WEBVTT
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so first we have all ready looked at this
the idea of computing and in that we are used
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to these kinds of computers where the basic
idea remains the same which was concede way
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back in nineteen thirty six before the computers
actually existed showing that a computer essentially
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is a device which uses limitless memory and
a scanner to scan the memory backward and
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forward and read it by symbol to symbol and
write additional symbols to execute any computation
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thats the turing machine ok the whole idea
of this principle still works and its the
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same one which is given for this computer
so that is the basic idea behind the concept
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of any classical computer
ok so the classical computer has some limitations
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even the fact you can have limitless parallel
computers they which can do really complex
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work but they are basically complex
turing engines which employ multiple computing
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modules dealing with pieces of incoming data
which are chunks of bytes instru[ctions] instructions
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etcetera but there are problems which are
beyond the competence of a universal turing
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machine for example we cannot predict whether
a program will terminate and this is the halting
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problem so famous one and then there is this
classes of problems which have been delineated
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and are solvable for example in polynomial
or exponential time and those whose answers
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are checkable in polynomial time
so these are the classes which will be looking
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at so these are n so if anything can be
solved in n polynomials then its a problem
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which is to be looked at in this fashion but
as these so this is an n p problem but as
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these become larger and larger it becomes
more difficult so will be looking into these
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things more and more and we have also discussed
about moores law before which was given by
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gordon moore who is a founder of intel cofounder
of intel who who made an implicit statement
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based on the fact that he saw that the
number of transits for chips essentially doubled
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every eighteen months and this tendency of
the computer world has kept on going since
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the beginning of the computers for a very
long time and thats kind of very interesting
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because now we are talking about switching
which are at very smalt scales and the current
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v l s i the very small very large scale
integrated circuits thats v l s i can be a
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small as a few tends of a micron and even
smaller actually they have reached nanometer
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scales
so the principles of quantum mechanics are
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going to come and in the spirit of feynman
the mystery associated with quantum mechanics
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is something which you perhaps cannot really
explain but what you can do is you can just
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look at it how it works because you cannot
really in his words we cannot make the mystery
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go away by explaining it so thats how you
look at it and this was done this is this
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is famous statement with respect to the double
slit experiment where interference fringe
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sent things were found for the first time
so in terms of quantum mechanics the beginning
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or basically the use of quantum mechanics
in computers started in nineteen eighty
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one where this statement was made that the
classical turing machine would probably experience
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an exponential slowdown because of the probability
issues as you become smaller and the immediate
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other side of the story was there the quantum
system with many particles is described by
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hilbert space whose dimensions are exponentially
large in the number of particles which means
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that it can utilized for its advantages and
this statement that this has its benefits
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ops you know what happened sorry this as
its benefits where described in a conference
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called cleo in this nineteen eighty one
so its in nineteen eighty one cleo conference
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feynman first coined the term which is
quantum computer
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then david deutsch in nineteen eighty four
proposed the idea of universal quantum
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computer took a it was quite soon that this
idea was brought out and the qubits required
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were the number of particles in the system
thats how it was defined and in then it when
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through a lot of developments and about
a decade later satellite talked about a
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standard quantum computer which can be programmed
to stimulate local quantum systems efficiently
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so this was one of the important statements
that he was able to make and he was able to
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show he was a able to take a stock of all
the developments over the more than a decade
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by the time and showed that it quantum computers
has been extend extended to larger classes
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of quantum systems
so well quantum mechanics essentially are
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systems which involve electrons protons neutrons
photon quarks neutrinos any of them can be
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our potential qubit in some sense and its
a system of laws that describes the behavior
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of such objects so this was predicted way
back then that by twenty twenty we will store
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one bit of data on objects of that size
not sure we are there yet but well not too
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far from twenty twenty either so let see what
happens
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so the one of the interesting concepts of
quantum mechanics is that the atoms size objects
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behave in unusual ways because their state
is generally unknown at any given point of
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time and changes if you try to observe it
now this is a very important aspect of quantum
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mechanics which is different from classical
works in a classical object whatever you have
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the solution is there in a quantum object
unless you measure it you dont know what it
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is and secondly if you measure it you cannot
be sure that its the same thing that you measure
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because the act of measurement changes it
so there is also this principle of standardization
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which exists in this quantum world
and simultaneously several properties of the
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systems of such systems can be manipulated
and measured simultaneously thats one of the
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other advantages of this so anyway these are
the motivations as to how you would be going
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and why you would be going quantum and we
have done this in class i am just going to
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re iterate here a it to bring you up to speed
one is that as we go smaller to avoid waste
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wastage in time energy etcetera because as
we go smaller the time taken to travel goes
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lesser so wastage of time is lesser energy
required in dissipation goes down and so eventually
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go to quantum so thats the thats the way of
looking at it the other option is that in
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order to be energetically most favorable you
want to act reversibly that comes from thermodynamic
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principles that gives minimum energy loss
and the biggest important point is turing
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machine is not reversible but the quantum
evolution is reversible ok
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so a classical bit again the other important
thing which is different is a classical bit
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an either be one or zero however a two state
quantum system can be in an arbitrary number
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of superpositions a all such states can be
processed at once in quantum operations right
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so for example this continuously the system
getting tipped is an example of a superposition
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principle so while classical essentially means
either one or the other in a spin system in
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quantum systems it is a continuous tipping
kind of a situation where in the middle wherever
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whatever you are you can see them so those
are our superposition states and we can used
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them
so there is this concept of hidden variables
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in physics which have been used quite often
to even talk about these a superposition and
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in between conditions but we not need not
worry about this right now it was assumed
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that quantum systems essentially meant that
they were lots of variables which you cannot
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see and they were called hidden variables
is one of the concepts of quantum physics
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or foundations of quantum mechanics but
it was found that just by thinking that there
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are hidden variables you can explain quantum
mechanics that didn't work similarly a quantum
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process also therefore will not perhaps be
reduced to a turing machine
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but on the other hand we will be discussing
something called a quantum turing machine
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which will be an analog to the world of computers
that we know it ok so the definition of quantum
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computing which will use is manipulation of
quantum mechanical systems for information
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processing and so there are two terms here
so let us be clear one is quantum computing
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and the other one is quantum computer please
remember they are not the same thing its like
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noun and adjective you dont want to use
a noun implies noun objective and vice versa
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a quantum computer is the noun and the computing
is actually the adjective so the act of manipulation
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where do have actually act of manipulation
is the quantum mechanical system for information
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processing that is quantum computing whereas
computer quantum computer is a device that
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processes information in a quantum mechanically
coherent fashion and so it could do it could
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performs certain types of calculations now
its important to mention this certain types
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of calculations not that we have going to
at least at this point of time say that for
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every class of calculation it will be better
but for certain parts it can be far more efficient
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and this is again the example of how they
look different in bits you can either be here
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or there whereas in quantum bit or qubit you
could be anywhere in between and all possibilities
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of these betweens exist which is sought of
not an issue for the bit case
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so interestingly way back in nineteen eight
two feynman essentially looked at quantum
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computing from a different perspective he
was not thinking in terms of a computer that
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can be faster or something he basically looked
at it saying that since all natural processer
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processes are at the base or at the fundamental
level quantum mechanical you can by using
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quantum computing realistically stimulate
quantum systems that was his basic principle
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and his basic interest being a physicist of
his kind so that was his main idea that you
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could stimulate quantum systems in the most
appropriate fashion once a quantum computer
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is here that's his way of looking at it the
first realistic application game as late as
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nineteen nineteen four by shor peter shor
showed that it is possible to actually
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find the primes or basically factorize
integers into their primes in a much faster
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way then it is possible by any foreseeable
classical computer because there is the place
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where you can see an exponential speedup by
using a quantum computers
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so who will do this particular problem and
the other important place right after this
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which became very popular is the grovers
algorithm which basically searches database
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now you might thing that you know this speedup
of this is not as much as a shors algorithm
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but it is usually popular because most of
the problems that you have in a quantum computer
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finally boils down to a such problem because
in most cases what happens is you come down
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to a point when you would say the answer exists
once you say an answer exists its essentially
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a search problem of finding the answer ok
and the best part is when you can say that
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there is only a single answer and if you say
that then this is perhaps the most important
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problems the most important algorithm that
you need need to use and therefore grovers
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algorithm although maybe not the most effective
way of quantum computing but its also one
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of the most popular approaches of quantum
computing
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however the i should mention that these nineteen
nineteen four work of shor is a by far the
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most complete application of quantum computing
and also is one of the most important basis
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of cryptography some of the applications today
that is being already done in the defense
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labs is some of the countries around the world
is due to the fact that this factorization
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to give raise to this extreme security
is possible because of this particular principle
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of shors algorithm which gives raise to exponential
speedup
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ok now the basic idea of qubit is that the
case where the quantum system with exactly
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two degrees of freedom ok now let me also
point out that i am giving you the picture
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which is an evolved picture in the nineteen
eighties people were not really concerned
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about just the two degrees of freedom when
the first idea came and the ideas of superposition
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and entanglement came became clear people
wanted to use many degrees of freedom but
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soon it turned out that in order to maintain
the language which can be continuous between
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the computer scientist as well as the rest
of the community developing this so this has
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a lot of people in this area your q c essentially
is represented in one side by computer scientist
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mathematicians
and on the other side by mostly physicist
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and some very odd people who are interested
in physics for example setloit he is a mechanical
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engineer so all of these different people
who had an inclination of physics understanding
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or interest in physics have also contributed
hugely in this but in order to make sure that
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the community understands each other quite
well more to say that the computer scientist
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are able to finally able to use and make it
or compare it with the classical computing
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it was decided that the two degrees of freedom
should be the benchmark because classical
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computers essentially still realize this based
on that binary principle so we will be using
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the same principle to go between zero and
one i mention this right away here because
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that is a large and we will perhaps look at
it in some point of the other is a large number
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of work body of work which is also parallely
developed into having multiple states or
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multiple degrees utilized for quantum computing
very often those are known as qudits
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ok anything more than two one when it goes
is called qudit that's a typical area of
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development which is also happened and people
have shown that you could use qudits to do
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computing without any problem its like
saying if you have is like having a decimal
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system verses a binary system and so on and
so forth you can have more than just two so
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just to clarify now once you say two degrees
of freedom then there are some very common
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ones to look at one is say for example the
hydrogen atom ground in the excised state
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these are the quantum states spin half system
and that can be in electron positron a nuclei
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any of these would work right so thats the
situations the state of the qubit is generally
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as we mentioned superposition of two basis
states
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so here the starting point is our basis states
so thats coming from quantum mechanics
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we have been looking at the basics of quantum
mechanics before so we have all ready know
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that the everything starts off with basis
states and here we are basically confining
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ourselves to the two basis states that we
start of with the rest state and the excited
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state are the basis states of hydrogen atom
for example the rest or initially ground it
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can be anything for example again in the spin
case the spin up and spin down are the states
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that a basis states of the spin half electron
now based on these you can have the vector
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picture which we discussed which is essentially
the idea of dirac and he used this bracket
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notation a essentially representing the arrow
going one way verses the other and very effectively
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using this pictorial principles to develop
the entire shorthand math associated with
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this entire process a the reason for the
rap to develop this was he was the first want
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to do the entire relativistic quantum mechanics
and with the help of using relativistic quantum
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mechanics he was able to for the first
time show the spin quantum number to appear
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and not adhoc put in as was done in the non
relativistic scheme scheme
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so those of you who have done relativistic
quantum mechanics you all ready know what
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i am talking about so for relativistic quantum
mechanics this was critical to be able to
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develop the ideas of shorthand notation so
it was important to develop the shorthand
21:06.390 --> 21:11.570
notations to take care of the entire math
that was going to go ahead with it simply
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put its a much easier way of looking at the
entire problem there are certain notions
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that you go by doing this one is to say that
the basis states for each qubit can be written
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as zero and one for instance and then each
of them is associated with some probability
21:30.299 --> 21:38.230
or some amplitude factor and they can be all
complex so for example alpha and beta as long
21:38.230 --> 21:43.810
as they the square of them add up to one you
are allowed to take any complex number for
21:43.810 --> 21:48.549
them
so thats how you bring in quantum mechanics
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so at the very beginning whoever is entering
this field for them its important to notice
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the main difference between the classical
system verses a quantum system even here in
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terms of writing out the linear way of writing
everything out is the fact that we invoke
22:07.740 --> 22:17.970
complex numbers for the amplitudes ok and
so these numbers by themselves really have
22:17.970 --> 22:23.480
no physical understanding no physical meaning
only when you square them then they start
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having some meaning ok
so you get the feeling as to how probability
22:29.399 --> 22:37.519
is associated with the square and not just
the amplitudes so we can use any kind of a
22:37.519 --> 22:43.470
basis sets instead of zero and one we can
use any other one as long as we can distinguish
22:43.470 --> 22:52.919
them the basis states are typically are chosen
in terms of orthogonal vectors the reason
22:52.919 --> 23:00.370
being the inner product of the two vectors
then are determined or are confirmed to be
23:00.370 --> 23:06.690
always zero when you take orthogonal right
this is always going to work
23:06.690 --> 23:16.080
so you setup the problem this way so you could
also use a plus minus notations for this and
23:16.080 --> 23:22.120
you can go from one basis set to the other
this is basis set transforms again something
23:22.120 --> 23:29.270
each you have done quantum mechanically many
times for simple simple cases its just
23:29.270 --> 23:36.260
a rotation for instance here we are showing
an example of a forty five degree rotation
23:36.260 --> 23:43.470
from the original orthogonal states of
one and zero to plus and minus which are off
23:43.470 --> 23:49.480
by forty five degrees and you can just easily
rotate one to get to the other ok so this
23:49.480 --> 23:55.380
is the qubit rotation in some sense so this
in a itself whenever you look at in a answer
23:55.380 --> 24:02.860
or situation like this you are going to visualize
what it means so here is our first example
24:02.860 --> 24:14.460
of an operation why is it important because
these are the operations that form the basis
24:14.460 --> 24:21.971
of how you are going to develop your computer
so just this very simple idea that you can
24:21.971 --> 24:27.929
actually rotate a basis set gives you the
first principle idea of an operation in a
24:27.929 --> 24:36.970
quantum sense
now when you have more then one qubit then
24:36.970 --> 24:44.679
their internally arrangement gives raise to
many more conditions so for instance if we
24:44.679 --> 24:49.250
know the individuals states of electrons of
a system given below so these are three electrons
24:49.250 --> 24:56.400
let say they are individual representation
on the basis of how they are would give raise
24:56.400 --> 25:06.120
to the three different possibilities of representing
them right so i could always write my different
25:06.120 --> 25:14.370
way functions psi one psi two psi three with
amplitude factors in such a way that will
25:14.370 --> 25:20.760
give me raise to three different way function
they can be as simple as zero and one amplitudes
25:20.760 --> 25:29.000
which give raise to either one of the other
state as it is being represented here or it
25:29.000 --> 25:35.929
could be much more complex so if we write
it like this very simply and we ask the question
25:35.929 --> 25:44.539
what is the overall state of the three particle
system then will be able to look at their
25:44.539 --> 25:51.720
composite products as their solution
ok and these composite products are tensor
25:51.720 --> 25:58.220
products they are not anything else but tensor
products so this is again a definition here
25:58.220 --> 26:09.919
so the inner product was the term where we
used the basis states to tell that they were
26:09.919 --> 26:15.470
orthogonal right so zero and one when they
formed orthogonal set their inner product
26:15.470 --> 26:25.630
was zero but when i actually taken outer product
which is representing the final state of all
26:25.630 --> 26:33.340
these different qubits that i have taken that
is my outer product and so this is so my final
26:33.340 --> 26:39.559
state psi is essentially an outer product
of all these three different states that
26:39.559 --> 26:45.600
i have which is a tensor product and that
gives raise to the state and so this is very
26:45.600 --> 26:52.901
different from the inner product
for any two states if i take an inner product
26:52.901 --> 27:01.350
that is not going to be the same as the outer
product so if i want to write if i take the
27:01.350 --> 27:06.120
phi and psi for instance if i want to write
this is my inner product as i have mentioned
27:06.120 --> 27:13.789
here then the outer product will essentially
be phi and psi this is my representation which
27:13.789 --> 27:26.010
corresponds to right thats what it means so
the tensor products are written in this way
27:26.010 --> 27:36.650
i could also as well represent this psi one
psi two psi three this is the shorthand notation
27:36.650 --> 27:49.000
which goes outer product inner product and
this is how they go ok
27:49.000 --> 27:56.399
now in this particular case where this was
just set it up like this we will finally
27:56.399 --> 28:03.590
find that my state psi is going to be just
represented by a state which is one zero one
28:03.590 --> 28:15.389
which is a tensor product of the states corresponding
to each of them ok
28:15.389 --> 28:25.700
now thats how you represent states next important
thing which we would like to understand
28:25.700 --> 28:35.130
is the concept of gates so the first thing
we discussed was qubits first thing is quantum
28:35.130 --> 28:41.600
mechanics and classical mechanics concept
to computer the very basics of it these are
28:41.600 --> 28:50.279
how we built then we looked at what is the
qubit we defined qubits and we specifically
28:50.279 --> 28:56.710
mentioned the way we are going then we came
to a point where we looked at the way the
28:56.710 --> 29:03.280
qubits are defined in terms of when you have
multiple of them and then its also looked
29:03.280 --> 29:08.009
at to say whether we are going to take in
a products on outer products what does them
29:08.009 --> 29:15.519
what do they mean inner products typically
are projections outer products are basically
29:15.519 --> 29:22.450
the final state ok when you have multiple
of them and thats what we looked at
29:22.450 --> 29:27.769
now when we are going to see what can you
do with them thats when we are going to do
29:27.769 --> 29:36.690
the quantum gates now it can be parallel to
the idea of a classical gate in terms of the
29:36.690 --> 29:42.920
classical computing a gate is an operation
on a unit of data where in our particular
29:42.920 --> 29:50.690
case it is a qubit in this particular case
right a quantum gate is represented by a matrix
29:50.690 --> 29:55.460
that may be applied to a state vector now
you can already see why it is a matrix because
29:55.460 --> 30:01.370
you are now looking at states which were being
represented as vectors which are also essentially
30:01.370 --> 30:10.700
single order matrices so if you want to
operate on them and change to another one
30:10.700 --> 30:18.490
you need a matrix most of in this square matrices
so quantum gates are essentially often square
30:18.490 --> 30:27.519
matrices which can give raise to the vectors
changing into the states that you would like
30:27.519 --> 30:32.080
now we will talk about this is more detail
later but for now lets look at some of the
30:32.080 --> 30:37.100
examples of commonly used gates now this is
important case so let us start with some
30:37.100 --> 30:41.549
of the gates here one of them is called the
hadamard gate and they are represented by
30:41.549 --> 30:49.289
different notations unfortunately the
hadamard gate representation is h which is
30:49.289 --> 30:54.870
very close to an hamiltonian therefore
please note for this class we will be using
30:54.870 --> 31:05.059
the hamiltonian in the cursory manner ok
so this will represent hamiltonian because
31:05.059 --> 31:11.320
we are used to using hamiltonian for generating
energy whereas for the hadamard gate will
31:11.320 --> 31:20.120
be using the h regular h capital h then there
are these very important pauli gates so what
31:20.120 --> 31:24.700
are these pauli gates ok so i will come to
them the next one is the pauli gate which
31:24.700 --> 31:32.409
is also very important and the first gate
which is of consequence to an actual computing
31:32.409 --> 31:41.710
is the controlled not gate ok so we will go
by them quickly to understand how these goes
31:41.710 --> 31:47.360
so as an example for the classical case so
this is a parallel that you are building the
31:47.360 --> 31:57.110
simplest boolean gate is not so for the simplest
case of a single qubit what is it we put in
31:57.110 --> 32:04.059
one case and we get out the opposite of them
as a not gate whatever you put in the opposite
32:04.059 --> 32:09.370
comes out so if you go up it will come out
as down and so on and so forth so zero one
32:09.370 --> 32:19.419
goes in and you get one zero now one of the
very important things to remember is that
32:19.419 --> 32:28.169
in quantum gates however you have define
it only not only on the basis of the equivalents
32:28.169 --> 32:31.820
of zero and one but also on the basis of their
superposition
32:31.820 --> 32:37.230
now this is one very important point that
you have to remember just by looking at the
32:37.230 --> 32:43.309
final outcome which is the classical case
you could have done or made your gates but
32:43.309 --> 32:49.960
in quantum mechanics thats not just the case
every other possible superposition also coming
32:49.960 --> 32:57.730
from the same condition should also be following
this rule so thats the idea so here is an
32:57.730 --> 33:06.860
example you have a state zero and you have
state one so if you want to define a not gate
33:06.860 --> 33:11.990
what you want to do is you want to have say
not operating on zero it should give one and
33:11.990 --> 33:17.470
not operating on one should give zero ok thats
what you expect
33:17.470 --> 33:24.070
so the action of quantum not gate on a superposition
must then also have a situation where if i
33:24.070 --> 33:29.870
if i do this then it should work which means
that this very important point is going to
33:29.870 --> 33:35.919
be always maintained incase of quantum mechanics
there all the quantum operations are going
33:35.919 --> 33:44.789
to be linear ok so this linearity now it should
not come to you as a surprise because you
33:44.789 --> 33:49.000
know that when we did quantum mechanics or
when we have been learning quantum mechanics
33:49.000 --> 33:55.470
we have always mentioned the quantum mechanics
is always going to work in a linear way all
33:55.470 --> 34:01.210
the operations in quantum mechanics are linear
the allowed operations or the ones which you
34:01.210 --> 34:06.879
actually call are measurable operations always
linear so whenever you use a hamiltonian the
34:06.879 --> 34:12.110
hamiltonian is linear whenever you use a momentum
operator the momentum operator is linear the
34:12.110 --> 34:20.550
position operator is a linear ok so thats
the point that if it works for any particular
34:20.550 --> 34:25.220
set then all the superposition states also
should be following that which means that
34:25.220 --> 34:30.850
a requirement of quantum operations are that
they are going to be linear
34:30.850 --> 34:37.950
now how do you get that so this is the logic
behind why do you have a matrix for a quantum
34:37.950 --> 34:45.280
gate the logic being that the quantum operations
have to be linear right so we can represent
34:45.280 --> 34:52.780
the action of a quantum gate by a matrix so
the quantum not gate or or its also known
34:52.780 --> 35:02.010
as the pauli x gate is written by this form
the x is equal to a matrix is zero one zero
35:02.010 --> 35:09.421
one zero this one whenever you operates so
here is a way how it works the quantum state
35:09.421 --> 35:18.560
is represented by a vector so zero is one
zero whereas one is zero one and the any psi
35:18.560 --> 35:24.440
which is a linear combination of this two
can be written in this form
35:24.440 --> 35:30.820
so you can now express you can now express
the not operation as a on a generalized qubit
35:30.820 --> 35:36.820
by using a matrix manipulation method and
you will find that this always works so thats
35:36.820 --> 35:50.280
why your not operator x is working properly
as a quantum gate ok and you can immediately
35:50.280 --> 35:57.500
see that not was a good one to take because
not is one of the classical gates which is
35:57.500 --> 36:03.060
reversible so i specifically choose the one
to start with which is the reversible one
36:03.060 --> 36:09.130
its make sense everything goes well so us
let us look at another important single
36:09.130 --> 36:15.000
qubit gates so this was a single qubit gate
we will just use one qubit the other single
36:15.000 --> 36:20.310
qubit gates which are important the that was
the pauli x gate and this works on one qubit
36:20.310 --> 36:26.280
the other common single qubit gates are the
pauli z gate so the representation therefore
36:26.280 --> 36:34.020
would then be in terms like similar to the
way that you write the classical ones you
36:34.020 --> 36:41.360
will be writing out how they are pauli z gate
is zero one zero zero minus one then pauli
36:41.360 --> 36:49.240
y gate is a multiplication of the z and the
x and the hadamard gate is a essentially a
36:49.240 --> 37:01.090
square root of the not gate ok
so you can for your own you can do these molecule
37:01.090 --> 37:07.840
excises to prove it to you that this is how
they work so if you just apply y what does
37:07.840 --> 37:14.440
the y look like you should write it out similarly
the z is anyway given but the fact that
37:14.440 --> 37:19.090
the hadamard gate can be actually written
in this form by taking a square root of not
37:19.090 --> 37:25.730
just simple exercise you should just refresh
your matrix manipulation by doing this
37:25.730 --> 37:38.400
x ok now the summary of the simple gates therefore
are these x gates basically the not gate alpha
37:38.400 --> 37:45.380
one zero converts to one beta one converts
to zero then the y gate which basically is
37:45.380 --> 37:52.920
a complex conversion let as a part which takes
you like this and then the z gate gives you
37:52.920 --> 37:58.540
that finally the hadamard gate now this hadamard
gate is actually very important important
37:58.540 --> 38:06.070
as will find out later because what it is
producing is a superposition of the two
38:06.070 --> 38:14.740
states that are involved with the individual
amplitudes
38:14.740 --> 38:25.540
right so basically your sort of equalizing
the two states right i mean if i take a
38:25.540 --> 38:34.870
if i take my basis now to be zero and one
superposition of zero and one any one of them
38:34.870 --> 38:40.170
then i will get a fifty fifty combination
of them so thats the reason why it becomes
38:40.170 --> 38:48.850
useful now the reversibility requirement is
we have being doing that but now this is the
38:48.850 --> 38:56.370
point where we should look at it since
all the quantum operations have to be reversible
38:56.370 --> 39:01.660
we have this principle that our gates have
to be designed in a particular way so that
39:01.660 --> 39:07.950
they are always reversible this is not true
for the classical case and that is what is
39:07.950 --> 39:13.880
written in next line which is that boolean
operations are not necessary reversible
39:13.880 --> 39:18.900
a reversible operation is always given by
a unitary matrix for which so all our quantum
39:18.900 --> 39:23.830
gates will be written will be possible to
be written in terms of this kind of a unitary
39:23.830 --> 39:28.290
transforms
so that is one other point which we have any
39:28.290 --> 39:35.170
gate that we device will have a unitary transform
so it will be possible to get this inversion
39:35.170 --> 39:44.580
very easily ok so any operation that you
have you have its inverse available right
39:44.580 --> 39:50.870
so thats the advantage of it so now let us
look at the hadamard gate because its an important
39:50.870 --> 39:57.530
gate and as i said it places it in a superposition
of one and zero so this is the gate which
39:57.530 --> 40:07.580
takes one qubit and puts it in the superposition
state of its own two states
40:07.580 --> 40:13.770
so this is important because you will find
many practical uses of this hadamard gate
40:13.770 --> 40:20.570
later on and this is the representation that
you will so in in shorthand notation when
40:20.570 --> 40:27.020
you write circuits which is analogues to
computer circuits you will be having this
40:27.020 --> 40:32.390
kind of scenario where we feeding in the states
and you will be having an output state like
40:32.390 --> 40:37.320
this with just the circuit diagram given like
that so you should be able to interpret these
40:37.320 --> 40:42.380
kinds of simple circuit diagrams to you thats
one of the learning that we do in this because
40:42.380 --> 40:48.700
we will be soon going to cases where i will
be simply showing you pictures like this and
40:48.700 --> 40:54.090
will not be writing down these other states
if i give you this then you should be immediately
40:54.090 --> 41:01.820
able to recognize the that this is how it
is or if i just give you this there should
41:01.820 --> 41:05.560
be able to understand how this is happening
this is important because these the language
41:05.560 --> 41:10.370
that will be following for building up quantum
circuits
41:10.370 --> 41:16.840
so the hadamard gate for instance can give
you very interesting results so with zero
41:16.840 --> 41:27.280
as an input you will be getting a superposition
of the sum with one you will get the superposition
41:27.280 --> 41:36.140
of the difference when you put in the superposition
state in there your actually get a superposition
41:36.140 --> 41:49.010
of the individual components with equal probabilities
of theirs ok or another way of writing
41:49.010 --> 41:53.660
is this one where we are actually taking in
some sense this is the very important gate
41:53.660 --> 42:04.150
for basis transform
this is very important because in quantum
42:04.150 --> 42:11.670
mechanics is all about taking the problem
to a basis where it is easy thats the reason
42:11.670 --> 42:16.940
why quantum mechanics is quantum computing
is also going to take advantage of that and
42:16.940 --> 42:25.290
the design of a quantum algorithm or anything
else that you can think of is ninety percent
42:25.290 --> 42:30.440
relying on this idea
so how smoothly can you come up with ideas
42:30.440 --> 42:37.350
with which you can do your basis transform
to a condition where it is the easiest to
42:37.350 --> 42:42.280
solve the problem and then you come back to
the state or the transform back to the case
42:42.280 --> 42:48.340
where you wear so that you can get your result
ok thats the point of this entire place so
42:48.340 --> 42:53.740
hadamard is almost nineteen nine point nine
percent you will always find a hadamard gate
42:53.740 --> 42:59.490
existing in any computing you cannot avoid
it because you always need to do basis transforms
42:59.490 --> 43:03.900
that's one thing
similarly if you input this superposition
43:03.900 --> 43:09.300
you will get back one so this is basically
complement so you its shows that its a
43:09.300 --> 43:14.400
completely reversible condition so if you
if you believe in this than you can immediately
43:14.400 --> 43:23.290
get the other and so on so forth the other
important gate is the pauli gate once again
43:23.290 --> 43:32.650
its important because the pauli gate is
in important for phase shift which means that
43:32.650 --> 43:44.440
the phase of the two states can be reversed
for example this is their zero and one
43:44.440 --> 43:50.210
are in the same direction sum but the reversal
will make one is plus and one is minus so
43:50.210 --> 43:59.490
this is the phase shift whereas the sigma
x is a bit flip so we started of by saying
43:59.490 --> 44:08.270
these were the z x and y gates more formally
in terms of the pauli gates they are more
44:08.270 --> 44:16.250
known as the sigma z sigma x and sigma y and
to complete the identity goes along with it
44:16.250 --> 44:28.320
so these are our final set of four sigma four
pauli gates and these are all very important
44:28.320 --> 44:35.700
in terms of anything to do with spin ok
so thats the part which is very important
44:35.700 --> 44:44.020
about these pauli gates so one of them
so you can see there is an understand the
44:44.020 --> 44:50.030
z is only one dimension zee sigma zee thats
only phase shift if you do a bit flip thats
44:50.030 --> 44:56.180
along the x axis if you do both phase shift
and bit flip which means that you are doing
44:56.180 --> 45:03.980
y which is a composite of zee and x will
find this and identity doesnt do anything
45:03.980 --> 45:09.340
but this is necessary to complete the gate
z thats what the pauli gates are
45:09.340 --> 45:15.000
now these are the corresponding pauli matrices
they are also often written as zero one two
45:15.000 --> 45:23.780
and three zero is nothing but the identity
sigma zero then sigma one is the x sigma
45:23.780 --> 45:31.040
two is the y and sigma three is the z zee
i have given you all the different notations
45:31.040 --> 45:36.470
which are used in different books ok each
books uses different notation and to be clear
45:36.470 --> 45:40.670
i have put them all in this particular slides
so that there is no confusion wherever you
45:40.670 --> 45:47.810
see these you know these are the most popular
single qubit gates right ok
45:47.810 --> 45:55.490
now the final the first gate which we will
do which is more than the single qubit gate
45:55.490 --> 46:05.820
is the control not gate this is a two qubit
gate ok so why you need more than one in
46:05.820 --> 46:12.370
this case is because you need a bit for the
control ok so its more often known as the
46:12.370 --> 46:18.240
so this this is our other bit requirements
so the control not gate or the c not gate
46:18.240 --> 46:23.210
as is popularly known is the standard two
qubit quantum gate and its defined in this
46:23.210 --> 46:30.430
fashion when you are only operating on
zero zero then its just going to give back
46:30.430 --> 46:38.040
the same one because your control parameter
essentially the zero so whenever you have
46:38.040 --> 46:46.560
zeros then its leaving it so your operation
and the control essentially gives raise to
46:46.560 --> 47:01.140
what is going to happen
so the one flips the bit and the zero is no
47:01.140 --> 47:09.960
change so thats your control the first bit
is your control that is working on the other
47:09.960 --> 47:18.560
one to do the not right so thats why its a
control not so the state of the first if it
47:18.560 --> 47:26.640
is going to be one will let will give rise
to a flip if it is going to be zero it will
47:26.640 --> 47:35.340
not let it will not make any changes right
so thats why its a control the first bit decides
47:35.340 --> 47:44.910
as to whether it will act like a not gate
or not so thats why its a control not
47:44.910 --> 47:53.780
so it has a notation this is its notation
a plus with zero inside inside the zero
47:53.780 --> 48:01.751
there is a plus so its a generalization of
the xor classical xor ok if you they call
48:01.751 --> 48:07.500
it x or xor whatever way you want to call
it so the classical x or is essentially these
48:07.500 --> 48:14.080
where you have this principle and here you
are using this two qubit case where the second
48:14.080 --> 48:22.190
one is second bit is going to behave based
on what the first bit is going to be its drawn
48:22.190 --> 48:26.560
in this fashion this is also important to
know because you would like to see so basically
48:26.560 --> 48:31.960
the first bit remains as it is so whether
it is zero or one remains as it is the second
48:31.960 --> 48:37.480
one is going to change and depending on whether
the control is going to do something or not
48:37.480 --> 48:44.950
its going to show off as a result thats how
it is
48:44.950 --> 48:54.220
so the first so obviously if it's a two qubit
gate then it has to be a four by four matrix
48:54.220 --> 48:59.080
because for the one qubit gates we were using
two by two matrices so this is a two qubit
48:59.080 --> 49:06.580
gate this will be a four by four matrix so
the simplest matrix corresponding to the c
49:06.580 --> 49:10.511
not will therefore be this so here here you
can see that this is my c not gate and and
49:10.511 --> 49:18.220
here is an example of applying the c not gate
for a for a matrix which is y for a for
49:18.220 --> 49:34.660
a state which is y and you get the solution
now this c not gate along with the single
49:34.660 --> 49:45.650
qubit gates are universal for quantum computer
ok irrespective of how you formulate your
49:45.650 --> 50:03.240
problem these will be always a complete closed
set ok as you go and make an bigger gates
50:03.240 --> 50:11.280
they can often be broken down into these basic
gates but these gates cannot be broken down
50:11.280 --> 50:17.740
further so thats what is known that you know
if you can go to the very basic states thats
50:17.740 --> 50:23.580
how they are ok
so how how can you actually use them so the
50:23.580 --> 50:30.990
first important thing about the quantum computing
is that once you have learnt your qubits and
50:30.990 --> 50:34.900
you have learnt some operations which are
you gates then you can actually write out
50:34.900 --> 50:43.300
your circuits so we have been doing that but
here is a here is a basic principle of
50:43.300 --> 50:54.870
how to go about doing something which is
combination of more than one applied gates
50:54.870 --> 51:02.010
to get to where you want to so for example
we would like to just show a quantum swap
51:02.010 --> 51:09.860
circuit so i would like to go from a to be
and b to a ok so the point is when you if
51:09.860 --> 51:17.970
you if you apply only one gate it will be
not possible to swap both is a c not will
51:17.970 --> 51:23.330
only swap one of them and the other one remains
constant but if you apply it twice then you
51:23.330 --> 51:26.290
end a producing a swap gate a quantum swap
circuit
51:26.290 --> 51:31.880
so this is exactly an example of that so by
using just control nots in series you will
51:31.880 --> 51:38.210
be able to get this happen right so you can
actually start building your interesting circuits
51:38.210 --> 51:47.640
right from this very basics that we have just
learnt ok so this is a circuit let see
51:47.640 --> 51:54.090
so here is the point that is being shown step
by step so the what does the first one do
51:54.090 --> 51:59.190
first one generates a c not so you get this
next one you have not applied anything on
51:59.190 --> 52:04.230
this part so its only going to be the next
and then the second one is going to be this
52:04.230 --> 52:10.940
which is going to go to become b and then
that one again remain same whereas the other
52:10.940 --> 52:19.690
one finally goes and applies this one to get
back there so how many c nots did i use one
52:19.690 --> 52:29.110
two and then the third one now if you tell
somebody who does classical computing will
52:29.110 --> 52:34.880
just look at and you say come on you know
is a swap qubits they equal swap qubits for
52:34.880 --> 52:40.210
that you have gone through three not givens
three controls and three gates to do this
52:40.210 --> 52:45.450
but one very important thing to note here
however is that this is reversible i can as
52:45.450 --> 52:51.300
well come from this side and i will have the
same result thats not true when you do a classical
52:51.300 --> 52:53.450
computing
in classical computing yes you can actually
52:53.450 --> 52:58.230
swap them just like that but you are r a you
are expending energy you want to go the other
52:58.230 --> 53:02.810
way round is expending energy because it is
not reversible its not the same way how it
53:02.810 --> 53:11.940
works ok so thats the idea so that brings
it to a very important point what are the
53:11.940 --> 53:19.700
basic features that we are now looking for
in a quantum circuit we cannot do a few things
53:19.700 --> 53:29.240
that we have just now realized no loops are
allowed ok quantum circuits are acyclic you
53:29.240 --> 53:37.540
know why ok so there are no loops which are
allowed quantum circuits are cyclic acyclic
53:37.540 --> 53:47.630
because you cannot paralyze loop circuits
even we when you do parallel prosing in
53:47.630 --> 53:53.550
a classical computer the most difficult part
is the nested loops so generally speaking
53:53.550 --> 53:58.880
you are not really supposed to have nested
loops if you have trying to do parallel prosing
53:58.880 --> 54:02.280
so for example fan in you can understand that
54:02.280 --> 54:07.040
why fan in is not allowed you are going to
loose qubits you cant have that and similarly
54:07.040 --> 54:11.340
you cannot gain qubits so you cannot have
fan out what we have going to do is that we
54:11.340 --> 54:18.090
have going to take on more from this point
on into for the lectures for today let us
54:18.090 --> 54:23.990
stop here and we will continue on with this
in the next lecture
54:23.990 --> 54:29.950
thank you