So, let us ah start a new subject ah or rather
ah new module in ah the study of quantum mechanics
that we have engaged in um and it is called
as the Quantum Computing and ah [vocalized-noise]
possibly it is one of the most interesting
topics that keep the scientist busy over the
next ah decade or so because there are lots
of developments that are happening and lots
of open problems which need ah to be solved
and attended.
So, in this particular lecture or the section
of the lecture, I will give a [vocalized-noise]
brief introduction to this quantum computing
and sort of this will be more like a popular
introduction which should be acceptable or
rather understandable for more general public
and then, I will go on to the details of various
subtopics that we will mention in this ah
lecture [noise].
So, ah once again we go back to this Schrodinger's
cat which we have ah talked about earlier.
So, just to remind you of the situation that
there is a cat which is left alone in a room
and nobody is watching it. There is hammer
that you can see which is ah hanging on a
green colored ah portion kept in a bottle
[vocalized-noise] and ah this hammer is connected
to a pulley. So, if the cat comes and [vocalized-noise]
disturbs the pulley, the hammer will fall
onto the glass ah tube and the portion [vocalized-noise]
will come out and the portion is assumably
poisonous. So, the cat will die and if he
decides to stay away from, then it will be
alive and if it does not ah you know I mean
tamper with the pulley, then the hammer will
be where it is and there will be no spilling
out of that poison and it will be safe.
So, unless one does a measurement, one does
not know whether it is dead or alive. So,
it is in the superposition of state in the
mind of an observer before an observation
is made [vocalized-noise]. So, it is [vocalized-noise]
in a state which may be given by [vocalized-noise]
a superposition ah superpose state ah which
is a psi alive [noise] plus psi dead and just
to have ah proper normalization, we can write
1 by root 2, so that the probability of ah
being it alive is half and the probability
of being dead is also half.
Now, as soon as one does a measurement, it
collapses into one of the available states
which is either dead or alive and say it is
alive, then ah the amplitude that is associated
with alive becomes 1 and the other amplitude
that is associated with the state dead is
equal to 0 [vocalized-noise]. So, as long
as one has not made any measurement, it is
in the superposition of state.
Let us talk about this coin, tossing of a
coin and bias coin. So, it will ah be ah when
it spins goes up and then, come down to the
ground either between your palm or on the
ground. It is found in one of the states that
is head and tail. So, once you make a measurement,
once you want to decide say who would bat
first in cricket, it collapses into one of
these head or tail. So, quantum mechanical
coin in a sense is always spining because
it is in the superposition of state and never
comes down. The moment it comes down and you
make a measurement, it becomes classical.
So, measurement actually in some sense kills
quantum mechanics.
So, there are ah various such ah [vocalized-noise]
examples of ah these ah two state systems
just like it is ah mentioned here and we are
given example of coin tossing giving head
or tail or it could be just the two states
of a spin half particle 
which are ah given as up down [vocalized-noise]
or take as on off stat of a switch or it could
be the ground voltage B equal to 0 and an
excited voltage which is say V equal to sum
V [vocalized-noise]. So, V can also be taken
as a two state system. So, we will show that
this two stage [vocalized-noise] system and
the superposition that ah goes along with,
it forms ah an important part of this quantum
computing or the quantum information has name
course.
So, ah we will ah discuss what are called
as qubits which are nothing, but just an amalgamation
of the two words quantum and bits. A bit are
similar in the context of ah [noise] classical
computers. Bits are 1 and 0 and every number
is formed by the combination of 1 and 0 and
when we discuss it in the context of quantum
mechanics, then it is called as qubit. So,
we are talking about particularly ah two level
quantum system. We can actually talk about
multi level quantum system if we just briefly
will touch, but the two level systems are
most ah important and convenient for us to
discuss.
So, as we have just said that there are ah
two [noise] level quantum systems, many of
them such as photons having ah the polarization
of photons, the spins of the electrons, the
nucleus spins and and a large variety of them
as we have just seen [vocalized-noise]. So,
ah let us just a draw two coordinate axis
and label them as 1 and 0. You can also label
them as ah dead or alive, you can also label
them as up or down in the context of the spin
of particles [vocalized-noise], but let us
just talk about 1 and 0 just like the classical
bits are being introduced [vocalized-noise].
So, these are the 0s and 1s are the computational
basis states which are going to be useful
for our discussion.
So, we are just looking at a vector which
[vocalized-noise] is formed by the combination
of the superposition of 0 and 1. Just recall
that we have written down state like this
in the last ah slide of the cat being dead
and alive. So, the 0 state corresponds to
dead and 1 corresponds to alive with their
individual ah amplitudes given by alpha and
beta. So, this is a vector that is drawn in
that space which is spanned by 0 and 1. There
is a course on normalization condition which
is what we have seen ah there is a half possibility
of the cat being dead and the half of it being
alive.
So, the amplitude square ah should be normalized
that is the alpha [vocalized-noise] mod alpha
square plus mod beta square should be equal
to 1. So, this is a statement made by Richard
Feynman. He said that all we do is draw little
arrows on a piece of paper [vocalized-noise].
So, you see that ah with 0 and 1, one could
write an unique classical number which is
usually the numbers that are you know fed
into the classical computers that we all are
familiar with, however in this particular
case, there can be infinite number of numbers
that can be formed with different choices
of alpha and beta and the only constraint
being alpha square plus beta square of the
mod alpha square plus the mod beta square
equal to 1.
So, say for example, alpha equal to or alpha
square equal to 0.99 and beta square equal
to 0.01 would give rise to a particular state
and [vocalized-noise] another distance that
can be obtained by taking alpha mod alpha
square equal to 0.98 and mode beta square
equal to 0.02. So, they are distinct quantum
states and could be made to carry ah distinct
information, ok and there are infinite number
of such combinations that are possible. So,
just by using bits 0 and 1 ah in a super positions
sense, we can actually ah built an infinite
number or rather store infinite number of
numbers.
So, in some sense this talk about ah postulate
or other piece of information which is important
in this context. So, it is associated with
any quantum system. There is a complex vector
space known as State Space, ok [vocalized-noise]
and the state space that we are aware of are
all complete which means they are orthogonal
to each other. Each entry is orthogonal to
the other and they are also normalize. So,
they are orthonormalized and in that sense
they are complete and ah an infinite vector
space, he has a special name which is called
as a hilbert space after the name of a mathematician
called Hilbert. The state of a closed quantum
system is a unit vector in the state space
just as in the slide that we have seen that
this is that vector. So, a state of a system
is represented by this vector which is alpha
0 and beta 1 where clearly 0 and 1 from the
coordinate axis for this particular problem.
We will of course mainly work with qubits
as I told which have a state space which we
write as C2 because these are complex vector
spaces. So, we write it with C which denotes
a complex ah vector space and 2 denotes a
two-dimensional ah ah complex vector space
[vocalized-noise]. So, a qubit alpha and 0
alpha on 0 and beta on 1 is actually written
[vocalized-noise] by two column vector which
is alpha beta. So, this is a notation that
is somewhat universally accepted and one can
simply write within like a column vector alpha
beta. It would mean that we are talking about
qubit with amplitudes alpha and beta corresponding
to 0 and 1 state.
Few more ah conventions if ah would actually
help us to understand things better ah [vocalized-noise].
So, we write vectors in state space as psi,
ok ah which is equal to alpha ah multiplied
by and 0 cat plus beta multiplied by 1 cat.
So, is called as a cat notation which you
are aware of and we nearly assumed that ah
all are physical systems are finite dimensional
ah state spaces. Of course, ah for the sake
of learning quantum mechanics, we have seen
that ah they are actually they could be infinite
dimensional. In fact, all these x and p, the
position in the momentum the basis in which
ah they are written are actually infinite
dimensional [vocalized-noise], but ah for
[vocalized-noise] ah the needs for this particular
ah subject, we can think of finite dimensional
state spaces and they are ah orthogonal ah
orthonormal to each other [vocalized-noise]
and not only that if they are not orthogonal,
they can be organized by a technique which
is called as a Gram Schmidt Orthogonalization.
So, ah we can actually think of dimensional
space instead of two-dimensional space and
ah we can write down the state space or a
state vector psi as alpha 0 0 and alpha 1
1 alpha 2 2 and all that and all the way going
up to alpha d minus 1 multiplied by the cat
d minus 1 and in a shorthand notation, it
will ah be alpha 0 alpha 1 alpha 2 and all
that 2 alpha d minus 1 and as we have learnt
earlier that this corresponds to d dimensional
complex vector space. So, this is called as
qudit, instead of qubit dit for d dimensional
[vocalized-noise] space.
How these ah concepts of quantum entanglement
and various other things ah would be ah elaborated
ah later in the next [vocalized-noise] lecture
on the next discussion, but let us priory
define them and try to at least ah get ourselves
motivated that these are important things
to study in the context of quantum information
[vocalized-noise]. So, quantum entanglement
is uniquely quantum mechanical phenomena and
has no classical analogue and this is what
it means [vocalized-noise]. It is a property
of a multi qubit which is a tensor product
of individual qubits and can be thought of
as a resource in quantum computing [noise].
The key to entanglement is the property that
the state space cannot be decomposed into
components spaces. So, the composite space
remains composite and no combination or rather
no choice is of some ah complex corporations.
We would be able to write that as a product
of two ah spaces or two qubits [vocalized-noise].
So, in summary if you want to just say before
we do things in details, in summary by measuring
1 qubit it is possible to affect the measurement
of other qubits or qubits in the system and
this also has relevance to EPR paradox. This
EPR are the first letters of the names of
Einstein Podolsky and Rosen [vocalized-noise]
and ah these are, so Einstein actually question
that can quantum mechanics be complete because
this quantum entanglement actually gives rise
to the possibility of faster than light communications
and since relativity clearly states that there
cannot be anything which moves faster than
light. So, is quantum mechanics is in a confrontation
with relativity and whether it is complete
and Einstein had this idea of that there are
hidden variable, however they are ah negated
[vocalized-noise] by Rosen and by Bell's inequality.
There is another ah very interesting concept
ah in [vocalized-noise] in physics which has
found a lot of mentioned in the science fiction
ah books and movies. It is ah called as Quantum
Teleportation. So, quantum teleportation is
about carrying fixed amount of information
through the system of qubit [vocalized-noise].
So, this certain amount of information is
being carried, ok and basically this if is
this ah transmission of information happens
at a speed which is faster than the speed
of light, then we can say that ah there are
you know ah teleportation of information that
happens and it means that to replace the state
of qubit with that of another, ok and the
interesting name, the person name is very
interesting is quantum teleportation.
This name ah is derived from the fact that
the state is transmitted by setting up an
[vocalized-noise] entangled teeth space of
3 qubits and then, removing two of them from
the entanglement via measurement we will of
course these are this let them remain as words
[vocalized-noise] complicated that too [vocalized-noise],
however we will ah make sure that [vocalized-noise]
one understands what is being conveyed here.
So, the information of the source qubit is
preserved by this measurements. The information
lands up in the third that is the destination
qubit. So, this is all about ah quantum teleportation
and you would find them as I said that mentioned
in the science fiction movies or books.
Then, we would also discuss ah what is called
as a super dense coding which is in some sense
it is the rivers of quantum teleportation.
So, the ideas to send two classical bits of
information by sending 1 quantum bit or qubit.
So, it works by first [vocalized-noise] free
communication of this EPR pair. So, EPR pair
are those variables which one, when one being
determined the other automatically gets determined.
That is called as EPR pair that is shared
between the receiver and the sender.
Let us just introduce the quantum logic gates
which are ah analogous to the classical logic
gates. So, if you have a quantum not gate,
so an input qubit would pass through a quantum
not gate and would give me. So, if ah it acts
on 0 will give me 1 and if it acts on a 1
it will 0. So, the output quit will be if
the input qubits are 0 and 1 as it is written
on the left hand side of those two equations,
then the output qubit would be what appears
in the right of those equations that is namely
1 and 0 corresponding to 0 and 1 qubit.
So, the question is that now if we want to
ah [vocalized-noise] do this operation on
a qubit which is given by a combination or
a super position of alpha 0 and beta 1, then
what happens? So, alpha 0 actually goes to
alpha 1 and beta [vocalized-noise] 1 goes
to beta 0. So, one actually ah gets the interchange,
interchanging of the ah probability aptitudes
or the amplitudes of this state spaces.
So, the matrix representation of such a thing
can be written as 0 1 1 0, so that 0 is converted
into 1 and 1 is converted into 0 and so on
and so forth and this is nothing, but if you
see that this is like the Pauli matrix. The
x component of the pauli spin matrix correspond
to [vocalized-noise] spin equal to half. So,
the general dynamics of closed quantum systems
including the logic gates can be represented
by unitary matrix. We have given the definition
of unitary matrix. Unitary matrix is defined
as u p or u daggered u is equal to an identity
matrix. So, that is the definition or rather
that is the test for unitary matrix [vocalized-noise].
So, let us ah look at ah this ah [vocalized-noise]
pauli gates which are sigma x or sigma 1 and
that you know just acts like not gates. So,
it ah converts 0 into 1 and 1 into 0 and ah
also there is y gate which is ah written as
sigma y or sigma 2 and this has a particular
form which is 0 minus i 0. Do not mistake
ah any of these having complex entries to
be non-hermitian operators. They are all hermitian
operators and they have really eigen values.
One can check that. So, this is y gate also
known as the sigma y or sigma 2. As I told
you and this is z gate also known as sigma
z and sigma 3 and that just returns back the
states excepting for the one state, it returns
a minus 1. So, it is written as 1 0 0 minus
1 and this is nothing, but the z component
of the pauli matrix, ok.
So, as an exercise you can actually show that
x into y [noise] is equal to i into z. So,
[vocalized-noise] this take, this matrix and
multiplied with this matrix. So, [vocalized-noise]
let us do that for the moment. So, x y equal
to 0 1 1 0 and 0 minus i i 0. So, this is
equal to 0 and i. So, then 0 minus i and 1
0. So, this is equal to 0 1 0, this is also
equal to 0 1 minus i. So, this minus i and
this is equal to i and this is 1 0 0 minus
1. So, this is i into z, ok [vocalized-noise]
so on.
So, these are easy things and there is also
another 2 by 2 matrix which is of importance
ah here, that is sigma 0 which is equal to
the identity matrix and it is important for
you to know from the mathematical point of
view that any 2 by 2 matrix call it a b c
d, they can be written in terms of ah [noise]
of ah sigma x sigma y sigma z and sigma 0
by properly choosing coefficients any 2 by
2 matrix [vocalized-noise], also as an excercise
one can show that the x square equal to y
square equal to z square equal to an identity
matrix.
So, let us ah give very brief introduction
to the quantum computing power [vocalized-noise].
Why quantum computers will take a center stage
in ah [vocalized-noise] in research and in
learning? For the next ah you know maybe a
decade or even more than that is because the
integer factorization which is impossible
for digital computers to factorize very large
numbers which are the product of two primes
of nearly equal size and ah quantum computer
with 2 n qubits can factor numbers with lengths
n bits which are binary lengths [vocalized-noise].
Quantum database search as an example one
can see that to search the entire [noise]
library of congress for a particular name
and given an unsorted database in classical
computers, it will take 100 years where as
in a computer quantum computer, it should
take half a second and that is a miraculous
speed that we are talking about.
So, what is the quantum computer? The computer
that uses quantum mechanical phenomena to
perform operations on data through devices,
such as superposition in entanglement which
is a little of it we have already seen. We
will see more of that and classical computer
ah as a post to the quantum computer. It uses
voltages that ah flow through circuits and
gates and can be ah calculated entirely by
classical mechanics [vocalized-noise].
So, one of the very important thing about
quantum computation, quantum computation is
factorization of very large numbers and there
was an algorithm ah put forward by Peter Shor
in which it is this algorithm is used to factor
numbers into their components which can potentially
be prime just like we have seen ah just a
while back and it does this in roughly order
of n cube quantum operations, ok. So, the
best known classical algorithms are at the
best exponential. Since the difficulty of
factoring is believed to be exponentially
hard, it forms the basis of most cryptosystems,
ok. This factoring in polynomial times in
so basically because a quantum ah computing
does it in polynomial times which is n cube,
it has attracted ah significant amount of
attention.
Let us ah look at these Grover's search algorithm
which says that in some sense you know the
[vocalized-noise] parallel version of the
classical computer is already in built into
that. So, if you want to understand how ah
[vocalized-noise] the classical computers
work, they actually work ah with the flow
in time which means that one job gets over
and the next job starts [vocalized-noise]
like if you think about the multiplication
of two large matrices, it will multiply the
row of the left one to the column of the first,
the second one and then, store it as the first
element some all these entries and store is
as the first element of that resulted matrix
and then, it will go on to do it for the ah
[noise] the second row and the second column
[noise], however all these things are done
in a parallel fashion in a quantum computer
[noise]. Let us see how that ah happens [noise]
ah.
So, ah let us say that we are [vocalized-noise]
I am thinking of a number between 1 and 100
and say have thought about the number which
is 3. So, the classical computer will keep
asking that is the number 1 is the number
2 is the number 3 is the number 4 and all
that and then, once it [vocalized-noise] gets
the answer, yes it stops its operations.
So, it is like this the classical computer
you keep [vocalized-noise] going 1 no, 2 no,
3 no and so on [vocalized-noise]. So, there
will be n such queries by which it will be
resolved that actually the number that I have
thought about is 3, however in a quantum computer
this is done very differently way. It is put
as the [vocalized-noise] entire you know 1
to 100 is put as superposition and so, there
is an input that is given to the quantum computer
is 1 plus 2 plus 3 plus 4 and so on. So, it
will keep doing it no no no and then, yes
for third and then, no no no no and for the
rest ah um you know 97 entries or 96 entries
[noise]. So, this is called is the Grover's
Search Algorithm. So, at the quantum computer
will find to the query in very lesser number
of you know queries as compared to the classical
computer.
So, the practical ah quantum computer applications
is that the quantum mechanics simulations
ah in physics, in chemistry, material science,
nanotechnology, biology, medicine, computers
can simulate millions ah of variables at once
and all are limited today by the slow speed
of the quantum mechanical simulation and then,
there is something called as cryptoanalysis
is capable of cracking extremely complicated
codes and this RSA encryption as a name goes
and ah typically uses numbers with 200 digits
[noise].
So, let us have ah brief at the history and
the main ones. In fact, this the quantum computing
history in 1993 Alexandra Holevo published
a paper showing that the qubits can carry
more than ah classical mechanics bits of classical
bits of information. 1981 Richard Feynman
determines that it is possible to efficiently
simulate a evolution of quantum system on
a classical computer. 1985 David Deutsch of
the University of Oxford described the first
universal quantum computer. 1993 Dan Simon
at Montreal, he invented the Oracle program
or problem for which quantum computer would
be exponentially faster than the conventional
classical computer. This algorithm introduced
the main ideas which were then developed in
Peter Shor's Factoring Algorithm. We will
give a detailed account of this Shor's algorithm
and Shor's problem, the factorization problems.
In 1994 Peter Shor at T bell labs, it discovers
an algorithm to allow quantum computers to
factor large integers quickly. Shor's algorithm
could ah theoretically break many of the cryptosystems
in use. Today in 1995, Shor proposes the first
scheme for quantum error correction. In 96
Grover which is we ah [vocalized-noise] ah
mentioned his work and Bell labs invents a
quantum databases search algorithm. So, that
is Grover search algorithm. In 97 David Cory,
A. F. Fahmy, Timothy Havel and ah Gershenfeld
ah and Isaac Chuang published the first paper
on quantum computing.
So, it is not very old. It is about 20- years
old. When the first paper had come out in
1998, the first working 2 bit 2 qubit NMR
computer demonstrators at the university of
California at Berkeley [noise], 1999 first
working 3 qubit NMR computer demonstrate is
at IBMS Almaden Research Centre. The first
execution of Grover's algorithm was achieved
in 2000, first working 5 qubit NMR computer
demonstrated as IBMS Almaden Research Centre
and in 2001, first working 7 qubit nmr computer
demonstrated as IBMS, again Almaden Research
Centre [vocalized-noise] and of course, these
activities are going on. It has not stopped
at 2001, but I have just given a brief ah
history of that how it developed in the initial
days and so are the quantum computing problems
[vocalized-noise].
The current technology or 40 qubit operations,
operating machines needed to rival current
the classical equivalent errors are caused
by the decoherence. There is the tendency
of the quantum computer to dk from a given
quantum state into an incoherent state as
it interacts with environment. Interactions
are ah unavoidable and induced breakdown of
information stored in the quantum computer
resulting in computational errors and error
rates are typically proportional to the ratio
of the operating time to the decoherence time.
So, that is the errors rates or one can estimate
errors by that and the operations must be
complicated, much quicker than the difference
time.
So, these are some of the problems that are
going to come up or that are already come
up, however there are also efforts to [vocalized-noise]
negotiate or mitigate those difficulties and
problems [vocalized-noise]. So, with this
I will end this somewhat popular introduction
to quantum ah computing, however I will ah
take on each of the topics or most of the
topics in detail for a better understanding
of the subject.
