240
is the basic idea here and individual photon
processes a few degrees of freedom each of
26:28.240 --> 26:32.910
which can in principle be used to carry the
information under appropriate experimental
26:32.910 --> 26:40.350
arrangements these degrees of freedom include
internal polarization orbital angular momentum
26:40.350 --> 26:49.660
spatial mode emission time frequency etcetera
here and before we have talked about these
26:49.660 --> 26:57.880
particular aspects which is the polarization
encoding which involves horizontal versus
26:57.880 --> 27:12.210
vertical polarization it could also have counter
clockwise versus clockwise circular polarizations
27:12.210 --> 27:21.120
either way it will have the polarization qubits
the path encoding could also involve their
27:21.120 --> 27:27.800
presence where they are here and there thats
the momentum approach and they can have spatial
27:27.800 --> 27:35.650
qubits which are based on their special modes
which could be the different modes of the
27:35.650 --> 27:46.280
qubits or the photon that we are looking at
now the aspects of photon polarization and
27:46.280 --> 27:54.650
its encoding as many different ways of looking
at it the quantum states of photons can be
27:54.650 --> 28:00.050
easily manipulated by simple linear optical
elements as we have been discussing it's not
28:00.050 --> 28:05.700
only interesting in its own right but also
has this high precision of about ninety nine
28:05.700 --> 28:13.260
point nine percent accuracy it's easily realized
with any single qubit rotations so robust
28:13.260 --> 28:20.720
to environmental noises photons have no charge
so they do not interact and create a problem
28:20.720 --> 28:27.200
for the other they are also the fastest information
carriers which is important for quantum communication
28:27.200 --> 28:36.060
and distributed quantum information processing
however the challenges are also not that simple
28:36.060 --> 28:41.870
difficulty of realizing two qubit gates for
photons is due to the lack of photon photon
28:41.870 --> 28:48.660
interaction the very process which makes
it robust also makes it difficult to scale
28:48.660 --> 28:58.580
it up so there are many newer approaches which
rely on utilization of nonlinear media
28:58.580 --> 29:04.950
and we will get into that also and the
other very important part is the storing of
29:04.950 --> 29:11.430
these photons for a reasonable long time
for this particular approach so there has
29:11.430 --> 29:16.370
always been this question as to whether it
is possible to scale up or do other things
29:16.370 --> 29:20.640
into the future but as demonstration purposes
this will also remained as a very important
29:20.640 --> 29:31.550
approach and in principle a lot of work
happened in the early two thousand were they
29:31.550 --> 29:39.080
have managed to show that it is possible to
show non deterministic quantum logic operations
29:39.080 --> 29:48.180
can be performed using linear optical elements
where in addition ancilla photons which are
29:48.180 --> 29:51.950
additional photons which are not participating
29:51.950 --> 29:58.060
in the actual process of the computation are
going to give the strength to this process
29:58.060 --> 30:05.690
and the post electron based output of single
photon detectors can also be utilized for
30:05.690 --> 30:14.750
further processing and robustness of the process
so this group and several others about their
30:14.750 --> 30:21.230
time were also able to demonstrate the success
rate of quantum logic arbitrarily close by
30:21.230 --> 30:25.480
using additional ancilla and detectors and
this has been a trend in the recent years
30:25.480 --> 30:31.850
a lot of developments beyond this has also
happened the non deterministic quantum
30:31.850 --> 30:36.570
logic gates based on out linear optics can
be used as a basic block for quantum information
30:36.570 --> 30:43.810
protocols even for efficient quantum computation
so certain part of the quantum computing development
30:43.810 --> 30:55.230
certainly benefits from the optical approaches
there are many different advantages the very
30:55.230 --> 31:03.220
important process of the laser cavity itself
has been utilized and while studying about
31:03.220 --> 31:09.780
different aspects of laser we talked about
kerr lens mode locking for making short pulses
31:09.780 --> 31:18.040
and that in itself has been found to be advantageous
for doing certain applications of quantum
31:18.040 --> 31:24.460
or demonstrating global search for instance
which we will do in this hopefully within
31:24.460 --> 31:32.690
this week the scheme itself is complicated
and in terms of the linear optical approaches
31:32.690 --> 31:37.750
because it may use complex interferometers
and it is often resource consuming because
31:37.750 --> 31:42.770
they are being linear often the number of
resources necessary for this processes quite
31:42.770 --> 31:47.340
high
however it's a real break too and the motives
31:47.340 --> 31:52.300
where many subsequent studies on linear optical
in quantum information protocols have been
31:52.300 --> 32:03.490
applied the there has been some recent
reviews and other work but what we will do
32:03.490 --> 32:08.890
in this particular lecture is to show you
how to implement a linear optical cnot gate
32:08.890 --> 32:17.380
we will demonstrate in some process here in
some sense this is a something which we discussed
32:17.380 --> 32:27.670
in the last class where we setup the universal
set of quantum logic gates and then we applied
32:27.670 --> 32:36.260
the hadamard in terms of the beam splitters
to be able to show that we were able to combine
32:36.260 --> 32:42.240
the input gates into a process where we would
be able to use them and similarly we would
32:42.240 --> 32:47.520
be doing the graphical representations of
the hadamard and the c not gates
32:47.520 --> 32:55.640
and since it's a process where many of this
has been already looked into let me end today's
32:55.640 --> 33:01.309
class because we have already come to a point
where we have covered most of these aspects
33:01.309 --> 33:08.490
before and so let us close this lecture by
mentioning that linear optical approaches
33:08.490 --> 33:15.110
to quantum computing and the various gates
that have been designed in this process based
33:15.110 --> 33:24.580
on the photon properties seem to be very effective
in many ways and we can utilize them to benefit
33:24.580 --> 33:32.550
and demonstrate quantum information processing
one of the very different approaches
33:32.550 --> 33:39.870
to quantum information processing with optics
has also come by in terms of using a laser
33:39.870 --> 33:46.040
cavity itself to demonstrate grover's algorithm
which we will do in the next class and i think
33:46.040 --> 33:51.440
you will enjoy that a lot so with this let
us close today's class and we will see you
33:51.440 --> 33:51.820
next week