so this is the final week of our course on
implementation of quantum computing so what
we will try to do in this week is ah ah go
back to the ah basic ideas that we started
from and ah give you all the summary of the
implementations and the aspects of the quantum
computing that we have learnt as a part of
this course so ah this perhaps this week we
will have a longish in terms of the number
of lectures and the amount of material just
to make sure that we are able to cover most
of the materials that we have presented in
this course also we will try to um look at
some of the difficulties that some of you
had in your ah responses in the forum and
other places so there may be places where
i also referred to the areas of difficulty
that you had while doing this course so that
we can sort of ah finish all that ah parts
which who had not possible to be addressed
during the course
so i have set it up in such a way so that
ah we go from the point of the background
where we started this course which was classical
computation wherein ah we gave examples of
how classical computer can be essentially
looked at in terms of the church turing thesis
which essentially means that it is anything
that can be done in a turing machine these
definition coincides with our intuitive ideas
of computation addition multiplication binary
logic etcetera
so in terms of a turing machine we have discussed
it as a machine which has a way to read write
data on an infinite tape and have a control
module which is in a finite state automation
so that it can achieve the computation on
the input data to give rise to ah the result
which is the output so in this principle problems
which can be posed only in a way which can
undergo specific answers are the ones which
can be addressed so there is a that is the
part which we discussed in terms of the concept
of computing
so universal statements for instance are not
information which can be used for doing computation
and so on and so forth so we had done quite
a bit of discussion on this area of a computation
now the other important part of the computation
lies in the concept of complexity which was
also importantly addressed in this course
where we have discussed the levels of complexity
that appears in computation some problems
are more difficult than the others generally
speaking all the computable problems that
i addressed through the turing machines have
polynomial hierarchy and ah such all turing
machine equivalent computers have identical
hierarchy
so there is this ah typical way of presenting
how the computer complexity looks like the
part which is solved quite regularly by turing
machines or our computers are the ones which
require polynomial time to solve which is
the complexity in terms of p and then the
ones which have ah difficulty starts from
the parts which are non determinate polynomials
and ah so there are two parts to it one which
is known as non determinate polynomial and
which require times which are definitely exponential
to solve which are known as n p complete whereas
the others which are exponential requirement
in time but under certain conditions it can
be transformed into certain levels of polynomial
times so those are the two different parts
of the difficulty levels so it is either n
p non determinate polynomial or n p complete
which is the hardest part of the difficulty
in terms of doing this work
so this is how we had discussed in terms of
classical computing we also refer to the fact
that some important problems do not have the
known classical polynomial algorithm and or
a known place in the hierarchy so in this
picture of polynomial hierarchy for instance
a factoring is a problem which works in exponential
time for instance the best known algorithm
to factor n digit number takes time which
is about exponential n to the power one third
so those are the ones which go out of this
polynomial hierarchy and ah they are classically
difficult problems to be done there is also
this problem of graph isomorphism best known
algorithm to compare two node graphs is also
n exponential problem and similarly such problems
can often run into regions which are not quite
in the polynomial regions and they could be
also non determinate polynomial in some sense
in this case we mentioned quantum systems
can be of huge help to us quantum systems
is a system of elementary particles like photons
electrons or nucleus governed by the laws
of quantum mechanics so we deviate from the
concepts of classical nature that we were
discussing when we discuss in terms of quantum
mechanics so in this regard of complexity
one of the concepts that we introduced was
the idea of using quantum systems when the
systems becomes small such as a quantum system
is a system of elementary particles such as
photon electrons or nucleus which are governed
by the laws of quantum mechanics and therein
the concepts of classical mechanics are not
quite applicable and thereby we can think
of computations which are no longer in the
classical domain that we were discussing until
now
so ideas was to take the advantage of it so
the parameters in this particular case or
the system may include positions of particles
momentum energy spin polarization the quantum
system can be characterized by it's state
that is responsible for the parameters the
state can change under external influence
which could be fields laser impulses etcetera
it could and finally those are to be realized
or found out the results in terms of the measurements
so these are the basic principles of the quantum
system coming into the picture in place of
the classical nature of objects that we just
discussed
the distinction of quantum mechanics lies
in certain points for example in terms of
super position if a system can be in either
of the two states simultaneously it also can
be in a super position of them this is distinct
from what a classical nature of a system is
supposed to be some parameters of elementary
particles are discrete for example energy
spin polarization of photons and that's how
they are quantum in nature the changes in
this particular case of quantum mechanics
are all reversible that's actually a very
important parameter also and a critical part
of quantum mechanics lies in the fact that
the parameters are undermined before the measurements
and so this plays a very vital role in terms
of the application of quantum mechanics into
the classical areas that we are discussing
another important aspect of quantum mechanics
that's distinct from classical mechanics is
that the original state is destroyed after
the measurement there is nothing which can
be cloned and so that comes under the no cloning
theorem which says it's impossible to create
a copy of unknown state and the point here
is that once you make a state known then it
doesnt remain as quantum anyway because it
becomes classical finally there is this concept
of quantum entanglement and quantum teleportation
all of this has been a part of your course
in great detail we are here just revising
and refreshing it in order to make sure that
we are completing all the parts of the course
that we have covered in this series of lectures
that we have done
so one of the term which we introduced in
this was the idea of a qubit which is the
quantum bit 
so that is our unit of quantum operation in
general one qubit simultaneously contains
two classical bits and that is one of the
important aspects of quantum mechanics which
is inbuilt into the system qubits can be viewed
as a quantum state of one particle photon
or electron qubits can be modeled using polarization
spin or energy levels all those which assume
discrete values qubit can be measured as a
result of measurement we get one classical
bit either zero or one so until the point
of measurement as has been correctly pointed
out quantum mechanics ensures the result or
the information can be embedded in terms of
qubit which can contain both the classical
information simultaneously therein comes the
concept of quantum computer whenever we use
quantum mechanics quantum computers uses properties
of elementary particles that are predicted
by quantum mechanics and so that is how we
arrived at this whole concept of quantum computer
or quantum computing
usual computers as we have been showing in
the very initial slides even today the information
is stored in terms of bits whereas in quantum
computers information is stored in terms of
qubits quantum bits theoretical part of quantum
computing is developed substantially as we
have shown however practical implementation
is still quite a big challenge although we
have discussed in this course some of the
even commercial approaches of quantum computing
so this point the practical implementation
point though is still an issue it's not really
a difficulty which has not been overcome as
we yet
there is a strong connection of quantum computers
with probability that is also another thing
that we have discussed in length in this course
when the quantum computers gives you the result
of computation this result is correct with
certain probability quantum algorithms are
designed to shift the probability towards
the correct result that is one of the most
important aspects of quantum algorithms running
the same algorithm sufficiently in many times
gets the correct answer with high probability
assuming that we can verify whether the result
is correct or not the number of repetition
is much smaller than for usual computers and
that is the power of quantum computer
so here is a short history of quantum computing
which we have discussed in this course earlier
also from about nineteen seventy onwards the
beginning of quantum informations theory started
at in nineteen eighty yuri manin first set
forward the idea of quantum computations paul
bennioff at about the same time discussed
with richard feynman on the concept that as
the computers becomes smaller errors in computers
become larger due to the probabilistic concept
of quantum behavior and the solutions would
becoming more and more probabilistic feynman
thought about the problem and he independently
proposed the use of quantum computing to model
quantum systems since quantum systems are
the essential building block of nature he
thought that was one of the best ways of addressing
quantum systems by building quantum computer
rather than dreading the idea
he also described the theoretical model of
a quantum computer based on this third process
however it was not until nineteen eighty five
that the concept of how such a principle would
become useful was given by the idea of david
deutsch who described the first universal
quantum computer and then came the big breakthrough
in nineteen ninety four where peter shor developed
the first algorithm for quantum computer which
is the factorization into primes now this
was one of the most important developments
perhaps in the area of quantum computing which
essentially made people realize that it could
really have a huge impact because of the break
of the r s i codes and other kinds of things
so this was a very important development and
it took almost a decade to get there because
a lot of error solving and other kinds of
problems were being worked on during this
period of initial development of quantum computer
in nineteen ninety six lov grover developed
an algorithm for searching in unsorted database
and this was also extremely important because
it turns out that for many problems it is
known that the solution exists but it's more
of a searching of a solution which is the
biggest step in computation and therefore
getting an algorithm which would be able to
search for a solution as a huge impact in
this area in nineteen ninety eight the first
quantum computer on two qubits based on n
m r was built in m i t stanford and i b m
in the year two thousand quantum computer
on seven qubits based on n m r was built los
alamos
another major development in the year two
thousand one was when the number fifteen was
possible to be factored on the seven qubit
computer by i b m and this was achieved by
chuang and his group in two thousand five
to two thousand six experiments with photons
and quantum dots fullerenes and nanotubes
as particle traps started becoming popular
which resulted and developed this field further
to finally the first commercial computer quantum
computer which came out in two thousand seven
were d wave announced the creation of a computer
on sixteen qubits
so looking a little bit into the basics here
a model of a qubit as has been discussed several
times essentially can be represented by a
vector with a ket representing vector having
the states zero and one such that the coefficients
a zero and a one can be complex numbers with
only the constraint that a mod squares add
up to one this state or the qubit psi ket
is a superposition of the basis states zero
and one the choice of the basis states is
not unique which means that several possibilities
exists the measurement of the wave functions
psi ket results in say the bit zero with probability
a zero mod square and in bit one with probability
a one mod square after the measurement the
qubit collapses into the basis state that
corresponds to the result and this particular
part is essentially classical so the point
of measurement brings back the classical information
as we have been discussing so here is an example
just to show that the the coefficients basically
give raise to the probability of the measure
of individual states
so this idea of using computation with qubits
how does it affect the concept of computer
so in terms of classical computation the data
unit is as mentioned either one or zero so
the valid states are either zero or one and
depending on how the switch is whether it
is pointing towards zero or towards one we
get the values as zero or one in terms of
quantum computer however the data unit which
is the qubit has the valid state in the superposition
of both zero and one and at any point of time
before measurement it can assume any of the
values that are possible in both cases logical
operations are to be performed
however in classical computation logical operations
need not be reversible so for instance not
gate where that is a one input and one output
is a one bit process is perfectly fine however
it also admits two inputs and one output in
terms of the and gate in terms of quantum
computation which is definitely going to be
only unitary and reversible the valid operations
therefore involve for example in terms of
one qubit the poly matrices as we have discussed
throughout this course as well as the hadamard
operation which basically produces an equal
superposition of the states concerned in two
qubit case there is this control not which
enables the switching of a bit based on the
control however almost all the however it
is important that all the logic gates in terms
of quantum computation be reversible in case
of classical computation the measurement is
deterministic if the state is a zero then
the result of the operation then the result
of the measurement would be zero whereas if
the state is one then the result of measurement
will be one
on the other hand in quantum computer the
measurement is probabilistic or stochastic
as a result we could land up having the measurement
of either zero or one as well as the case
where both zero and one occur with a fifty
fifty probability so these are the major distinctions
which can be immediately looked at when we
look at computation with qubits versus the
computation with when we are using several
qubits the subsystem of n qubits contain two
to the power n classical bits or the basis
states so here comes the exponential benefits
in terms of the classical computing the potential
of quantum computer grows exponentially we
can measure individual qubits in the multi
qubit system
for example in a two qubit system we can measure
the state of the first or the second qubit
or both the results of measurement are probabilistic
as we have just showed after the measurement
the system collapses in the corresponding
state
so here is the how it is looks like we are
in terms of quantum computer working in the
hilbert space for a single qubit we have already
seen how it works when we have two qubits
they work as tensor products and so there
is an exponential growth in the space for
any arbitrary state for a single qubit the
coefficients are just a two coefficients of
the two states however for two qubits there
are four possibilities which is two to the
power n in this case n is equal to two and
the operator is supposed to be always unitary
which acts on the given state as has been
shown here
so we have basically explored the entire process
by using the quantum circuit model in most
of the cases throughout the discussion in
this course where for example we have shown
circuits which are lines which are drawn with
the operations and the path and finally we
have used measurement as the final result
so whenever we looked at the inputs we had
the inputs provided and then we had certain
operations for example in this case there
is a one qubit operation of a poly x and then
after the two qubits are in to the c not gate
which is the two qubit operation the c not
gate operates and we can have the measurement
with gives rise to whatever the processes
happened so here is the ah principle of applying
how the state goes so we have the particular
state coming in spin put into this ah one
qubit operations which switches it to one
the other unit state as zero continuous after
going through the c not because the control
is a one it does it's not operation so it
switches the last two bits and that can be
measured as the final result
so here is another way of looking at operations
on one qubit quantum not gate for instance
it switches the values of the operations so
zero goes to one one goes to zero and not
operation is therefore is just a matrix which
looks like this the hadamard gate as i mentioned
before essentially creates an equal superposition
of the two states and so here is an operation
of hadamard which is shown here in terms of
the matrices so we have this equivalent approaches
either through vectors or through matrices
which go through this ah processes
there is another representation of the controlled
not operation where we have used the matrices
and the gates ok vectors in this operation
procedure and as can be seen this can be represented
in many possible ways and these are the ways
that we have discussed in the classes we have
gone through so here is another example where
instead of the two pure states coming we have
one state zero and the other one is hadmard
state which comes in and as we go through
the operations how they go and change it depends
and so in this particular case for example
it's a fifty fifty mixture of either getting
a zero or a one so the states can be of two
kinds where one case where they can be separated
and can be written as a tensor products as
has been shown here those are called superposition
states
however the other kinds of states where the
states cannot be written as tensor products
then those are known as entangled states as
we show it here so these states are entangled
states and in case of entangled states our
results are always going to be probabilistic
so the interesting consequences result of
this quantum concept is the principle of reversibility
because quantum mechanics is reversible dynamics
are unitary quantum computations are reversible
the other is the quantum superordinacy where
all quantum computations can be performed
by a quantum computer however one very important
thing is there is no cloning possible it's
impossible to exactly copy an unknown quantum
state one of the important things to realize
however that although quantum superordinacy
exists it is not true that every principle
of a classical computer will be more efficient
when we go into the quantum computer domain
it might take more steps because of the requirements
of the reversibility of the quantum computing
process however at the end of the day we all
know reversible computers are going to be
more efficient at least energy wise so it
has it's advantages
so we may ask what is a quantum computer good
for this is one of the things which we have
looked into many practical problems require
too much time if we attempt solve them by
usual computers for example it takes more
than the age of the universe to factor a thousand
digit number into primes because this is an
exponential problem the increase of processor
speed slowed down because of limitations of
existing technologies this is one of the facts
which has happened theoretically quantum computers
can provide truly parallel computations and
operate with huge data sets so these are the
major aspects of quantum computing advantage
why there is a huge research to a quantum
computation so with this we would like to
actually end this first lecture where we basically
bring out the essence of why we were studying
quantum computations and it's implementation
in the next lecture we will go into some more
details of the very basics that we introduced
here in terms of where this entire idea of
quantum computer started how we dealt with
it and we have benefited from learning more
and more about the aspects of quantum developments
over these years and how we have managed to
go to a point where we can do implementations
so from the next lecture on we look more into
these kinds of aspects that we have already
discussed in this course
thank you
