in the last lecture we looked at the n m r
principles of quantum computing as that was
the first one to have the practical implementation
problem now let us get into the optics part
and let us start with the single photon qubits
the single photon qubits have the advantage
that it can deal with the single aspect of
measurement instead of ensembles which has
been one of their difficulties in terms of
doing the quantum computing with m n r one
of the popular ways of looking at single photon
qubits have been the idea of the polarization
where the photon polarization could be either
horizontal or vertical and they are often
denoted into the two different states by using
this particular polarization principle
so a horizontal h photon represents a logical
zero and a vertical v photon represents a
logical one so the ket zero would represent
h ket and ket one would represent v ket any
arbitrary state can be then plotted on the
block or pointier sphere examples of diagonal
would be like ket d which is equivalent to
a superposition of zero and one ket vectors
similarly anti diagonal would be say the ket
a which is a difference of zero and one the
other possibilities of right circular and
left circular can also be taken and these
are all pictorially represented in these pictures
that we show here application of any gate
would then involve using optical elements
which can perform the transformations that
we are interested in
so for example single qubit gates can be easily
realized by using birefringent wave plates
that retard one polarization by a fraction
of the wavelength relative to an orthogonal
polarization causing a rotation of the state
on the block sphere with the axis of rotation
determined by the orientation of the wave
plate so for example hadamard gate could therefore
be achieved by using a half wave plate which
would have the horizontal and the vertical
polarizations be equally mixed up once the
wave plate is set at half of forty five degrees
which is twenty two point five degrees so
any arbitrary rotation would therefore be
possible if we placed series of wave plates
as it is shown here by either using a quarter
wave and half wave and quarter wave combinations
converting between polarization and path encoding
requires a polarizing beam splitter p b s
as it has been shorten here which transmits
the horizontal and reflects the vertical and
a half wave plate oriented at forty five degrees
which transforms the vertical into horizontal
these are essentially mirrors which are turning
the laser beams as we are looking at
so single photon qubits can be quite attractive
in terms of providing the principle of cubit
for doing computation one of the most important
part in this process is the concept of generating
single photons which involve lasers which
we have also discussed it at length in our
course and the single photon source would
essentially involved reducing the number density
of the photons coming out of the light source
to a point where the detector starts giving
results which could be associated with the
single photon essentially clicking the detector
in terms of the number of photons which are
received by the detector so these are principles
which we have discussed at length in class
and here is essentially the brief summary
of how single photons can have single photons
have become useful as quantum computing devices
the other important development in terms of
quantum computing has been the trapped ions
where an atom have been isolated into an environment
where it can be kept in an isolated condition
so that it can be made to change it's property
in a quantum mechanical way so electrically
charged atoms or ions have quantum energies
that depend on the location of electrons tune
blazers can cool and trapped ions and put
them in superposition states so here is an
example where a laser can essentially put
the electron in a superposition state between
the ground and the excited state so if this
is the ground state and this is the excited
state of the atom or the ion concerned
the use of a laser can essentially create
a superposition state between the ground in
the excited state the advantage of such isolated
ions is that they have very long longevity
greater than thousands of seconds and the
logic success rate because it uses laser switch
attuned precisely it could be as high as ninety
nine point nine percent and by using this
technique until now fourteen entangled ions
have been shown to be trapped is it a very
high number in terms of qubits and as we discussed
in the class presently a new company have
been floated it is called ionq which is going
to look at building the commercial version
of this development the advantage in this
technique as i mentioned before it's a very
stable it has the highest achieved gate fidelities
ninety nine point nine percent however it
has difficulties in terms of it's slow operation
and for making any of these processes to go
on many lasers are needed
in terms of commercial aspects in the race
to build quantum computers companies are pursuing
many types of quantum bits or qubits each
with it's own strengths and weaknesses however
that is still limited as compared to the academic
researchers who used far more varieties and
options so we will first look at the commercial
ones as we have seen ionq is gearing up to
get into the commercial mode for this one
of the new information which is come out as
late as december first two thousand sixteen
as i mentioned in class it was the fact that
the university of maryland college park physicist
chris monroe along with jungsang kim from
duke university have floated this company
ionq to setup and ion trapped based quantum
computing
so this the quantum computing startup that
examines equipment that keeps ions trapped
in long lasting quantum states so that is
one of the areas this particular development
is going the other commercially viable approach
that has been looked at is the superconducting
loops for quantum computing where a resistance
free current oscillates back and forth around
a circuitry an injected microwave signal excites
the current into superposition states here
the difficult is the longevity is in terms
of a very short time five into ten to the
power minus five seconds the logic success
rate is still quite a ninety nine point four
percent and the number of entangled qubits
have been about nine there have been quite
a few companies who are interested in this
and they have been supporting this area of
research which include google ibm quantum
circuits etcetera
the advantage here is that it's fast working
and it builds on the existing semiconductor
industry that's one of the biggest advantage
and that's why the companies are interested
however the difficulties are also quite pronounced
that it can collapse easily and must be kept
quite cold the advantage of the semiconductor
industry in terms of fabricating and working
on this essentially has made this an area
where a lot of effort has going on in terms
of commercial aspects
the main commercial quantum computers is from
d wave systems which is a company based in
canada so they started their venture as early
as january nineteen two thousand seven when
they announce the creation of a prototype
of commercial quantum computer called orion
according to d wave adiabatic quantum computer
orion uses sixteen qubits and can solve quite
complex practical problems now this was way
back in january two thousand seven ten years
ago and they use the principle of adiabatic
quantum computer to do this heroic feet and
initially they had this difficulty that they
were not disclosing any technical details
of their computer which caused a significant
criticism among specialist however their stand
has changed over the years and it has been
put to use in many places including ibm and
recently the company received seventeen million
investments
so one of the later versions of the d wave
quantum computer is known as the d wave two
x quantum computer which came out about a
couple of years ago and had a lot of nice
review on it and we discussed at length about
this this is from the website of the company
which is given here again and their main advantage
has been the fact that they have managed to
isolate the system where the quantum processor
works to a temperature which is colder than
interstellar space and this feet which they
were able to achieve has managed to isolate
their system quite effectively and given them
the power of the quantum processor so they
give quite a bit of a detailed nowadays about
how they go ahead about doing their computation
and this is quite impressive reducing the
temperature of the quantum processor to near
absolute zero is required to isolated it from
it's surroundings so that it can behave quantum
mechanically
in general the performance increases as temperature
is lowered the lower the temperature the better
the d wave two x processor operates at a temperature
of fifteen milli kelvin which is approximately
hundred and eighty times colder than interstellar
space the refrigeration system used to cool
the processor is known as a dry dilute refrigerator
it uses liquid helium in a closed loop cycle
in which it is recycled and re condensed using
a pulse tubes cryocooler the closed loop refrigeration
removes the need of the onsite replacement
of liquid helium and makes a system suitable
for remote deployment and that has been their
biggest achievement in terms of commercial
viability while dilution refrigerators are
not uncommon in research environment d wave
has advanced technology to ensure long life
and high reliability as the cooling power
available at such low temperatures is extremely
low d wave has taken great care to minimize
heat loads and effectively manage the heat
transfer within the system and given their
development they have managed to ensure that
despite the extreme environment inside the
system the d wave quantum computer can be
located at a standard data centre environment
so these have been some of the major technological
developments which the company was able to
achieve which made sure that their processor
was able to be placed at near zero kelvin
temperatures so that it could behave in a
manner close to quantum conditions so this
is the slide that i used in the lectures with
the only exception that i have upgraded the
fact that in this year the number of qubits
now they have advanced to two thousand so
they had one thousand qubits until last year
where this particular d wave two x system
could search through two to the power thousand
possible solutions simultaneously and the
search principle that they use is based on
simulated annealing concept so their specific
idea is quoted here solving problems with
the d wave two x system can be thought of
as trying to find lowest in a landscape of
peaks and valleys every possible solution
is mapped to coordinates on the landscape
and the altitude of the landscape is the energy
or cost of the solution at that point
the aim is to find the lowest point or points
on the map and read the coordinates as this
gives the lowest energy or optimal solution
of the problem so essentially this is looking
for the lowest energy point or energy minimization
principle what it takes advantage of is the
principle of quantum tunneling which allows
the quantum computer to explore the landscape
in a way that is otherwise not possible with
the classical systems the quantum tunneling
is explained in terms of say a layer of water
that covers the entire landscape as well as
running over the surface water can tunnel
through the mountains as it looks for the
lowest valley the water is an analogy for
the probability that a given solution will
be returned
when the quantum computations occur the probability
is pooled around the lowest valleys the more
water in a valley in terms of the analogy
the higher the probability of the solution
being returned a classical computer on the
other hand is like a single traveller exploring
the surface of a landscape or one point at
a time so this is the analogy to finding out
how they get to their best possible solution
in a search environment so they basically
rely on the idea that there is an adiabatically
cooled system so that the lowest possible
is possible to be seen however it would also
involve assigning very judiciously the probabilities
to all the values that come out as a result
of this so there are applications of search
computers however this is not the very straight
forward way of the developments of quantum
computer that we have been discussing or for
that matter how the evolution of quantum computers
have happened over the years it is one of
the places where certain levels of advantage
exist and so a lot of effort has gone into
developing this kind of quantum computer
some levels of skepticism still exist around
the d wave computers however they have had
certain applications and certain areas advances
with respect to classical computers have been
noticed but it is not possible to ascertain
a complete benefit of the number of qubits
that seem to exist in such systems we discussed
at length in more detail about the d wave
computing in our regular classes if you wish
to you can go back and revise those lectures
to ensure that you understand the little bit
more on the principle of their operations
and the technology that they have used in
terms of the superconducting circuitry and
connectivity to their quantum computing processor
that they have kept at very low temperatures
the other areas of commercially viable quantum
computing approaches include topological qubits
where quasi particles can be seen in the behavior
of electrons channel through semiconducting
structures their braided parts can encode
quantum information and this kind of work
has been supported through microsoft and bell
labs a lot of information on this particular
area is yet to be forthcoming the advantage
of this technique is that it's a greatly reduced
error process however the biggest difficulty
of this process is the fact that the existence
of this theoretical concept is yet to be confirmed
completely through experiments there is still
a lot of dilemma about this development
another area where a lot of effort has been
going currently is in the area where vacancies
in diamonds have been utilized for quantum
computing and nitrogen atom and a vacancy
add an electron to a diamond lattice diamond
as you know are made from carbon atoms put
together it's quantum spin state along with
those of nearby carbon nuclei can be controlled
with light it is a fairly long longevity about
ten seconds and it's logic success rate is
also not bad ninety nine point two percent
number of entangled qubits as of now is a
six however it is difficult to entangle six
is a heroic number in this particular case
the advantage is that it can operate at room
temperature that's one of the huge advantage
as of now because most of the techniques that
we discussed are not quite operable at room
temperatures and this is being supported by
the quantum diamond technologies in most of
these cases as had mentioned previously the
longevity is the record coherence time for
a single qubits superposition state while
the logic success rate is the highest reported
logic fidelity for logic operation on two
qubits and number of entangled state is the
maximum number of qubits entangled and capable
of performing two qubit operations so these
have been the benchmarks which have been utilized
for understanding or making a comparison between
these different techniques
another area where a lot of effort have actually
gone in although the number of entangled qubits
are low this is the area of silicon quantum
dots where the artificial atoms silicon quantum
dots which can be controlled by using microwave
radiation these artificial atoms are made
by adding an electron to a small piece of
pure silicon the microwaves control the electrons
quantum state longevity is not great it's
about point zero three seconds and the logic
success rate is about ninety nine percent
and the number of entangled states are still
not very large about two the advantage is
on it's stability and it is built on existing
semiconductor industry
the difficulty and the negative part is the
fact that only a few entangled qubits are
demonstrated and it has to be kept cold and
an intel is the company which is supporting
this effort given the fact that it's a connected
to the semiconductor industry based on these
discussions we would like to revisit the quantum
algorithms that such quantum systems have
been looking at since we already discuss the
grovers search algorithm and the shors factorization
algorithm earlier even in this week i am going
to start off by discussing these concepts
and in going forward with this let me first
give you the principle behind the grovers
algorithm where in one hour lectures before
we had even shown wave optical search engine
which did not have anything to do with quantum
systems but it was the wave nature of light
that was utilized to show at grovers algorithm
could be demonstrated in principle
so it is indeed possible to show that wave
optics principles can demonstrate superposition
as well as the principle of grovers algorithm
because it relies on the concept of superposition
however the resources scale higher in this
process of classical nature of the wave compared
to the quantum process so in a true quantum
computing scheme the space resources necessary
would be much less as compared to that what
is necessary for the classical wave operation
case so this was one of the cases that we
had discussed before and i thought i would
bring it back on to make sure that you understand
that just because classically it is possible
to do superposition it doesnt mean that i
just thought that i will bring it back to
you in order to remind you that although grovers
algorithm could work in a complete classical
manner in terms of having superposition alone
the idea of not having the quantum nature
in the process of developing it would give
some difficulty in terms of scaling the problem
so that was one of the things which i just
wanted to bring back to your memory
in typical grover search algorithm we are
looking for a solution to a problem by utilizing
the oracle or a black box that can check whether
a given answer is correct so for example if
we are thinking of a number for instance here
i think of three and we put in any other number
than three the oracle is supposed to say no
until and unless i put in the number three
and the oracle is supposed to say it's correct
so the idea of an oracle in this concept -[con]text
is basically to ensure that it can check the
correct answer so in terms of a classical
computer you need to have at least n by two
queries to be made on a data set for the oracle
to check the correct answer so the best classical
computer can do on an average is n by two
queries so this query concept is the principal
where the oracle becomes critically important
whereas in case of a quantum computer since
the superposition of all n possible inputs
are possible this can be achieved much faster
and that was the basic idea behind the grovers
algorithm which says that a quantum computer
can find the answer is root n queries
the advantage of this it is the fact that
it can be used on any unstructured search
problem even in terms of non determined polynomial
complete problems however the disadvantage
is that it is only a quadratic speed up over
the classical search the circuit is not complicated
because it essentially involves a bunch of
hadamard operations followed by policy operations
but it doesn't provide any immediacy intuitive
picture of how the algorithm works so we could
ask the question are there any intuitive models
for quantum search when we utilize shors factory
in algorithm it utilizes quantum fourier transforms
which is exponentially faster than classical
fourier transforms
so for example if you would like to find the
factors of fifty seven which is three and
nineteen you take at least exponential time
to get to the answer all known algorithms
for factoring an n bit number on a classical
computer takes time proportional to order
of n factorial which is an exponentially growing
problem however the shors algorithm for factoring
on a quantum computer takes time proportional
to order of n squared log n which is the polynomial
scaling the details of shors factoring algorithm
are more complicated than the grovers search
algorithm but the results are very clear with
classical computers the number of bits are
thousand twenty four as our computers become
more and more powerful we could achieve this
in let's say three days in the year twenty
forty two whereas if we increase the number
of bits to twenty forty eight or forty ninety
to six it just becomes more and more complex
to the point that it's not possible to really
make a much of say about it can be say factored
or not
but on the other hand with potential quantum
computers where the order is polynomial it
would still only take four and four point
eight hours if you get to a forty ninety six
bit system so this was essentially shown by
his work by hughes and it is straight out
of this ah write up that they had on this
work so this so this particular part of the
lecture was dedicated towards showing how
the commercial principles of quantum computing
and the developments have been utilized in
recent times to get to developments where
the most important aspects of quantum computing
developments in terms of shors and grovers
are possible to be shown by these techniques
and they seem to do a pretty good job in terms
of showing the scaling is in our favor in
terms of getting the use of quantum computing
so with that i would like to end today's lecture
we will see you next time with more details
on some of the other parts of the course that
i haven't discussed until now see you next
time
