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