Hello and welcome to this lecture on quantum
computing and quantum information this is
a continuation of our theme of discussing
experiments in quantum computing let us look
at what we have installed today.
Last lecture we discussed the DiVincenzo’s
criterion which where the set of constraints
that an experiment in system was fulfill to
be able to do quantum computing we understood
decoherence and we also discuss some models
of decoherence we actually considered NMR
nuclear magnetic resonance is an example of
such a system today let us see why there are
problems with the NMR qubit why it is unsuitable
for quantum computing and let us then move
on for an alternative which is trapped ions.
So let us see why NMR is a bad qubit NMR is
a bad qubit because the nuclear magnetic resonance
qubit is in what is called a pseudo- pure
state it is at an inverse temperature ß which
is given by 1/ KT so the initial state of
an NMR qubit can be written approximately
as 1 + ß??0IZ let me remind you that the
IZ is just half sZ and this tells us what
the problem is the problem is that even though
we have access to a single qubit in the form
of the sZ or the IZ part of the density matrix
almost all of the population is in this maximum
mixed state which is given by the identity
matrix.
So what this means is that this qubit actually
exhibits really bad signal to noise ratios
and this unsuitable for large-scale quantum
computing it is unsuitable for this scaling
relationship in quantum computing so let us
look at an alternative and as an alternative
let us look at trapped ions to consider trapped
ions as qubits we one must first discuss how
to trap ions.
So let me mention that typical ions that are
trapped are calcium, beryllium, magnesium
and for trapping one would like to take the
simplest approach possible so the simplest
approach is to see from electrostatic potential
is able to trap an ion after all it is a charged
particle this is in general not possible and
these are consequence of a theorem in electrostatics
called Earnshaw theorem let us take a look
at it.
Now Earnshaw theorem is just a simple statement
that a source free potential must obey the
Laplace equation ?2V = 0 now if you expand
the ?2 we can write this for a two dimensional
potential as ?2V/ dx2+b2V/dy2 = 0 and now
we see what the problem is the problem is
that if the curvature of the potential along
the x axis is positive let us say the first
term here is positive then the second term
has to necessarily be negative for these two
to add the 0 what this means in a picture
is that if we was the potential that I wanted
to consider and if I imagine that instead
of a charged particle I just have a marble
and that V is a gravitational potential.
Then if the marble were moved a little bit
along this axis it will roll back gently to
this place just like a marble would sitting
in the bottom of a hill but with respect to
this direction it is on top of a hill and
if I perturb the marble a little bit towards
this direction on the opposite direction going
down if you just roll away and so this requirement
that ?2V =0 basically translates to the fact
that there are no stable trapping potentials
electrostatic offers to us, so what is the
solution around it?
The solution actually is available even in
this analogy that I pointed out.
So let us stick with the analogy of a saddle
and the marble in a saddle and let us see
what can be done.
So now imagine that instead of a stationary
saddle we have a saddle that is spinning about
one of the axes that is indicated to it.
So consider this saddle to be spinning with
a frequency ?, if this marble stations at
the center of this potential is not displaced
a little bit, if ? is slow enough then the
marble simply rolls away, the saddle is turning
the entire time but the marbles motion is
much faster than the saddles.
So the spinning of the saddle has no effect
on the marble being in an unstable potential
this problem has not been resolved, now imagine
the opposite limit where the frequency with
which the saddle is fun is rather fast, if
the frequency is fast in comparison with the
typical dynamics of the marble let us the
amount of time that the marble takes to go
from here to up to a little bit far away from
here.
If this frequency is much faster than that
typical dynamics then what happens is that
before the marble has a chance to get away
too far, the saddle has spun to a place where
this potential is in the place of this potential.
So what this means is that while the marble
was trying to roll away the potential spun
around so that now it offers resistance to
the marbles motion in by increasing this height,
right.
So if you do this spinning quickly enough
what happens is that the marble finds itself
to be localized around the center and this
basically is the solution to the problem that
we are facing, this spinning saddle basically
the way it breaks Earnshaw's theorem to use
a colloquial world, is that it takes this
potential V (x, y) and introduces a primes
component to it, the frequency implies that
there is a time component.
And this time is, is crucial in this entire
picture because what it does is it allows
us this additional degree of freedom to control
this, this part.
So let us see how to translate that with trapping
ions.
So the spinning saddle idea inspired what
is now called the Paul trap, the Paul trap
consists of four electrodes which I have indicated
in red and green here and the ions are along
the z-axis on the center.
Now I described the fact that one must change
the potential in the time depend fashion and
if you look at what happens to these electrodes
we see that while is at time T = 0 along the
diagonal the two electrodes are positively
charged which is to say that if the potential
is + V0/2 cos (?t) along the anti-diagonal
the potential is negative which is to say
it is - V0/2 cos (?t).
In a short time what happens, is that the
polarity flips and along the diagonal the
potential is negative and along the anti-diagonal
the potential is positive, this spinning saddle
is now emulated by the Paul trap by simply
transforming the time-dependent potential
in this way.
What happens is that this prescription traps
the ions along the axial direction.
Let me point out that the axial Direction
is along this Z, to analyze this a little
bit further let us look at the equations of
motion involved.
Le t us start with the simplest equation of
motion which is the z equation of motion one
can derive that the Z equation of motion is
that of a simple harmonic oscillator Md2z/dt2
is proportional to the –z this proportionality
constant which is the frequency has terms
such as Q which is the charge of the ion,
M which is the mass of the ion, L which is
the distance between the end gaps and Vcap
which is the cap potential.
Here I am referring to two end caps which
are on both sides of this, of this diagram
which are also charged by an amount recap.
So this equation of course means that the
motion of the ions is that of a simple harmonic
oscillator.
A detailed analysis also shows that there
are additional modes along the z axis like
this breathing mode that I have derived here.
Let us now consider the motion along the other
two axis which is the x and the y axis.
One can again derive the equations of motion
along the x and the y axis and these equations
of motion you are given by what are known
as the Mathieu equations.
I will describe these equations to you the
differential equations have the familiar form
of d2x/dt2 is proportional to x and d2y/dt2
is proportional to y.
But there is an additional time dependent
term which is cosin(?t) that enters both equations.
We know where this cos(?t) comes from it comes
from the fact that plus or minus V0 cos(?t)
divided by 2 was the choice of the modulation
that we have chosen.
Again various constants enter this, enter
this equation.
Let me point out this constant a.
Which is related to the distance between.
The distance between the ions in the center
and one of these rods which is in the corner,
right?
So this ion and this rod so along this direction
which is in the xy plane that is the, that
is the distance a.
Now these equations can also be solved, but
let me actually show you a picture of what
is happening.
To do that let us actually draw the saddle
in a color block, so here what I am showing
you is a contour plot where red indicates
a lower potential which means that an ion
placed here and then gently push with this
run away in that direction that is the arrows.
And blue indicates a higher potential which
means an ion that is placed here and push
would, would gently roll back into the into
the center.
As we spin this potential very, very quickly
the ion basically executes motion about the
center, this motion looks like this.
What I have done here is of greatly exaggerated
both axis and I am describing to you two kinds
of motion, the first of these is what is called
the center of mass motion which is just the
fact that this ion just rolls around the center.
The second is an extremely rapid motion that
I have depicted here and this is called micro
motion and as a consequence of the fact that
the potential is spinning rapidly under it.
Micro motion contributes to the heating of
ions, because it increases the number of photons
in the vibration degrees of freedom.
So what we have are trapped ions that are
rather hot and this is no good for quantum
computing again.
So what do we do about this we have to cool
these ions?
So let me describe to you two mechanisms that
one uses to cool ions.
The first of these is what is known as Doppler
coding.
If you want to cool ions down to around 15
photons we can appeal to something called
Doppler cooling.
Doppler cooling is based on the Doppler Effect.
And is described by the cartoon that is given
here so imagine that an atom is moving in
a particular direction and if, if electromagnetic
radiation is incident on this item such that
it has a lower frequency than the transition
of the atom then we see the atom is moving
towards the electromagnetic field it can absorb
a photon when it absorbs a photon because
the photon is moving in the direction that
is opposite to that of the of the eye on the
ion loses momentum and hence queen by repeating
this over and over again one can cool these
ions to around 15 for 15 phonons it turns
out is not good enough for quantum computing
still.
One must get these ions as cold as possible
and to do that one has to do what is known
as sideband cooling.
So let me describe to you sideband cooling
briefly now.
Consider the following abstract problem so
imagine that you have an atom which has many
energy levels and which has a particularly
narrow energy gap furthermore imagine that
the damping rates of this atom are rather
slow as well now consider the following scenario
where the temperature sets the scale KBT which
is much larger than the atomic frequency gap
what this means is that the state of the atom
is a thermal state given by e to the - ß
H divided by the partition here H reference
to just the bare Hamiltonian of the atom consequently
an energy level EK somewhere in the middle
is occupied with probability e – ß EK this
is the Boltzmann distribution factor.
What this means is that this qubit is no longer
the ground state and it breaks one of the
assumptions that is made in the DiVincenzo
criterion listen remember that criterion 2
was that we must be able to initialize this
quantum system in the ground state and because
of this thermal occupancy this, this atom
is no longer than the ground state so we wish
to cool this atom what do we do what we should
do.
Is in fact to bring another atom which has
a much larger spacing and which is damped
really, really fast by somebody to cover now
because this atom has an energy level spacing
much larger than the temperature scale and
because it is damped really fast it is almost
always going to be in the ground state which
is to say that the density matrix of this
item is quite well approximated by the ground
state now what this means is that our target
atom has a lot of entropy in it where is the
auxiliary atom that we have just brought in
has almost no end to the image side band cooling
is a technique that allows us to shunt entropy
from the target atom.
To the auxiliary air to do this what we need
is some sort of coupling between these weapons
a typical coupling is of the form ab dagger
+ data will be what is the physical meaning
of this term ab dagger to say that will be
note that a is the annihilation operator for
this atom and v is the annihilation operator
from this happen further more remember that
the annihilation operator reduces the occupancy
of the foxed state by 1 right and the creation
operator which is a+ increases the occupancy
of the foxed state by 1.
So what is the meaning of this Hamiltonian.
The meaning of this Hamiltonian is that we
would actually like to reduce the number of
excitations in this atom and transfer the
excitations to this atom which will then subsequently
be transferred.
For Hamiltonian set we need to be and this
is conjugate of this which is a+ this explains
that Sideband cooling interaction Hamiltonian.
So how is this implemented for our physical
system which is the high on top, so let us
look at that.
In our physical implementation there are two
modes as well, the first of these is the electronic
states of the trap up for instance let us
consider calcium a good qubit is represented
below the vibration modes of this atom represent
the other degrees of it.
So let us see how both of these modes were
used to cool the atom.
Consider the electronic state 0 and the phononic
state 2 what this means is that the electronic
state is in the ground state and the phonons
which are quantization of the vibration modes,
so let us see how sideband cooling is implemented
with these two modes.
Consider the following transitions so consider
the state 02 where zero reference to the electronic
degree of freedom and 2 reference to the vibration
degree of freedom what we want to do is cool
this vibration degree of freedom.
So what we would like to do is transfer this
excitation to the electronic degree of freedom
so there is something called the sideband
transition, the red sideband transition in
this case which I will describe in the notes
for you which does exactly this it is the
a+ b + b+ a transition and what it does is
it takes 02 and it outputs 11.
Here this 1 refers to the electronic degree
of freedom having one, having an occupancy
of one and the other one reference to the
fact that the phononic degree of freedom still
has an occupancy of 1 which is still 1 phonon.
This is dumped quickly and we find ourselves
in the states 01 again repeated application
of the red sideband transition takes us to
the state 10, so this excitation is transferred
from here to here and it takes us to the state
1 and it leaves the phononic degree of freedom
in the state 0 and this again decays quickly
to 00.
And because this is now the ground state there
is no occupancy for this transition to go
anywhere and the cooling has been completed
so this procedure of combining doctor cooler
with sideband cooling allows us to get extremely
cold atoms which are then useful to do quantum
computing.
So let me briefly discuss how gates are implemented
in an ion trap quantum computer.
The route to implementing quantum gates is
represented in this slide we have a quantum
Hamiltonian which is proportional to the sX
and this allows us to make transitions between
0 and 1 further more to perform multi qubit
operations the following interaction Hamiltonian
is available which is also form as+ eip +
a+ s-e-ip note that this is exactly of the
same form as before it has the form a b+ +
a+ b but instead of the second harmonic oscillator
mode we have a single spin this I will show
in the notes.
Allows us to perform both single qubit gates
and two qubit gates let me summarize briefly.
In this lecture we saw that nuclear magnetic
resonance qubits are not good qubits.
We considered an alternative which was trapped
ions we consider the issue of trapping ions
and we saw that Earnshaws theorem implies
that electrostatic potentials cannot trap
ions.
We then consider the spinning saddle example
and motivated the fall trap.
We saw the vibration modes of the fall trap
and considered motion along the axis and in
the transverse direction.
We discussed heating and we discussed how
to resolve heating in cooling we discussed
two separate things one was doctor cooling
very briefly and the other was sideband cooling.
Finally we discussed how gates could be implemented
in the ion of quantum computer, thank you
very much.
