this week we have been looking at the basics of quantum computing once more as we have
been looking into the implementation angles from the last week we realized that they were
certain aspects of the basic part where a little bit more understanding or relook at
them would help in many ways to better understand and go forward with implementation ideas so
with this idea we have started relooking into the concepts of the very initial aspect of
the qubits their interactions and in this regard we just a address the idea of coming
together of two qubits we are going ahead with those and looking into their interactions
and gates and will be again going back to the implementations very soon as soon as we
mix and bring these ideas back on board so that they can be looked at in terms of the
implementations that we are talking about so in terms of two qubits the controlled not
gate is the gate which are which is going to be discussed here we have looked at these
kinds of things before also its just say revision in some sense of making sure that we are going
back to this idea and i just mentioned earlier in the lecture that these or gate is one of
the classical gates which is x or this is one of the reversible classical gates and
therefore it can be implemented quantum mechanically as in quantum mechanics we would like to only
have reversible gates so its also known as the addition modulo two in terms of the xor
gate so once a c not is applied on two qubits will be able to flipped the second bit based
on the property of the first qubit so the control axis on the first qubit which
dictates whats going to happen to the second qubit so here is the logic here so as long
as the first qubit is zero nothing happens to the second qubit however if the first qubit
is one the second qubit is flipped so that is the control not that we have seen so with
one as the first qubit the control bit our second qubit flips and so that's why you get
to see the logic operational here and that gives raise to the matrix where the first
part where the control bit is not operational is is not going to change is an identity where
as in the other case where the control bit is present is going to flip so its going
to have it in the not form so this part the first part of this looks like an identity
where as the second part looks like the not gate so that's the control not
measuring of the qubits in this case as before for all quantum system would be uncertain
before the measurement is done however after the measurement this state is certain it is
either one of these zero zero zero one one zero or one one like in the case of classical
two bit system we have already discussed this idea that whenever we have measurement made
then it follows two the classical condition so before the measurement this state of this
system consisting of two qubits are uncertain its a super position state which is being
subjected to the gate applied however after the measurement we can collapse into one of
the four possibilities so if we one to observe only the first qubit
what are the conclusions that can be drawn we expect that the system to be left in an
uncertain state because we did not measure the second qubit that can still be in a continuum
of states the first qubit can be zero with probability of alpha zero zero squared plus
alpha zero one squared and one with probability of alpha one zero squared and alpha one one
squared its we call psi zero superscript i as the post measurement state when we measure
the first qubit and find it to be zero and we call the other one where we are going to
find at one as psi i one then will be finding that these follow these formulas where each
of them will give raise to their probabilities and similarly if we label the second qubit
measurement in a similar fashion will be finding that they will follow this particular format
of probability that means the probability of measuring zero would be for the second
qubit with be of this kind is the probability of measuring one for the second qubit would
be of this kind for bell states which is a very special state of pair of qubits which
cannot be broken into the individual qubits that they come from the both these alpha zero
zeros and one one are one by root two and zero one and one zero are going to be zero
when we measure the first qubit we get the post measurement state as one one and zero
zero but as we we measure the second qubit we get the post measurement state as zero
zero one one this is an amazing result because the two measurements are correlated once we
measure the first qubit we know exactly the same result as when we measure the second
one so that is a very important result that the two measurements are correlated
so measuring the first qubit gives us the measurement that we have going to get from
the second one measure so the two qubits need not be physically constrained to be in the
same location and yet because of the strong coupling between them measurements performed
on the second one allow us to determine the state of the first one and so this is a very
interesting principle which enables the measurement of one correlated to the other
one so this arrives at the idea of entanglement which was discovered by schrodinger it can
be formally defined as follows an entangled pair is a single quantum system in a superposition
of equally possible states the entangled state contains no information about the individual
particles only that they are opposite states so this an important point that it does not
contain any information about the individual states or in other words an entangled pair
can never be decomposed back to the individual states which give raise to the entangled pair
so this is one of the principles which led einstein to loose faith on quantum mechanics
and he came up with this principle that it says spooky action at a distance because there
is no requirement of how close these states are to be for this particular property to
be maintained we have looked at classical gates and classical gates are there to generally
implement boolean functions they are not reversible we cannot recover the input knowing the output
which means that there is an irreversible loss of information when we are looking at
classical gate that is one important aspect of the classical nature of computing and these
are the typical classical gates which are essential for the basis of a computer to work
so these are the set of basis gates on the basis of which a computation can be performed
so as the not gate and gate the nand gate the or gate the nor gate and the xor gate
most of the computations can be belt on this particular few fundamental basis gates in
terms of classical computation so the ideas of these a very clear not gate
is essentially just opposite so y is equal to not of x and gate always involves more
than two states so we would have two states come in to give raise to the third state the
nand gate the not and gate also requires minimum of three states the or gate also has two inputs
to give raise to one output which is one of the two so not reversible not reversible not
reversible the not or gate is also not reversible because the input and the output are not the
same number however the xor gate is something where as we will see later on can be correlated
to a condition which can be made reversible but as it directly looks like here its also
a non reversible condition for the classical case the only reversible gate in that sense
is the not gate so the idea of the gate in terms of quantum mechanic where is the operation
of a square matrix on the qubits that we have so for a single qubit it will be a two by
two matrix which will operate on the input qubit to give raise to the final result so
this is the square matrix which defines the operation
so the basic one qubit gates are the identity gate which leaves the qubit unchanged the
x or the not gate which transposes the components of an input bit the y gate which rotates the
qubit around the y axis of the bloch sphere by pi radians the z gate which flips the sign
of a qubit and the hadamard gate which makes equal superposition of the individual qubits
now all of these qubit gates in terms of quantum mechanics are going to be reversible because
that's the basic requirement of quantum mechanics the other important aspects of these gates
are for identity transformations the pauli matrices are the once which work on the spin
basically they rotate or producer it identity matrix and the hadamard which makes an equals
super position so these three are the most important aspects of the gates that we are
looking at the identity transformation is the one which basically just keeps the same
qubit the x gate is the not gate this also a part of the pauli matrices the y which
has i described is rotation and the z gate which was the one which flips the sign of
the qubit and the hadamard one these are one basically next equals super position of
the two qubits the c not that we had looked at earlier also is a two qubit gate which
requires an control input as a result of which the target input is going to be undergoing
addition modulo two the control bit is transferred to the output as it is and that's the quantum
nature of this particular control gate going to be reversible so there are two inputs control
and target and there are two outputs one is the control as it is and the output which
is going to be addition modulo two the target qubit is unaltered if the control qubit is
zero and is flipped if the control qubit is one so that's the one that we have shown here
which keeps raise to this again the same matrix that we discussed before with the first upper
part to be the one which essential it is identity doesnt do anything keeps the same form but
is the other one flips it is a not gate the two input qubits of a two qubit gates
and the super position of the two states and then they can be put together as a matrix
multiplication form to give raise to the overall states which are going to undergo tensor multiplication
to give raise to the final results the state space dimension of the classical and quantum
systems are also quite different the individual states space of n particles combine quantum
mechanically through the tensor product so if x and y are vectors then that tensor product
is also a vector but its dimension is now the multiple of the two while is the vector
[duct/product] product has addition of the dimensions of the two so for example if dimension
of x and dimension of one dimension of y is ten then the tensor product of the two vectors
has dimension hundred while the vector product has dimension of twenty and this is the reason
for the exponential nature of the quantum computing process
parallelism and quantum computers in the quantum system the amount of parallelism increases
exponentially with the size of the system thus with the number of qubits in quantum
systems the amount of parallelism increases exponentially with the size of the system
thus with the number of qubits for example twenty one qubit quantum computer is twice
as powerful as a twenty qubit quantum computer so that's the exponential nature of the problem
that is the advantage in the quantum computers a quantum computer will enable us to solve
problems the very large state space that's the biggest advantage of the quantum parallelism
that we take advantage of in case of the quantum circuit as we have
been discussing if we have a given function f of x we can construct a reversible quantum
circuit consisting of say the fredking gates only capable of transforming two qubits as
follows the function f of x is hardwired into the circuit so this is how looks like which
is sort of the cnot that we looked at if the second input is zero then the transformation
is done by the circuit is given by as this one and we apply the first qubit through
a hadamard gate then produce state whereas the resulting state of the circuit is this
the output state contains information about f of zero and f of one the output of the quantum
circuit contains information of both f zero and f one this property of quantum circuit
is called quantum parallelism the quantum parallelism allows us to construct the entire
truth table of quantum gate arrays having two n entries at once in a classical system
we can compute the truth table in one time step with two to the power n gate arrays running
in parallel or we need two n times steps with a single gate array
if we start with n qubits each of the state zero and we apply walsh hadamard transformation
we are able to do this particular gate principle so we have seen all these gates before so
i am not going into a details of it walsh hadamard is the one which we use for the grovers
algorithm and so we can go an with these number of dimensions which go higher and we have
given this new principle where we have actually telling you how these dimensionality of the
problem is increasing because you are doing quantum way your dimensionality is going as
the product of the cases so here we can take the hadamard on a particular step and we can
repeatedly apply the hadamard on n qubits which are undergoing tensor products and as
result we can get the superposition of states which have undergone the hadamard operation
so we can apply this unitary transform on that as a result of this operation so a
hadamard operation of this kind so out of looks like this and this can be utilized for
looking at the deutschs problem once again the deutschs problem was the one were was
going to look for the balanced function whether the function is balanced or not so if we consider
black box characterized by a transfer function that maps a single input bit into an output
takes the same amount of time t to carry on each the four possible mapping performed by
transfer function f of x in the black box the problem posed is to distinguish if the
function is going to be equal for both f equal to zero and f equal to one or its going to
be unbalanced this is f zero is not equal to f one so the times cases involved in each
of these cases are are shown here the unitary operation which goes for this takes this into
the operation procedure where we can do all of this in one shot so the quantum circuit
to solve the deutschs problem can be given in this particular form which has been discussed
earlier also but here we are just telling you the details in some sense we have the
initial inputs zero and one which goes through first hadamard operation on both the inputs
and then it is undergoing a unitary transform where the essential part of the unitary transform
is to take one part of bit into the modulo function whereas the other one remains the
same and then the first part undergoes the hadamard transform while the second one is
essentially gives raise to the final answer so in terms of the matrice operations and
the tensor products these are the basic steps which undergo in terms of how this entire
process goes the overall gates that we are applying are multiple hadamard gates which
are getting applied in a tense of form which grows in terms of the size these are
then looked into by applying the overall gate so here is the g one gate which has been
created and that's been applied to the original hadmard transformed input which then gives
raise to the final result in first step and then that is been looked into as a combine
state which can be broken down into two different states as that then forms a basis sets of
x and y this is the first form of the part which goes in when we look at this point so
at one this is what we have just achieved this is the process which takes us from
zero to one so here this entire step goes from zero to one in our earlier slide here
next it undergoes an interaction with the unitary transform where it undergoes the
function modulo applying on the y qubit and it gives raise to a functional form [whic/which]
which would be of this kind so this is the part where we have the the form of the function
interacting with a y qubit and so the y qubit takes the this particular format follows the
principles that if f of x is equal to zero it will be the positive value whereas if
it is one it will give the negative value so the x and y are the ones that we have
started of with after we have gone through the hadamard transform then we have applied
the gate so these gates essentially take the first qubit into the unitary transform
which gives raise to the solution where as the second qubit undergoes the part were
the solutions would be of this kind and based on their individual results will be able
to have the different conditions that we give raise to them so in the end will be having
two conditions one when our function is equal in one case and it will be different
when our function is not equal and so by idea that our interaction for the part which
is the second qubit would give raise to this results will be able to understand how this
interaction goes on so in the third part is again the first
part goes to hadamard transfer and so that the upper qubit goes through a hadamard transfer
rate however the second part of the qubit remains as is so by measuring the first output
qubit we are able to determine whether f of zero addition modulo of f one is performing
a single evaluation or not and so if we go ahead with this result will be actually
getting either zero when f of zero is equal to for one and it will be one when f of zero
is not equal to one so that's the basic idea behind the grovers algorithm that we
had looked at before and in many cases this implementations have been utilized as we have
done in our last few lecture in this lecture we have been revisiting the aspects of quantum
computing basics in relation to the implementations that we have been undergoing over the last
few weeks in the next lecture we will be dealing with
some more of the basics and some other implementation aspects as we develop the basics in relation
to the implementations and i look forward to having you in the class
thank you