WEBVTT
00:14.320 --> 00:20.320
this week we have been looking at the basics
of quantum computing once more as we have
00:20.320 --> 00:27.930
been looking into the implementation angles
from the last week we realized that they were
00:27.930 --> 00:35.480
certain aspects of the basic part where a
little bit more understanding or relook at
00:35.480 --> 00:44.710
them would help in many ways to better understand
and go forward with implementation ideas so
00:44.710 --> 00:51.570
with this idea we have started relooking into
the concepts of the very initial aspect of
00:51.570 --> 00:59.559
the qubits their interactions and in this
regard we just a address the idea of coming
00:59.559 --> 01:06.619
together of two qubits we are going ahead
with those and looking into their interactions
01:06.619 --> 01:11.780
and gates and will be again going back to
the implementations very soon as soon as we
01:11.780 --> 01:17.990
mix and bring these ideas back on board so
that they can be looked at in terms of the
01:17.990 --> 01:23.110
implementations that we are talking about
so in terms of two qubits the controlled not
01:23.110 --> 01:27.520
gate is the gate which are which is going
to be discussed here we have looked at these
01:27.520 --> 01:34.360
kinds of things before also its just say revision
in some sense of making sure that we are going
01:34.360 --> 01:41.220
back to this idea and i just mentioned earlier
in the lecture that these or gate is one of
01:41.220 --> 01:48.119
the classical gates which is x or this
is one of the reversible classical gates and
01:48.119 --> 01:54.710
therefore it can be implemented quantum mechanically
as in quantum mechanics we would like to only
01:54.710 --> 02:04.550
have reversible gates so its also known as
the addition modulo two in terms of the xor
02:04.550 --> 02:16.180
gate so once a c not is applied on two qubits
will be able to flipped the second bit based
02:16.180 --> 02:24.389
on the property of the first qubit
so the control axis on the first qubit which
02:24.389 --> 02:30.670
dictates whats going to happen to the second
qubit so here is the logic here so as long
02:30.670 --> 02:39.190
as the first qubit is zero nothing happens
to the second qubit however if the first qubit
02:39.190 --> 02:49.040
is one the second qubit is flipped so that
is the control not that we have seen so with
02:49.040 --> 02:58.650
one as the first qubit the control bit our
second qubit flips and so that's why you get
02:58.650 --> 03:06.950
to see the logic operational here and that
gives raise to the matrix where the first
03:06.950 --> 03:14.940
part where the control bit is not operational
is is not going to change is an identity where
03:14.940 --> 03:22.230
as in the other case where the control
bit is present is going to flip so its going
03:22.230 --> 03:36.010
to have it in the not form so this part the
first part of this looks like an identity
03:36.010 --> 03:45.010
where as the second part looks like the not
gate so that's the control not
03:45.010 --> 03:55.229
measuring of the qubits in this case as before
for all quantum system would be uncertain
03:55.229 --> 03:59.840
before the measurement is done however after
the measurement this state is certain it is
03:59.840 --> 04:06.500
either one of these zero zero zero one one
zero or one one like in the case of classical
04:06.500 --> 04:15.030
two bit system we have already discussed this
idea that whenever we have measurement made
04:15.030 --> 04:20.350
then it follows two the classical condition
so before the measurement this state of this
04:20.350 --> 04:28.870
system consisting of two qubits are uncertain
its a super position state which is being
04:28.870 --> 04:34.690
subjected to the gate applied however after
the measurement we can collapse into one of
04:34.690 --> 04:40.400
the four possibilities
so if we one to observe only the first qubit
04:40.400 --> 04:46.890
what are the conclusions that can be drawn
we expect that the system to be left in an
04:46.890 --> 04:53.430
uncertain state because we did not measure
the second qubit that can still be in a continuum
04:53.430 --> 05:00.960
of states the first qubit can be zero with
probability of alpha zero zero squared plus
05:00.960 --> 05:06.640
alpha zero one squared and one with probability
of alpha one zero squared and alpha one one
05:06.640 --> 05:16.120
squared its we call psi zero superscript i
as the post measurement state when we measure
05:16.120 --> 05:24.940
the first qubit and find it to be zero and
we call the other one where we are going to
05:24.940 --> 05:38.150
find at one as psi i one then will be finding
that these follow these formulas where each
05:38.150 --> 05:45.150
of them will give raise to their probabilities
and similarly if we label the second qubit
05:45.150 --> 05:51.730
measurement in a similar fashion will be finding
that they will follow this particular format
05:51.730 --> 05:58.670
of probability that means the probability
of measuring zero would be for the second
05:58.670 --> 06:04.170
qubit with be of this kind is the probability
of measuring one for the second qubit would
06:04.170 --> 06:11.280
be of this kind for bell states which is a
very special state of pair of qubits which
06:11.280 --> 06:20.250
cannot be broken into the individual qubits
that they come from the both these alpha zero
06:20.250 --> 06:27.310
zeros and one one are one by root two and
zero one and one zero are going to be zero
06:27.310 --> 06:33.840
when we measure the first qubit we get the
post measurement state as one one and zero
06:33.840 --> 06:39.670
zero but as we we measure the second qubit
we get the post measurement state as zero
06:39.670 --> 06:45.639
zero one one this is an amazing result because
the two measurements are correlated once we
06:45.639 --> 06:52.570
measure the first qubit we know exactly the
same result as when we measure the second
06:52.570 --> 07:01.310
one so that is a very important result that
the two measurements are correlated
07:01.310 --> 07:07.280
so measuring the first qubit gives us the
measurement that we have going to get from
07:07.280 --> 07:12.590
the second one measure so the two qubits need
not be physically constrained to be in the
07:12.590 --> 07:16.900
same location and yet because of the strong
coupling between them measurements performed
07:16.900 --> 07:23.389
on the second one allow us to determine the
state of the first one and so this is a very
07:23.389 --> 07:33.940
interesting principle which enables
the measurement of one correlated to the other
07:33.940 --> 07:47.560
one so this arrives at the idea of entanglement
which was discovered by schrodinger it can
07:47.560 --> 07:54.530
be formally defined as follows an entangled
pair is a single quantum system in a superposition
07:54.530 --> 08:00.190
of equally possible states the entangled state
contains no information about the individual
08:00.190 --> 08:08.460
particles only that they are opposite states
so this an important point that it does not
08:08.460 --> 08:16.590
contain any information about the individual
states or in other words an entangled pair
08:16.590 --> 08:27.250
can never be decomposed back to the individual
states which give raise to the entangled pair
08:27.250 --> 08:34.899
so this is one of the principles which led
einstein to loose faith on quantum mechanics
08:34.899 --> 08:41.589
and he came up with this principle that it
says spooky action at a distance because there
08:41.589 --> 08:49.600
is no requirement of how close these states
are to be for this particular property to
08:49.600 --> 08:55.870
be maintained we have looked at classical
gates and classical gates are there to generally
08:55.870 --> 09:02.120
implement boolean functions they are not reversible
we cannot recover the input knowing the output
09:02.120 --> 09:06.900
which means that there is an irreversible
loss of information when we are looking at
09:06.900 --> 09:15.980
classical gate that is one important aspect
of the classical nature of computing and these
09:15.980 --> 09:27.580
are the typical classical gates which are
essential for the basis of a computer to work
09:27.580 --> 09:35.300
so these are the set of basis gates on the
basis of which a computation can be performed
09:35.300 --> 09:43.320
so as the not gate and gate the nand gate
the or gate the nor gate and the xor gate
09:43.320 --> 09:52.510
most of the computations can be belt on this
particular few fundamental basis gates in
09:52.510 --> 10:00.820
terms of classical computation
so the ideas of these a very clear not gate
10:00.820 --> 10:08.880
is essentially just opposite so y is equal
to not of x and gate always involves more
10:08.880 --> 10:17.580
than two states so we would have two states
come in to give raise to the third state the
10:17.580 --> 10:27.510
nand gate the not and gate also requires minimum
of three states the or gate also has two inputs
10:27.510 --> 10:36.630
to give raise to one output which is one of
the two so not reversible not reversible not
10:36.630 --> 10:44.300
reversible the not or gate is also not reversible
because the input and the output are not the
10:44.300 --> 10:56.820
same number however the xor gate is something
where as we will see later on can be correlated
10:56.820 --> 11:03.180
to a condition which can be made reversible
but as it directly looks like here its also
11:03.180 --> 11:12.800
a non reversible condition for the classical
case the only reversible gate in that sense
11:12.800 --> 11:19.960
is the not gate so the idea of the gate in
terms of quantum mechanic where is the operation
11:19.960 --> 11:30.089
of a square matrix on the qubits that we have
so for a single qubit it will be a two by
11:30.089 --> 11:39.210
two matrix which will operate on the input
qubit to give raise to the final result so
11:39.210 --> 11:44.810
this is the square matrix which defines the
operation
11:44.810 --> 11:51.920
so the basic one qubit gates are the identity
gate which leaves the qubit unchanged the
11:51.920 --> 11:58.860
x or the not gate which transposes the components
of an input bit the y gate which rotates the
11:58.860 --> 12:08.440
qubit around the y axis of the bloch sphere
by pi radians the z gate which flips the sign
12:08.440 --> 12:18.680
of a qubit and the hadamard gate which makes
equal superposition of the individual qubits
12:18.680 --> 12:30.000
now all of these qubit gates in terms of quantum
mechanics are going to be reversible because
12:30.000 --> 12:41.510
that's the basic requirement of quantum mechanics
the other important aspects of these gates
12:41.510 --> 12:49.870
are for identity transformations the pauli
matrices are the once which work on the spin
12:49.870 --> 12:57.300
basically they rotate or producer it identity
matrix and the hadamard which makes an equals
12:57.300 --> 13:03.580
super position so these three are the most
important aspects of the gates that we are
13:03.580 --> 13:08.339
looking at the identity transformation is
the one which basically just keeps the same
13:08.339 --> 13:25.330
qubit the x gate is the not gate this
also a part of the pauli matrices the y which
13:25.330 --> 13:35.529
has i described is rotation and the z gate
which was the one which flips the sign of
13:35.529 --> 13:44.260
the qubit and the hadamard one these are one
basically next equals super position of
13:44.260 --> 13:55.370
the two qubits the c not that we had looked
at earlier also is a two qubit gate which
13:55.370 --> 14:02.360
requires an control input as a result of which
the target input is going to be undergoing
14:02.360 --> 14:11.380
addition modulo two the control bit is transferred
to the output as it is and that's the quantum
14:11.380 --> 14:18.220
nature of this particular control gate going
to be reversible so there are two inputs control
14:18.220 --> 14:26.399
and target and there are two outputs one is
the control as it is and the output which
14:26.399 --> 14:31.760
is going to be addition modulo two the target
qubit is unaltered if the control qubit is
14:31.760 --> 14:38.399
zero and is flipped if the control qubit is
one so that's the one that we have shown here
14:38.399 --> 14:45.720
which keeps raise to this again the same matrix
that we discussed before with the first upper
14:45.720 --> 14:54.529
part to be the one which essential it is identity
doesnt do anything keeps the same form but
14:54.529 --> 15:03.980
is the other one flips it is a not gate
the two input qubits of a two qubit gates
15:03.980 --> 15:11.450
and the super position of the two states and
then they can be put together as a matrix
15:11.450 --> 15:26.390
multiplication form to give raise to the overall
states which are going to undergo tensor multiplication
15:26.390 --> 15:32.730
to give raise to the final results the state
space dimension of the classical and quantum
15:32.730 --> 15:38.270
systems are also quite different the individual
states space of n particles combine quantum
15:38.270 --> 15:45.501
mechanically through the tensor product so
if x and y are vectors then that tensor product
15:45.501 --> 15:51.700
is also a vector but its dimension is now
the multiple of the two while is the vector
15:51.700 --> 15:57.649
[duct/product] product has addition of the
dimensions of the two so for example if dimension
15:57.649 --> 16:03.959
of x and dimension of one dimension of y is
ten then the tensor product of the two vectors
16:03.959 --> 16:09.339
has dimension hundred while the vector product
has dimension of twenty and this is the reason
16:09.339 --> 16:16.190
for the exponential nature of the quantum
computing process
16:16.190 --> 16:22.290
parallelism and quantum computers in the quantum
system the amount of parallelism increases
16:22.290 --> 16:27.800
exponentially with the size of the system
thus with the number of qubits in quantum
16:27.800 --> 16:33.350
systems the amount of parallelism increases
exponentially with the size of the system
16:33.350 --> 16:40.820
thus with the number of qubits for example
twenty one qubit quantum computer is twice
16:40.820 --> 16:47.700
as powerful as a twenty qubit quantum computer
so that's the exponential nature of the problem
16:47.700 --> 16:53.060
that is the advantage in the quantum computers
a quantum computer will enable us to solve
16:53.060 --> 16:58.700
problems the very large state space that's
the biggest advantage of the quantum parallelism
16:58.700 --> 17:03.210
that we take advantage of
in case of the quantum circuit as we have
17:03.210 --> 17:09.890
been discussing if we have a given function
f of x we can construct a reversible quantum
17:09.890 --> 17:17.059
circuit consisting of say the fredking gates
only capable of transforming two qubits as
17:17.059 --> 17:25.049
follows the function f of x is hardwired into
the circuit so this is how looks like which
17:25.049 --> 17:30.890
is sort of the cnot that we looked at if
the second input is zero then the transformation
17:30.890 --> 17:38.909
is done by the circuit is given by as this
one and we apply the first qubit through
17:38.909 --> 17:45.859
a hadamard gate then produce state whereas
the resulting state of the circuit is this
17:45.859 --> 17:55.100
the output state contains information about
f of zero and f of one the output of the quantum
17:55.100 --> 18:00.240
circuit contains information of both f zero
and f one this property of quantum circuit
18:00.240 --> 18:07.059
is called quantum parallelism the quantum
parallelism allows us to construct the entire
18:07.059 --> 18:14.210
truth table of quantum gate arrays having
two n entries at once in a classical system
18:14.210 --> 18:19.659
we can compute the truth table in one time
step with two to the power n gate arrays running
18:19.659 --> 18:26.809
in parallel or we need two n times steps with
a single gate array
18:26.809 --> 18:33.159
if we start with n qubits each of the state
zero and we apply walsh hadamard transformation
18:33.159 --> 18:38.610
we are able to do this particular gate principle
so we have seen all these gates before so
18:38.610 --> 18:42.990
i am not going into a details of it walsh
hadamard is the one which we use for the grovers
18:42.990 --> 18:51.200
algorithm and so we can go an with these number
of dimensions which go higher and we have
18:51.200 --> 18:58.340
given this new principle where we have actually
telling you how these dimensionality of the
18:58.340 --> 19:04.919
problem is increasing because you are doing
quantum way your dimensionality is going as
19:04.919 --> 19:13.980
the product of the cases so here we can take
the hadamard on a particular step and we can
19:13.980 --> 19:23.100
repeatedly apply the hadamard on n qubits
which are undergoing tensor products and as
19:23.100 --> 19:37.070
result we can get the superposition of states
which have undergone the hadamard operation
19:37.070 --> 19:46.879
so we can apply this unitary transform
on that as a result of this operation so a
19:46.879 --> 19:53.359
hadamard operation of this kind so out of
looks like this and this can be utilized for
19:53.359 --> 19:58.179
looking at the deutschs problem once again
the deutschs problem was the one were was
19:58.179 --> 20:06.640
going to look for the balanced function whether
the function is balanced or not so if we consider
20:06.640 --> 20:12.929
black box characterized by a transfer function
that maps a single input bit into an output
20:12.929 --> 20:21.070
takes the same amount of time t to carry on
each the four possible mapping performed by
20:21.070 --> 20:28.520
transfer function f of x in the black box
the problem posed is to distinguish if the
20:28.520 --> 20:35.609
function is going to be equal for both f equal
to zero and f equal to one or its going to
20:35.609 --> 20:43.480
be unbalanced this is f zero is not equal
to f one so the times cases involved in each
20:43.480 --> 20:52.169
of these cases are are shown here the unitary
operation which goes for this takes this into
20:52.169 --> 21:01.820
the operation procedure where we can do
all of this in one shot so the quantum circuit
21:01.820 --> 21:06.820
to solve the deutschs problem can be given
in this particular form which has been discussed
21:06.820 --> 21:13.259
earlier also but here we are just telling
you the details in some sense we have the
21:13.259 --> 21:20.649
initial inputs zero and one which goes through
first hadamard operation on both the inputs
21:20.649 --> 21:28.229
and then it is undergoing a unitary transform
where the essential part of the unitary transform
21:28.229 --> 21:33.929
is to take one part of bit into the modulo
function whereas the other one remains the
21:33.929 --> 21:39.160
same and then the first part undergoes the
hadamard transform while the second one is
21:39.160 --> 21:44.379
essentially gives raise to the final answer
so in terms of the matrice operations and
21:44.379 --> 21:50.470
the tensor products these are the basic steps
which undergo in terms of how this entire
21:50.470 --> 21:58.919
process goes the overall gates that we
are applying are multiple hadamard gates which
21:58.919 --> 22:05.249
are getting applied in a tense of form
which grows in terms of the size these are
22:05.249 --> 22:12.919
then looked into by applying the overall
gate so here is the g one gate which has been
22:12.919 --> 22:18.919
created and that's been applied to the original
hadmard transformed input which then gives
22:18.919 --> 22:27.299
raise to the final result in first step
and then that is been looked into as a combine
22:27.299 --> 22:33.649
state which can be broken down into two different
states as that then forms a basis sets of
22:33.649 --> 22:42.129
x and y this is the first form of the part
which goes in when we look at this point so
22:42.129 --> 22:48.169
at one this is what we have just achieved
this is the process which takes us from
22:48.169 --> 22:56.419
zero to one so here this entire step goes
from zero to one in our earlier slide here
22:56.419 --> 23:02.460
next it undergoes an interaction with the
unitary transform where it undergoes the
23:02.460 --> 23:15.220
function modulo applying on the y qubit
and it gives raise to a functional form [whic/which]
23:15.220 --> 23:23.590
which would be of this kind so this is the
part where we have the the form of the function
23:23.590 --> 23:31.629
interacting with a y qubit and so the y qubit
takes the this particular format follows the
23:31.629 --> 23:39.000
principles that if f of x is equal to zero
it will be the positive value whereas if
23:39.000 --> 23:47.419
it is one it will give the negative value
so the x and y are the ones that we have
23:47.419 --> 23:52.389
started of with after we have gone through
the hadamard transform then we have applied
23:52.389 --> 24:03.019
the gate so these gates essentially take
the first qubit into the unitary transform
24:03.019 --> 24:10.549
which gives raise to the solution where as
the second qubit undergoes the part were
24:10.549 --> 24:16.559
the solutions would be of this kind and based
on their individual results will be able
24:16.559 --> 24:23.659
to have the different conditions that we
give raise to them so in the end will be having
24:23.659 --> 24:33.279
two conditions one when our function is
equal in one case and it will be different
24:33.279 --> 24:42.229
when our function is not equal and so by idea
that our interaction for the part which
24:42.229 --> 24:49.840
is the second qubit would give raise to this
results will be able to understand how this
24:49.840 --> 24:55.249
interaction goes on
so in the third part is again the first
24:55.249 --> 25:01.720
part goes to hadamard transfer and so that
the upper qubit goes through a hadamard transfer
25:01.720 --> 25:09.039
rate however the second part of the qubit
remains as is so by measuring the first output
25:09.039 --> 25:17.909
qubit we are able to determine whether
f of zero addition modulo of f one is performing
25:17.909 --> 25:26.269
a single evaluation or not and so if we
go ahead with this result will be actually
25:26.269 --> 25:33.340
getting either zero when f of zero is equal
to for one and it will be one when f of zero
25:33.340 --> 25:43.700
is not equal to one so that's the basic
idea behind the grovers algorithm that we
25:43.700 --> 25:48.399
had looked at before and in many cases this
implementations have been utilized as we have
25:48.399 --> 25:53.379
done in our last few lecture in this lecture
we have been revisiting the aspects of quantum
25:53.379 --> 25:57.889
computing basics in relation to the implementations
that we have been undergoing over the last
25:57.889 --> 26:02.159
few weeks
in the next lecture we will be dealing with
26:02.159 --> 26:10.149
some more of the basics and some other implementation
aspects as we develop the basics in relation
26:10.149 --> 26:15.950
to the implementations and i look forward
to having you in the class
26:15.950 --> 26:16.259
thank you