- All right, good morning, everyone.
It's my pleasure to be here today with you
and to share some lectures.
And Brian, how do we
pull up the lectures now?
All right, perfect.
All right, so thank you for
coming to this lecture to learn
how real qubits really work.
It's really my pleasure to be here today,
and as Brian said,
I'll pause every five or 10 minutes or so
to take your questions
because the next three hours
are really your time.
The next six hours of
my lecture will answer
what is a qubit?
The kind of superconducting qubit
you may have already logged into
in the cloud and experimented on.
How can you design, control
and measure a real superconducting qubit?
And also, I'd like to focus
on the question of why.
Why are these the right
important questions to ask?
I think these are valuable
and practical stepping stones
that you will need in
your career in quantum
to be successful.
And the next six hours of these lectures
will give you the ideas
and the skills needed
to grow in this field.
So what lies on the
road ahead for us today?
As you can see, it's a winny road,
and we'll start over here
at the very beginning,
and we'll chart a narrow path
but a very deep and long path, meaning,
we'll go pretty far but
start very introductory.
The first topic I'll get into is
what's the big picture?
Right going between your computer,
through a system in the cloud,
through some devices
superconducting qubits in this case,
all the way to the Bloch sphere.
Or in other words,
how do we take the Bloch sphere
and actually realize it in practice?
And that will bring us
to the second part,
which is quantum physics
and the physics of real atoms,
the quantization of energy levels,
and the fact that there
are actually no qubits.
There is no such thing
as a real qubit in life,
there are only two level approximations
where we can usually take two levels
from much larger system
and pretend that it's a qubit.
But it works really well.
In the case of the kinds
of superconducting qubits
that you have been experimenting
on and playing with,
that will take us into discussing
what is the Transmon qubit?
It's the prototypical or
the main bread and butter
superconducting quantum bit.
Which implements these two
levels and you play with.
And now also,
you can explore these higher levels using
some new features like qiskit pulse.
And that will allow you
to take advantage of
a lot more physics and potential
that we haven't really
been able to explore before
with just two levels,
but we can now leverage
the much larger space of possibilities.
Now to really describe what
a superconducting qubit is
or what a real atom is,
or an artificial atom.
I'll go back and review the
classical circuit theory
and the classical harmonic oscillator.
This will be very familiar to most of you.
But I'll try to add a bit of a twist
to how we see it
and try to build a little more
intuition which can help guide us
on our path
leaping into the quantum.
Unveiling, the quantum oscillator,
or the quantum harmonic oscillator.
Which will be the building
block of all the tools,
and all the superconducting devices,
both transmitted qubits
the way we read them
out the way we control
and manipulate them
and, as some physicists
like to think or say
they're really only spins and oscillators
and everything else just reduces onto
spin or an oscillator.
And finally, that will
lead us into the last hour
of the talk,
which will be on the details
of the transformed qubit.
Going from the basics of how do we move
from the harmonic oscillator
to actually a nonlinear
or so called an harmonic oscillator,
look at its wave functions,
the way it's described
and then It's experimental signatures.
How would it look like in
a spectroscopy experiment?
How do I control it to do a Rabi rotation?
How do I calibrate it?
Right all of the practical questions
all of the stepping stones needed
to actually build the logical circuits
that we implement the control, not gates,
the hadamards et cetera
And also to take
advantage of the more rich
and dynamic space in which
the qubit exists in libs.
And that will be for today
and tomorrow we'll pick
up with measurements.
Now, this lecture,
I view as both introductory
and skill reaffirming meaning
we'll take a big step back.
You don't really need
to know much going in.
But we will go far.
We will go quite far.
That means that
some sections will
preview advanced material,
I'll try to label those
and denote those with
Knuth's dangerous bend symbol.
So if this is the first
time you're going through
some of this material,
you can skip past those,
or just take a cautionary sign here.
And the idea is
that we'll use examples
simple, basic examples,
things you've probably already seen.
However, these examples
serve as stepping stones
to very practical
problems and the very practical way
that we actually realize qubits.
And the idea here is
to extract the macro from the micro.
The very simple things
we will illustrate today
really carry over and
describe most of the physics
of the entire field,
they're just the roots of it
and everything else grows on top.
And so, I will try to be a step by step
as possible in the derivations
and then the way go from
a classical circuit to a quantum circuit,
or from the quantum circuit
to the transition spectrum and so forth.
Of course, please ask questions
or write your questions down
and post them in the channel, Brian,
will get those to me as soon
as we can.
And finally, this is
very tightly integrated
with lab work by Dr. Nick Bronn and Co.
I'll try to avoid the
firehose of information,
and someone shared this graphic with me
when I was trying to prepare this lecture
and I thought it was
a good reminder to try
to chart a narrow path
and and focus on a few key points as much
as possible while exposing you
to the much larger picture.
So let's begin with the big picture.
I think all of you at this point
from the first few lectures
in the summer school have
seen that a qubit consists
of two energy levels,
typically denoted by a
zero kit and a one kit.
Energy levels were a picture
proposed by Niels Bohr in 1913.
Now you saw that this picture is a bit
of an abstract thing.
And the way that these
quantum states actually get represented,
or the really physical distinct
number of distinguishable
potentially states are infinite,
and we can represent
them on the Bloch sphere.
Now, finally,
how does this very abstract
picture of energy levels
and Bloch sphere actually
occur in the cloud?
All right now this is a
picture of a real device,
one of the devices I
think you may log into.
And this is a fridge.
Now not the kind of
fridge of course you have
in your home.
This is a very, very expensive fridge
that can go to very low temperatures
for reasons that we'll see
in a minute.
But the quantum bit,
the actual physical quantum
bit sits at the very bottom
of this fridge
and everything else is
Input Output wiring.
In the more recent days,
they have course turned
to look quite a bit more stylish.
And this particular
case housing this fridge
was done by the same company
that does the Mona Lisa and I thought
that was very cute.
Now what's inside of this fridge?
Let's take one of these
dilution fridges and zoom
unveil what's inside of it
by removing the outer can.
And then usually shows us a picture
that looks very much like this.
We call this the golden chandelier.
And this is a superconducting
dilution fridge inside
where you see a lot of
cables going in and out
it is actually gold plated.
The point here is though,
that everything that
is quantum really sits
only at the very bottom,
in this little shield at the
very, very bottom of the fridge
and it operates at 15 millikelvin.
And that's negative 273.13 Celsius
and we'll see why that
temperature has to be so low.
Now inside of this shield,
unveiling several other shields,
you see a picture of a chip like this.
This is a superconducting qubit chip.
And what you see here is a device
with a number of quantum bits qubits,
which we'll delve into today,
which are housed on this substrate
or chip back here which can be silicon
or Sapphire or some other dielectric.
And a number of readouts.
These quickly superconducting
coplanar waveguides
indicate readout resonators.
And as I'll discuss in
the rest of the stock,
the superconducting qubits
are also a type of resonator,
a type of microwave resonator
and in this lecture today we will focus on
how do we create and define
the quantum bit, or qubit.
If I zoom in
on the picture here,
it's this little structure composed of
these two metallic pads connected
by some sort of inductor
or nonlinear wire,
that it actually supports
the physical picture
of the Bloch sphere.
And we'll talk a little bit
about an intermediate step
of the representation
because these at the end
of the day are oscillators,
microwave oscillators
or electromagnetic oscillators,
oscillations of the electromagnetic field
which we can represent using circuits.
Perhaps the kind of circuits
you studied way back one.
Except that now these are going
to be quantum circuits.
And we'll see how we can make a transition
from the classical world
to the quantum world
and describe the very
unique features of quantum
with the circuits.
And how they allow us to
support the Hilbert space
in which we do the logical computation.
And at the end of this lecture,
we will derive the Hamiltonian,
which describes exactly
how this qubit originates.
That brings us to circuit
quantum electrodynamics
and qubit in the cloud.
So, let me give you the
big picture of the summary
of the flow,
you have a laptop which
you might log into,
which can then send some
information that controls a fridge
inside of this fridge at the very bottom
of this dilution refrigerator
sits the qubit chip.
This qubit chip then allows you to support
in the abstract sense here,
we're now moving to the abstract
electromagnetic circuits
with electric and magnetic
fields which themselves allow you
to create this type of Bloch sphere
and then to control it and
manipulate it and so forth.
And they also allow you to measure it
so that information can flow back
from the Hilbert space all
the way through the circuit,
up the fridge and to your laptop and back.
And so this closes the feedback loop.
And today we'll focus on
how the qubit originates
from this circuit as well as
how we can really connect
the very physical picture
to the very abstract Hilbert space.
And I think this is
where I'll pause for questions.
So let's see Brian if
there any questions yet.
(mumbles)
- [Brian] I want to remind everyone,
if you have questions
about things not related
to Zlatko's lecture,
such as labs,
final project any of that,
please use the discord
or email me we're not gonna be able
to spend time answering
those during lecture.
So, a first question we have here
Zlatko this is kind of a
personal question for you,
but the question is
what's your personal favorite flavor
of superconducting qubit?
And why?
And maybe you could also add
in if there's any cutting edge kind
of superconducting qubits,
that aren't fully functional yet,
but that you are really
interested in maybe share
some weight on those as well.
(laughs)
- [Zlatko] That's an unexpected question.
Thank you.
I have two flavors.
I have the flavor that I like to sit down
and write everything about,
which is the one I will describe
in this discussion today.
It's called the transmon qubit.
It's perhaps the simplest qubit.
And I like the simplicity of it,
because it allows you
to illustrate pretty much all the most
of the general ideas.
Very simplistically, more generally, yes,
I like very exploratory
superconducting qubits, though,
and most of all the ones
we haven't anticipated yet,
including the hybrid ones,
of course the flick, Sonia
is a favorite of mine,
but I have measured and played with a lot.
And now people are very excited
about the zero pi qubit.
And variations of it.
So, that gives you a
little bit of a flavor.
I don't get the ice
cream with the one scoop,
but with the several scoops.
- [Brian] I love it.
I think that for now we'll let you go back
to your lecture I'll bring
more questions later on
- [Zlatko] All right, thank you.
And I'll try to point out some differences
between maybe the very
simplistic transmon keyword
we have here,
which is the bread and butter
and actually would you
log into through qiskit.
But maybe also some of the variations
to what other types
of qubits look like.
Alright, so that brings
us from idea to reality.
Let's get into some of the concepts.
Again, this qubit implements two levels,
and a Hilbert space, right?
These are two pictures
that are interconnected.
For the more advanced
the view of left a little
question slash note down here
if you're interested in this.
Now for the rest of us,
how do we realize
these two levels in practice?
And so that will get me to talk
about real super real atoms.
And let me back up to get everyone
on the same page.
So, for the more advanced
students, I apologize.
But I hope this will get
us all on the same page.
And all the way back to the
origins of quantum physics.
Imagine a cloud of atoms,
right each of these pictures
here is an individual atom.
And this is essentially
how all experiments
in physics quantum physics
were done until circa 1980.
Now you can take this cloud of atoms
put it under subjected
to some excitation such
as the voltage source here
and it will glow.
It will shine.
You can feed that light through some lens
and thin ribbon in a prison.
The prison will split out
the different frequency
components of the light.
And what physicists
observed is that the light
that gets split out from the cloud
of atoms comes out only
in certain colors,
such as red, green, purple, and so forth.
And so if you look at a spectrum
of the light and this is an
actual measured spectrum of H2,
you'll notice that
you get very discrete
distinct colors lines,
this motivated people to
look at very various atoms
and they noticed that
each atom had a different structure
hydrogen had one set of the screed lines
helium had a second set
neon had a third set
sodium a fourth and so on.
That led, Bohr and company
to propose that atoms
are different from classical mechanics
in that the energy can
only take certain discrete
amounts, or levels can only
be in certain discrete levels.
This helps explain why
the transitions appear
in these discrete
but not continuous emissions.
So, the atom can only emit
at a particular frequency
because the energy can
only be either zero or one
not 0.5 nothing in the
middle, so to speak,
this the colloquial language to say it.
Of course, you'll recognize this in
the recent July 4 fireworks,
some of you may have seen,
and each of the different fireworks rep
has different atoms,
which of course give off
a different set of frequencies, all right?
And that brings us to
the atomic energy levels and transitions.
An atom quite often for the electron
has a potential energy landscape
of one over r.
And so this is
where you can write down
the potential energy say
as a function of the radius
and it typically goes
like something one over r
where r is the radius between,
say the atom and
the electron over here the atom nucleus.
And this potential,
normally in a regular system,
the atom can have any energy,
it could be a particle here, here here.
The idea is that in quantum of course,
these energy levels are quantized,
denoted by these red lines.
Now, they're not just quantized
in any way,
they're quantized in a very special way,
what we call anharmonic.
Anharmonicity means that the energy levels
are not spaced equally.
What that means is that if you apply
a laser that shines
some microwave light,
or optical light at the atom
and the atom starts say
in the ground state,
and is subjected to a force that tries
to steer the atom from the ground state
to the first excited
state once the atom is
in the first excited state,
because here I'll assume
that the frequency
of this laser is tuned exactly
to the transition frequency
between the first and zero level.
The atom can no longer steer the atom
from the first state to
any other higher level here
because of the anharmonicity,
meaning that there's no
level that the frequency
of the laser can take this atom to
that's up above.
Where the height of the cell represents
the frequency of the laser.
That's very special,
because what it means is
that when we shine this atom,
it really only acts or interacts,
when we shine light,
it only interacts with the atom
when it's in the lowest two energy levels.
And this allows us to
isolate a qubit subspace.
And for a real atom to really just work
with the lowest two levels.
So that's the basic idea
of then how you take
one of these
atoms into a two qubit subspace.
It's also very important
that this atom is isolated
from the environment.
So the thermal fluctuations and noise
in the environment doesn't
destroy the quantum coherence
and quantum properties of the atom.
And the Ns energy level diagram allows
for very specific control
and also for very high in good readout.
The Atom in that sense is low loss.
However, again, my main message here is
that it's not a two level system
and there are no two level systems.
In reality, there's always a coupling
to some other degree of freedom,
which really creates more than two levels.
And that will be exactly the case
as we move into the superconducting world.
Atoms are very nice,
however, they come with
certain properties.
And we'd like to be able
to engineer everything
about the atom.
So, my personal flavor of
atoms is artificial atoms,
created by electromagnetic circuits.
Which can mimic the
electromagnetic properties of atoms
and also have quantized levels.
And so that will bring us
into very quickly the next section.
Brian, are there any
questions at this point?
- [Brian] Yeah, I am gonna go through
and pick up we got a lot
of good questions,
but I think one that maybe we wanna tackle
right here at the beginning is actually,
I saw four or five people
asked the same question
about the chandelier itself.
You showed the gold plated
chandelier and you explain that
this is our quantum computing setup.
But a lot of people
actually didn't understand,
there's a lot of loops,
the gold itself.
Yes, exactly that picture.
So I was hoping if you could walk through
and explain to people why it's gold plated
and what all those loops actually do.
- [Zlatko] Yeah, great question.
Thank you.
So, it's gold plated.
Because we like gold,
it's gold plated,
because gold doesn't rust,
because gold is an
incredibly good thermal conductor,
and because it's soft.
Basically the gold has very
nice material properties,
which are very nice
for things like
thermalization and preventing.
So why thermalization?
Notice that at the very bottom here,
I said this operates at 15 degrees,
so 0.015 Kelvin, above absolute zero.
Up here, at the very top,
we operate at 300
Kelvin, room temperature.
There must be some source that's cooling,
this thermal load at the
bottom to this temperature.
And that's in the mixing
chamber, which is,
I think, somewhere here in the back
in order to create a thermal link
that can take away the
heat from this element
to that cooling source,
we need to have a good conduction,
something that doesn't intrude
in that doesn't isolate
between the very cold and the very hot.
And so gold is a very
good thermal conductor
and it allows you to press
two metals very hard together
and get a nice thermal contact,
whereas a carbon metal usually
doesn't allow you to do that.
It also doesn't oxidize very much.
So it it has a very thin oxide layer
which can break through easily
which helps with the conduction.
All the loops also have
to do with thermalization.
When things get cold,
they tend to change in size.
And as the fridge changes in size,
as you heat it up and cool it down,
it stretches and if you have
a very taunt, metal cable,
it can get snapped quite physically
it can get pulled apart
from its connectors.
And so the loops allow you to give
some mechanical flexibility
or playability to the cable
as you have thermal
shrinking and expansion.
Thank you, Brian.
Was that answering the question?
- [Brian] Yep, perfect.
- [Zlatko] Perfect.
All right.
Okay, so, artificial atoms.
These artificial atoms will consist
of capacitors and inductors.
And I'll spend quite some
nice time talking about that.
Now, very much like the
harmonic degree of freedom,
the harmonic oscillator that is set up
by an inductor and capacitor,
which allows the electromagnetic energy
to slash back and forth
between the capacitor which
say has negative charges
and positive charges
flowing back and forth
and We'll get into that in a minute.
It has a potential energy landscape
that looks very much like
that of the atom in a way.
Except that it's typically quadratic.
In, say the charge or
in what I like to think
of here is the magnetic
flux across that coil.
This quadratic energy landscape
also has only finite levels,
the quantum physics allows you to get into
and we'll derive exactly what those are.
If we now coupled this circuit
to an input output transmission line,
something just like the laser
that allows you to guide waves
in and out of the structure
to subject it to forces and control it.
You notice that if I send in a microwave
or a wave at a particular frequency,
let's say omega 01,
that's the frequency of this wave up here.
If the atom starts in the ground, state.
Then microwave light has energy,
it can transfer photon that takes it
from the ground state over here
to the first excited state.
Now, once it's in the excited state,
if I keep shining the light,
that laser can again provided
another photon of energy
that can stimulate it to go up
to the next level,
and up to the next and so forth.
In other words,
I have no individual
control that allows me to
isolate the lowest two energy levels here.
I cannot isolate this
level and this level,
just like we did in the atom.
And that's because the spacing between
these lines you notice is equal
or harmonic in other words,
and that's why in this talk will
introduce a non linearity
which will open up the
potential and make it softer
as the particle climbs in the energy.
And so we'll make this inductor oops,
a tunable inductor,
a nonlinear inductor
tunable in the sense that
its inductance will change as a function
of how much energy is stored in it.
And that will open up this potential
and allow us to isolate the atom
of the qubit in the lowest two levels.
And that's really
the core idea of everything we'll discuss
in the next hour two hours and a half.
Except we'll go through a lot more
technical rigor in detail,
and physical pictures of how
this actually happens and
how do we describe it,
and what are the consequences.
And how this really shapes
what a logical qubit is for us today.
Okay, so then to summarize
the very big picture overview,
there's the idealization of the qubit,
which has two energy
levels say zero and one.
This Hilbert space is really realized
and harmonic oscillator,
which has transition frequencies
between energy levels that are not equal
to each other
due to a non linearity.
The way that that an harmonic oscillator
is created and modeled
is through a circuit.
Which really describes a physical device,
real electrons moving
in real metals and real,
real supported by real dielectrics.
And so we move from the
level of idealization
to the level of physical reality.
And in the following,
I'd like to spend a little bit more time
discussing the physical reality
and then take us all the way
back up to the idealization.
And the rest of the talk for now focus
on these two parts.
And that brings us
to circuit quantum electrodynamics,
where I'll give a bit
of an macroscopic overview.
Now you've noticed that we've moved away
from spins quite a bit more
and more towards oscillators
and like to discuss this
joke with my friends.
There are two kinds of physicists,
those who believe all physics spins,
and those who believe
all physics is oscillators.
Maybe you can guess which
one your host today is.
Now, as we saw the ingredients
of circuit quantum
electrodynamics really consist
of microwave oscillators,
which come in a variety
of shapes, forms, sizes,
and so forth,
such as these coplanar
waveguide oscillators,
these squiggly lines here,
which we will describe
today at the quantum level,
as well as these transformed qubits
which implement the nonlinear
and harmonic oscillators
which provide the actual
superconducting qubit.
All of these might all of
these physical devices we can represent
with very simple lumped
element model circuit elements,
such as capacitors inductors,
which you've probably seen
in your high school physics class,
and of course, Josephson tunnels junctions
a key ingredient to the non
linearity which is low loss.
And in very well behaved,
that I'll get into in the later part,
as well as the input output elements
that can allow you to
control and measure these.
And the goal of circuit
quantum electrodynamics
is to combine these elements just like
you can combine the same ingredients
and make them into a burrito or a taco
or a different flavor of food
in various in unique ways,
and essentially form the
basis of most of our field.
Now, at this point,
you might be asking yourself,
"Wait a minute, these are just circuits,
"I've seen circuits,
"what's so special about them?"
"How are they quantum?"
Well, is your microwave quantum?
And you might say
"you need low loss."
"Quantum systems are very precarious."
"I need to have very little dissipation."
Well, a microwave is actually not too bad.
It has a quality factor of 10
to the four.
So the microwave mode
in there is pretty high.
But it also needs to be
isolated from the environment.
Thermal fluctuations need
to be very, very low relative to
the frequencies of interest.
And this is why a microwave
isn't the quantum object,
because the temperature is so much higher
than the energy spacing
between those levels.
So essentially, all quantum mess gets
washed out very quickly.
A microwave is also not quantum
because it lacks nonlinearity
and doesn't have a lot of
and we'll see that one more aspect
that's important for certain
quantum electrodynamics
is large quantum vacuum fluctuations.
And I'll define what
those are in a minute.
So to get all of these nice properties,
we need superconductivity
which will allow us to
have small dissipation,
isolation from the environment.
It works at low temperatures already,
so that gets taken care of.
However, we also need
to have Nonlinearity.
Something that creates
a nonlinear potential.
So we can have infirmity.
And for that we'll use the
nonlinear and very robust effect
of the Josephson tunnel junction
for which Brohm Josephson and
which branches and predicted
and won a Nobel Prize for.
And to give you a more of a picture of
where does this all physically sit
in the large scheme of things.
Here's a picture you
may have seen from NASA,
which describes the wavelength
of electromagnetic light,
the frequency of electromagnetic light,
which goes all the way from
sort of radio frequency,
up to microwaves to Wifi
and goes into the visible light
and all the way towards X ray
and gamma rays up here.
And we can relate the frequency
to an effective temperature of the body,
through blackbody radiation
or maybe through the simple equation hear.
And I can give you an order of magnitude
of where things sit.
The fridge has to be
at the lowest temperature the environment
in which the superconducting
qubit operates
sits at about 10 to the minus two Kelvin.
So roughly here in the temperature range
very very close to absolute zero.
The superconducting
circuits will operate with
have frequencies resonant frequencies
on the order of 10 Gigahertz,
which is about half a Kelvin,
which is much larger
than the fridge temperature,
but as you can see is really,
really low energy relative to
visible light or really most
of the other electromagnetic
phenomena we see
in our everyday life.
And the wavelength of these is
about three centimeters.
The optical transitions
you see in fireworks or
are get controlled in
atomic nuclei tend to be
in the visible spectrum
and have a much larger range.
But the photons in the
energy we'll talk about
today and microwave oscillators is isn't
that sends a very low energy.
And in the next slides,
I'll get into a lot more detail,
and for those of you who want
to see more depth or follow up
with more reading,
here are a few references.
You can pause in the
playback and dive into after
this lecture if you're interested.
And that brings us to the world
of circuit quantum electrodynamics.
And Brian, I think I'll pause
and take any questions.
- [Brian] Okay, so one question
before we dive into any
of the hard science questions,
which I know are coming so if you've asked
one of those already, just hang tight.
But just one final question
kind of about the setup
that we at IBM have chosen to use.
One person was asking
with all the different candidates,
why is it that IBM decided
to go with superconducting qubits?
- [Zlatko] That's a good question.
I guess there's a story there,
which may be written up somewhere.
But the research group here, of course.
Well, IBM has a very long history
with superconductors
with superconductivity.
It really pioneered some of the early work
on superconducting classical processors
and has quite spectacular fab
for superconductivity in a lot
of expertise in this field.
So there was a natural point of connection
and transition there from the early days
of Josephsons junctions
and a lot of the key research
back in the day was done
at the facility that we work at today.
And in the 80s, in earlier days,
quantum effects in those junctions,
and those nonlinear circuits
were completely ignored
and really out of reach.
But with the advent of
technology and science,
we were able to, isolate
those circuits minimized
to noise reduced temperature enough
and start to understand
their quantum behavior
and show that in fact,
a superconducting circuit,
a circuit, which consists
of billions of electrons
can actually behave macroscopically
as a single degree of freedom.
Just like the position of the
electron around the nucleus.
And these effects,
I think, then led to
a very exciting new field which said,
"Wow, you can suddenly
create your atoms as you want
"artificial atoms in a way
"you can engineer the energy level diagram
"in any way you want
"something you can't
really do with real atoms."
They come in the varieties
that they're given to you.
And superconducting circuits have,
as I'll show at the end of this talk
had a tremendous trajectory in terms
of their growth and coherence times
and control parameters.
And of course, they're in
the way that most analogous
to classical computers so you
naively you can see a path to scale it
to scaling to large
architectures very quickly
because they use printed
or they use micro
lithography nano fabrication
and very much the same way
that conventional computers do.
And so that motivates a nice analogy
between classical computers
done with CMOS technology
and quantum computers,
which that when done with
superconducting circuits
are also on silicon chips
and metals and so forth.
And I think that they
are particularly exciting
and very promising.
So, maybe that gives
you a bit of the flavor.
- [Brian] Perfect.
(mumbles)
- [Zlatko] Okay.
Good.
That brings us to the world of circuits
and to do quantum,
you often have to start with classical
and in part that's because
in quantum, a lot of things
have a really nice analogy
that can be really understood
in classical physics very well.
So we're gonna spend some nice time
on a classical oscillator,
which will really explain a ton
about the quantum oscillator.
And we'll go over some very basic things
you may have seen,
but with a twist.
Here's the transmon qubit.
In the picture of a real device,
let me show you now a picture
of a transmon qubit that I've drawn here
as an illustration and zoomed in.
Where you see that we have a substrate,
which is some sort of
dielectrics such as silicon.
There are two metal pads,
just pieces of metal on top of it
just like a classical circuit so far,
that you might find in
your iPhone or Android
or what have you.
And somewhat specially
these two metallic pads
of the superconducting transmon qubit
are connected by some sort of inductor,
which here I've just depicted
for illustration purposes as a box.
And in general,
this will be some kind
of nonlinear inductor.
Such as a Josephson tunnel junction,
which we'll get into later.
Now, let me give you a
bit of a physicl picture
as to what actually happens
and why this is an
oscillator in the first place
and what's actually oscillating.
These metal pads,
of course have many electrons.
And by applying an
electric field across them,
you can create a displacement,
where you shuttle some of the
positive or negative charges
from one island to the other island.
so these green circles here
indicate a net positive charge.
The red circles indicate
a net negative charge,
typically in one of these metallic islands
that are about 10 to
the 12 mobile electrons
to give you an idea of how many electrons
are involved in this.
Now in practice,
this is going to be a
very low temperature,
which means that these electrons condense
into superconducting pairs,
so called Cooper pairs and
they move without dissipation.
We can label the charges in one island
as plus Q,
and they can vary on time.
And the charges on the
other island as minus Q.
Of course there's conservation of charge
charge doesn't get destroyed or created.
So we can only slosh
charges back and forth
between the two islands
through this little bridge.
This variable Q which
will describe the charges
has support on the full rail line.
Of course, an electric charge
and that electric charge
creates an electric field
which in general depends
on position and time
it can oscillate.
And if you have an electric field
that electric field can then
be used to define a
voltage between one pad
and the other pad.
So I can say that the voltage here
and this is not a detail
you need to get very acquainted with
I just want to illustrate
the bigger point.
But the voltage here is related to
how much electric field lines link
one pad to the other pad
and it goes from RA,
which is Pad A
to RB, which is pad B.
doesn't matter which way you drive.
And the voltage is again a number between
minus infinity and infinity.
Now there's something very special about
this particular structure.
And that's that,
because the frequencies will operate
at are relatively low compared
to the inner modes
of the structure.
In other words,
the wavelength will be very
large compared to the dimensions
of this little transform,
which by the way is,
the dimensions here
about 100 microns across
so they're relatively small.
This two metal pads are in the
so called lumped element regime,
and we can model them
as a very simple capacitor
you've seen before.
Fineman has a very nice discussion,
of how this happens
between Maxwell's equations.
And these this very
simple relationship here,
we can say that the charge in one island
is related to the voltage by a value C,
where C is called the capacitance
and it's a number between
zero and infinity.
And of course,
we denote that in electrical language
with a symbol like this.
If a charge moves from one
island across the bridge
to another island,
that creates a current
defined by the positive test charge here
moving across, of course,
it leaves behind depletion
and moving across
to this island,
it changes the charge,
because charge is conserved
we can of course write down
this universal relationship that
is valid for all quantum circuits,
which says that the
rate of change of charge
on the island is equal to the current
in this case the current
through this junction.
Or equivalently,
that the charge is equal to the integral
from the start of the world so to speak,
the start of the circuit all the way
to the time t,
it's just integrating how
much charge has gone left
to right through this
bridge keeping track of it.
In circuit theory,
we keep the initial conditions
fixed at minus infinity,
t minus infinity is just a label,
it could be t equals zero,
it's some reference state.
But we'll keep those
initial conditions zero.
And the current course
is also on the support
of minus infinity to infinity.
Now for the more advanced
of you who've seen
some of this material before,
I'll just draw a cautionary tale
that we have chosen the reference
direction of the voltage
and current opposite to each other
and that's important for
later writing the Lagrangian
in a nice way.
Now what happens when a
charge tunnels or travels from
one side of one island to the other side
of the other island,
it can create an electromagnetic field.
Moving charges create magnetic fields,
and a lot of magnetic fields
can create a magnetic flux.
The magnetic flux is we denote
by this variable phi of t.
It can be related to the voltage across
these two metal pads by a
very simple relationship.
This is essentially a
lumped element version
of Faraday's law of induction.
And it's a universal relationship one
that holds for all circuits
for all branches independent
of their nature constitution.
The magnetic flux is
again the time integral
from the reference state to time t
of the voltage,
just like the charge.
And similarly, the rate of change
of magnetic flux,
or how many of these electric
magnetic field lines are here.
Depends on the voltage
between these two metal plates
and we'll pick again for
the initial conditions
that the flex is at
time zero equal to zero.
And the magnetic flux lives
on an infinite support
between minus infinity and infinity.
And emphasizing here the
support of these variables
where they're actually
mathematically defined.
Because quite often in circuit theory,
there's a question as to
whether the variable across
these two pads is compact
or non compact,
meaning does it go between zero
and two pi,
because these are superconductors.
Or can you just use a variable that goes
from minus infinity to infinity.
And there is a very
simple again relationship
that I recommend fireman's
book on which describes
that you can in the limp element regime
link the magnetic flux phi
to the current that flows
through this junction.
With L L being the inductance.
And Brian, maybe I should pause and take
any questions at this point.
- [Brian] Yeah, I'm looking through
we have a lot of questions coming in
at a pretty wide spectrum,
if you will have answers us.
But I've seen a few people asking
about the bridge that
you're discussing right now.
In fact, I know this is probably
a straightforward question,
but they want to know is
that bridge itself acting
as an inductor?
- [Zlatko] Thank you.
Absolutely.
Yes.
So, great question.
So this bridge here,
I have purposefully not specified
what kind of bridge it is,
because it could be anything
it could be linear inductor,
just a piece of wire.
It could be a Josephson tunnel junction,
which is a superconducting element.
It could also be a nano wire,
it could be an atomic point contact.
It could be a variety of things.
But the essential thing
is that whatever it is,
it has some relationship
inductive relationship
between the current that flows through it
the amount of charge.
And the magnetic flux
that gets established
and that constitutes an inductor.
So the the orange bridge
here is the inductor.
- [Brian] Awesome, thank you.
- [Zlatko] Good.
And I see that we're coming up
on 10 o'clock.
So, I think maybe we'll take
one or two more questions
and then head for a 10 minute break.
- [Brian] Good I'll ask
you one more question,
going through right now.
I'll give you the top voted question.
Can we use a quantum computer
so it's a little off topic, I apologize,
but can we use a quantum computer
to find the path of least action
in a lagrangian mechanics?
I'm sorry for mispronouncing
that so poorly?
- [Zlatko] Lagrangian
mechanics I presented
- [Brian] Yeah,that's the word.
- [Zlatko] Yes, that's is
an interesting question.
Well.
Yes, you can set up an
optimization problem.
I guess it depends maybe a little bit
on how complicated your problem is.
But in principle, yes,
we will get into lagrangians in a couple
of slides after the break,
which will be our path and stepping stone
towards a hamiltonian description
first the classical hamiltonian
which will very quickly turn
into a quantum hamiltonian.
And that will lead us to describe exactly
this structure you see
here in much more detail.
Of course, underneath all the theory,
what I haven't described,
or one described in great detail
is that we're optimizing
an action integral
or a particular path.
Although I will introduce
the classical action angle
variables very briefly,
which we'll talk about that
which will be the analog
of the quantum bosonic ladder operators.
The raising and lowering
operators in a diagram.
