welcome back this being the final week we
are looking at all the aspects of the course
that we have covered in this particular area
of quantum computing and quantum information
in the last couple of lectures this week we
have covered the basics now let us look at
how we have gone ahead to look at the implementation
aspects of quantum computing some part of
quantum information transfer which is also
critical in terms of ah quantum information
transfer and ah secured communication was
discussed in the last lecture in this lecture
we will be looking at the comp computing part
so the [vocalized noise] basic point is how
the quantum computer works thats the main
part where we start from the routine essentially
involves initialization
for example all the qubits are in say state
zero quantum computations which involve the
gates and the different algorithms which are
applied and then finally reading the result
which is our main important part which brings
it back to the classical world which is measurement
an ideal quantum computer therefore must be
universal which can be capable of performing
arbitrary quantum operations with given precision
it must be scalable this is one of the biggest
ah difficulty in terms of quantum computing
as of today as we have discussed several times
in this course and it must be able to exchange
data and thats another part where it becomes
critical that we are able to understand how
quantum data can be connected or communicated
and in that regard quantum tele portation
or associated methods are extremely important
it is important to understand at quantum computing
is not analog quantum word essentially implies
that we are looking at discreet aspects and
when we are evolving the state or the property
we are essentially using the quantum nature
of the system which means that we are utilizing
schrodinger equation or associated similar
equations in the statistical concept which
is a liouville equation where we have instead
of the wave functions we have the density
matrix
all of which in terms of evolution essentially
involve a linear equation governing the quantities
which in these cases essentially the amplitudes
that are not directly observable and therefore
it is far from being analog thats one of the
most important point which often comes out
as a misconception should be cleared and the
power of the computing therefore is not to
be misjudged with the fact that the principle
of analog computation and its power can be
related to how a quantum computer works so
we should be clear on this point all the time
however we should in fact that is one of the
very important aspects which means that this
fact has many profound implications
such as the fault tolerance theorem which
means that absurd precision in amplitude is
not necessary for scalable quantum computing
so this is actually one of the aspects which
ensure the advantage of quantum computing
and definitely makes it benefit over hat fact
that it is not analog so in terms of the complexity
that we had originally mentioned what we are
looking at is we essentially have the capability
of the exponential space or exponential way
of looking at the problem in terms of how
the schrodinger equation looks like and thats
the parallel nature of the problem and that
is one of the reasons why some of the problems
which are otherwise difficult to look at can
be thought of in the exponential domain in
practicality however as feynman pointed out
most of the problems that we will be dealing
with are going to be super polynomials in
the sense that we can paralyze the system
in such a way that we get the advantage of
working in a domain where the quantum system
itself evolves because we are using a quantum
processor and so the laws of physics and that
was the basic vision of feynman would be better
described and processed by using the quantum
computations bqp by the way is bounded quantum
problems or bounded quantum polynomial so
ah these are basic principles i just wanted
to mention once more before we go ahead with
the implementation concepts because we have
to be clear n how we are going to go ahead
with this implementation part where we are
going to go and apply quantum computation
and quantum principles
the quantum algorithms which have been the
most celebrated once as we know are the shors
algorithm which enables factorization into
primes and this can work in polynomial time
with respect to the number of digits in the
representation of an integer so this is basically
the principle of the factorization that is
achievable through shors algorithm given this
advantage can actually be able to break the
rsa encryption because the rsa encryption
is based on the principle of factorization
of a very large number which is typically
takes a very large time and so prime factorization
principle application into encryption essentially
means that we chose a large enough number
whose factorization would be extremely difficult
in proper classical computers and therefore
it becomes encrypted in quantum sense however
this could break the rsa encryption because
of the advantage of factorizing using the
quantum algorithm the other very important
algorithm which is useful for quantum computation
is the grovers algorithm which is the best
application in terms of database search and
since most of the problems that we deal with
essentially involves finding the solution
from the set which has the solution already
present it amounts to essential database searching
the brute force technique for doing a database
search requires about n operations where n
is the number of records in the database and
as which means that as the problem size becomes
larger it becomes increasingly difficult to
do or to address a problem grovers algorithm
on the other hand which utilizes quantum super
position principle can be done can be achieved
in about root n operations some of the major
problems that we have quoted over and over
again in terms of quantum computing quantum
bill formation quantum teleportation and associated
properties of using quantum concepts is one
of the major aspect being ah decoherence quantum
system advantage or the parallelism often
is attributed to the fact that the system
behaves coherently and does the work in a
way so that it can all be processed together
wherever the coherence part of the problem
gets into compromise because of various difficulties
we have difficulty of losing the quantumness
of the system quantum systems are extremely
sensitive to external environment so it should
be safely isolated so one of the biggest issue
about quantum computing implementation is
the concept of safe isolation it is hard to
achieve decoherence time that is more than
the algorithm running time this is another
big problem of these entire field which is
that ah it is hard to achieve the decoherence
time that is more than the algorithm running
time the other important aspects which are
related could be attributed to also decoherence
part is the fact that it requires corrections
and the corrections could be due to loss of
coherence due to many other errors and so
it requires the principle of error correction
which means that it would essentially require
more qubits then it is required to just simply
look at the problem from first principles
and thats the reason in most of the implementation
aspects we have been encountering ancillary
qubits such ancillary qubits are critical
in ensuring that the error correction part
of the problem is properly taken care of but
the requirement becomes harder as more and
more qubits are necessary for achieving fault
tolerant computation physical implementation
f computations have always been one of the
challenges new quantum algorithms to solve
more problems is another area where a lot
of efforts have been dedicated as we know
there are only a few limited quantum algorithms
which take advantage of the quantum nature
of the system as far as the data transfer
processes concerned as we mentioned for quantum
systems entangled states are necessary which
are critical in ensuring quantumness of the
problem and so making of entangle states and
preserving them for data transfer is another
very important area of quantum computing which
needs ah to be addressed while quantum computing
parts is been looked at so these have been
ah some of the basic problems or hurdles in
the path of achieving quantum computing to
the levels that we are actually interested
in still a lot of practical implementations
we have discussed in this course and some
of them are listed here one of the first implementation
of quantum computing in a very successful
manner has been the use of nuclear spins and
that have been utilized in a machine known
as nmr which is the principle which is used
in terms of measuring or manipulating nuclear
spins known as the nuclear magnetic resonance
spectroscopy
since this technology has been already available
in spectroscopic sense for identifying molecules
and has been utilized very often in chemical
and biological systems and has made a huge
advancement n the area of spectroscopy this
particular technique became the most handy
way of showing the first few implementation
principles of quantum computing we have also
talked about electron spins and quantum dots
in the process of implementations the energy
levels of ions and ion traps are the other
very important areas of quantum computing
implementations which we have discussed he
use of super conductivity is another very
important area where quantum computing has
been brought into and adiabatic quantum computers
are the ones which are finally made some of
the commercial implementations possible in
most recent times so by using these two principles
of super conductivity adiabatic quantum computation
principle as well as isolation from the environment
the first commercial quantum computing came
into be one of the most important features
which enabled the principle of quantum computation
has come due to the effort put in by divineenzo
where in he was able to come up with the stability
criteria that are necessary for minimally
ensuring quantum computation can be achieved
so he had a few set of rules which are often
known as the five sets of rules provided by
divineenzo and those are listed out here the
machine should have a collection of bits which
is the basic requirement of starting the quantum
computing process recommended bits for doing
the large computing would be in the range
of thousand qubits which we know that we are
not yet there in that terms in reality to
use a thousand qubits in all practical senses
commercial processor cleaning thousand qubits
are now presently available however for most
of the application experiment that have been
made possible with them as we have discussed
earlier all the thousand qubits have not been
effective utilized as we just mentioned in
the previous slides that ancillary qubits
are an extremely important part of quantum
computing to ensure error corrections can
be done so currently although about thousand
qubits or even more could possibly be available
the actual use of thousand qubits for effective
computation in terms of the power coming from
thousand qubits is not really available in
that sense however the first point that the
machine should have a collection of bits in
our particular case qubits is definitely the
first criteria secondly it should be possible
to set all the memory bits to zero in other
words initialize 
before the start of each computation this
is also extremely important as it is important
to know how to go about the computation cycle
otherwise once computation cycle over a particular
problem is over the computer becomes un usable
he third important parameter laid down by
divineenzo was that the error rate should
be sufficiently low and in some sense there
was some numbers given by them n their initial
several work which comes out to be about less
than ten to the power minus four to make sure
that the computation becomes effectively fruitful
it must be possible to perform elementary
logic operations between pairs of qubits to
ensure that computation is possible otherwise
just having a lot of qubits which could be
initialized is not going to really have any
computation go ahead with any kind of advantage
there should also be a reliable output of
the final results otherwise the computational
concept does not become useful so this last
part is associated with measurements so in
other words essentially it can also be simplified
to availability initialization or initialization
of qubits then logic operations which imp
implies principles of gates logic gates or
operations which could lead to computations
and finally after the process of computation
is over we would like to have the measurement
made so that we can know we can get to the
results which could be called computation
so these are the basic set of five rules which
are often used nowadays and they go by divineenzo
rules pictorially such a system can be shown
in this kind of a graphical term wherein we
have the input which goes in to the unitary
transformation systems which can be controlled
if necessary by classical concepts and then
finally to get to the output which can be
the result leading to computation so that
is the basic idea behind the quantum computation
that can be utilized to ensure that it is
a computation of use
now in order to realize how this is in relation
to todays computer here is a sort of a comparison
todays computers essentially are based on
turing machines which is the theoretical device
that consists of a tape of unlimited length
that is divided into little squares each symbol
each square can either hold a symbol one or
zero or be left blank so that is very basic
concept of the classical turing principle
of a computer the classical computers work
by manipulating bits that exist in one of
the two states either a zero or a one one
and zeros are carried and turned on by states
of electrical current that is basic idea behind
todays classical computer in terms of quantum
computers are not limited to two states zero
or one ike the classical computers they encode
information as quantum bits or qubits which
can exist in superposition superposition essentially
means are the quantum computers can represent
both zero and one as well as everything in
between at the same time the qubits can be
carried as atoms ions photons or electrons
and their respective control devices that
are working together to act as computer memory
and a processor so the net result essentially
results in a quantum computer can work on
a million computations at once while our desktop
pc works on one
so that is the basic promise of a quantum
computer the extreme parallelism but as we
have been saying all the time this has to
be harnessed and that is the basic or the
major issue about the implementation of quantum
computation so let us start with the very
first one in terms of implementation which
we looked at which definitely is the concept
of the fourier transform nmr machine which
was able to be used to show the some of the
most important steps in quantum computing
so the principle of fourier transform nmr
device lies on the fact that the system can
interact with the applied radio frequency
excitation to raise the proton spins to the
upper level so in order to make the proton
spins effected by this kind of an action what
is necessary is that the spins become non
degenerate because almost all nuclear spins
have similar energy unless and until they
are subjected to static strong magnetic field
which results in splitting of the states based
on the applied magnetic field so once the
static magnetic field be zero say is applied
it will be able to split the otherwise similar
nuclear states into two different states depending
on their aligned with the field or being aligned
opposite to the field so the one which is
aligned along the direction of the field will
be lowered in energy verses the ones which
are opposite the direction of the field will
have the higher energy
so once this separation of energy is set up
for the nuclear states the radio frequency
excitation can be utilized to resonantly excite
the nuclear field resonantly excite the nuclear
spin from the ground to the excited state
so this is the first step after this is done
this was the simplest case is the case of
the proton nmr as it can interact with the
applied magnetic field you have to have the
nuclear state in such a way so that it can
interact with the applied magnetic field so
all the nuclei which would have an interaction
possible with the applied magnetic field would
be able to go into this mode of separation
of their energies so protons happens to be
one of those which can undergo a plus minus
half separation into the two cases where the
spin is aligned n the direction of the magnetic
field or opposite to the direction of the
magnetic field so similarly carbon thirteen
also has similar properties and there are
many other nuclei that we have used and we
have discussed which have these properties
which can interact with the applied magnetic
field to split into two states so once the
proton is excited the relaxation signal of
the excited state can be amplified with respect
to time and this process when the system is
freely coming back to its normal state is
known as free induction decay fid and once
this signal is fourier transformed then we
get the proton nmr signal at only one frequency
because of the constant magnetic field
on the other hand if the added magnetic field
has a gradient then this can have different
frequencies with respect to position due to
the gradient magnetic field the other way
of looking at this principle is how you are
changing the property of the system whatever
be the principle the very idea that a signal
in terms of the fourier transform of the relaxation
of the excited state can be identified it
forms the basis of the nmr now this particular
process is very subjective to the environment
under which the proton or the nuclei is presented
and so although all the protons would have
otherwise given a single frequency signal
upon fourier transforming under this condition
considering the ideal case that all the protons
are the same they are change in environment
around the proton give raise to different
frequencies and we have discussed this principle
of nmr at length during our course
once the principle is understood it is possible
to understand that different molecules can
have protons for instance under different
environments due to the electron cloud around
them are different so for example just a carbon
carbon chain having single and double bonds
around them would have the protons facing
different environments of electron cloud around
them making the environment of the proton
different from where their otherwise and similarly
if they have other electron withdrawing or
electron donating systems around them they
will also lead to changes in the proton environments
that they will notice as they undergo this
simple transformation and that is one of the
basis behind the nmr spectroscopy which is
use for identifying the molecule per se now
the same principle which is otherwise known
as shielding is the one which is utilized
for this entire process so a few more words
about this principle that i just discussed
each nucleus of the same element may not be
in a similar chemical environment in a compound
this is the basic idea behind whatever i was
discussing until now the relative shielding
offered by the opposing induced magnetic field
created by electrons around the nuclei is
going to be different thus the local effect
of magnetic field is different thus the magnetically
non equivalent nuclei have different larmor
frequencies the other name for resonant frequency
thats attributed to the scientists who had
first notice that which produces different
peaks in the nmr spectra the induced opposing
electric field is proportional to the applied
one that is the induced magnetic field has
a relation to the effect of the surrounding
which is otherwise known as shielding and
therefore the frequencies will be dependent
on the effect of the environment and this
can shown to be in this manner
so in these regards the qubit can be represented
in this principle a single qubit would have
a computational basis which can be given by
zero and ones where the state can have a representation
as we know from all times a combined effect
of both the basis states and thats the part
which is the quantum part of the states that
we look at and point of time when we make
a measurement we get a probabilistic contribution
of one of the two spin up or spin down in
these particular cases for getting their result
and the constraint which we have for all these
cases the fact that the mod square of the
probabilities in the both the cases has to
add up to one thats because we are looking
at a probability and the probability of finding
the result is always going to be there and
so it is going to add up to one all the time
so similarly if we end up getting two qubits
which is basically two spins interacting we
can go back and see the same results that
we have been discussing from all the times
in terms of the contributions from their combined
effect and as before their sums have to always
land up to be one
so in case of n qubit quantum computers we
definitely would have the two to the power
n states where each of these states are essentially
being represented from zero one to all the
way to two to the power minus one states and
ultimately whatever we get is essentially
connected to the contributions of each of
these which corresponds to their probabilities
basically square mod square of each of the
amplitudes in terms of the nucleus spin hamiltonian
which is what is the result of all these interaction
we have in this particular case the two states
which have being affected because of the applied
magnetic field and which are embedded in the
splitting along the z dimension as we discussed
applied field being taken along the z or z
dimension and therefore we have this particular
hamiltonian which essentially looks like this
whenever we have multiple spins without qubit
qubit coupling then they would essentially
scale with respect to the applied magnetic
field the coupling term is the one which is
represented by their spin spin interaction
which goes along with them which is the j
j coupling in this particular case
so the basic steps involved in the nmr quantum
computing goes along with those five principles
that we basically discussed the first part
being initialization process of such spin
states to be able to get to a condition where
they can be put in the original starting phase
now in this process a lot of these have to
be treated by using the density matrix approaches
because we are looking at an ensemble of states
and therefore most of these nmr quantum computing
pictures essentially uses the density matrix
approach although we start of by sing the
fact that we are essentially using just one
spin which can have the two possibilities
of plus minus half 
but since we are looking at a system which
is going to have ensemble because most of
these nmr machines that we are discussing
now are in terms of liquids which have been
looked at and the advantage of studying these
in the liquid phase is the fact that he molecule
by itself can behave as individual identity
as the dipolar averaging around the molecule
essentially goes to zero as the rest of the
molecules around them can have all possible
orientations and they all essentially average
out to zero contribution so the individual
molecules itself can be looked at and so within
the individual molecule the density matrix
is found by the number of coupled spin states
that have been put together so in some sense
this is a average hamiltonian theory which
is applied for this entire process and we
have mentioned some of it during our discussions
a lot of deep theory we didnt get into because
we were mostly interested in the implementation
part by using the nmr method however its suffice
to say that the density matrix picture including
average hamiltonian theory is basically the
idea behind this principle
where before the computation is started qubits
are initialized to a certain well defined
state the information is then processed by
applying unitary transformations and so the
well defined state has the quantum register
information which is then undergoing the processors
steps which are the unitary transforms and
at the end of the computation the result is
processed in the read out process and the
readout in our particular case is the final
fourier transform giving raise to the measurement
of the signal the molecules used in nmr quantum
computing have been quite quite a few ah so
far example the grover ah and the dj algorithm
were able to be shown by using chloroform
where the each of these cases the red nuclei
are the once which have been used as qubits
so far instance here the qubits are the carbon
basically carbon thirteen and the proton so
two qubits the error corrections quantum error
corrections ave been shown to be achieved
y this kind of a molecule were we had the
carbon been used as the qubits only the carbons
not the protons so this way this is the three
qubits system logical labeling and grovers
algorithm again has been utilized by using
fluorine as the nuclei of choice for the nmr
application so one two three fluorine in terms
of teleportation a proton and two carbons
in this kind of a molecule and in terms of
error detection a proton and a carbon has
been shown
so several different molecules have been used
for quantum computing using nmr and grover
search algorithm quantum fourier transforms
shors algorithm dj algorithm deutsch jozsa
algorithm order finding error correction codes
and dense coding all these have been shown
to be possible to be done by using nmr quantum
computing has shown a great lead in terms
of major developments in quantum computing
and so it has been one of the many studied
areas of quantum computing the best feet in
terms of quantum computing was the implementation
of shors algorithm by seven qubit quantum
computer which was possible in ibm research
by chuang was working on that time over there
this use this molecule which is a seven qubit
molecule ah having an amino acid alanine so
the fluorine nineteen has been used as one
of the qubits in this case five fluorine nineteen
and two carbon thirteen makes the seven qubit
molecule for this particular case this has
been used for finding the factors of the number
fifteen which shors algorithm and so these
are the five fluorine and the two carbon atoms
as we mention and the programming in these
cases are essentially done with the rf pulses
because those are the ones which enable the
unity transforms and can be detected by nmr
technique
in spite of its great success one of the biggest
difficulties have been the scaling problem
in terms of nmr quantum computing and thats
one of the reasons why though nmr is a great
technique it cannot be an area which could
be a next quantum computer for us typically
the single spins in nmr are too weak to be
measured and as a result we have been always
talking about ensembles so that is one of
the issues where quantum computing has always
been more and more difficult and more and
more involved whenever it has been an nmr
quantum computer its so much simpler to talk
about quantum computing in terms of pure states
however in these cases the best could have
been done are known as the pseudo pure states
because they have been always cases where
it had to be dealt with in terms of ensembles
the quantum measurements are usually projective
and in nmr get the average over all the molecules
so once again the very idea of having one
to one correspondence is a little difficult
and so for smaller numbers of qubits while
there is always a correlation possible to
be found as the number of qubits become larger
and larger his correlation becomes more and
more difficult and as such it becomes very
difficult to implement quantum computing in
terms of scaling the qubits for nmr
there are also couple of others difficulties
for example the spins have an tendency to
align with the field for example the spins
tendency to align with the fields is very
is quite weak and that is a difficulty because
even at equilibrium most spins are random
and therefore most often we have to talk about
effective pure states which is what the principle
of pseudo pure states have been ah so all
in all it is always an issue where scaling
becomes a major problem for nmr quantum computing
and very often the signal to noise levels
go below the quantum noise levels and so it
cannot be scaled effectively so this was one
of the major aspects of quantum computing
which was the first one to basically show
the way that quantum computing can actually
be possible in reality to be implemented and
so nmr formed one of the basic backbone of
starting to realize and go ahead with quantum
computing so i first thought that we will
first finish that part then going to the other
implementation aspects and to the commercial
one in the next lecture
thank you
