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>>: Today we have Roman Lutchyn here from -- he actually works at
Microsoft Research at Station Q in Santa Barbara. And he's visiting
today. He graduated from the University of Minnesota and then spent
three years at University of Maryland before coming to Station Q where
he has been for a year. Just about a year. So he is going to talk to
us about topological quantum computing with Majorana fermions. Roman.
Roman Lutchyn:
Thank you very much for the introduction and for the
invitation to come here and give a talk. It's always a pleasure to
visit Redmond.
So this is Microsoft Station Q. We are -- you know, that is view from
our building. We are located in a very nice campus. UCSB campus. And
our group consists of mathematicians, physicists and computer
scientists.
And what we're trying do, we're trying to explore the -- this interface
between three different disciplines to build a quantum computer. And
the particular approach that we pursue is topological quantum
computing. And I'll explain today what this is about and why this
approach is very promising.
So most of the people in our group are theoreticians or -- but we also
coordinate with many experimentalists around the world, and we just ask
them to do some experiments to test our ideas.
So this is the outline for my talk. I will start with some
introduction of quantum computing and I will explain why it is
advantageous and then I'll explain what is topological quantum
computing and present some realistic systems where we think we can
implement this approach.
So as you know from every day world that most of the physics -- most of
our every day life can be explained in terms of classical laws.
Newton's law, Maxwell equations of electro magnetism, et cetera.
But if you scratch the surface, we can quickly realize that the world
is actually quantum mechanical. And these are devices that we use in
everyday life, and some of the things in those devices is actually
based on quantum physics and transistors tunnelling. Tunnelling is a
particular quantum phenomena. In lasers, the fact that you have a
discrete two-level system, so you have quantitization of levels is also
a quantum mechanical phenomenon.
In MRI, the fact that atoms have nuclear spins, nuclear spin is also
quantum mechanical object.
So I would say we use these properties in everyday life, but we use
them not in -- not to the fullest extent. And in order to do that, we
have to go back to the principle. In order to explore quantum
mechanics in depth, we have to go back and use the most fundamental
principle of quantum mechanics, the principle of superposition.
So if quantum mechanics postulates that if you have a state Psi, then
the state can be a superposition of two other states, zero and one.
And principle of superposition can be illustrated in this very nice
experiment, double slid experiment which we all know and love.
So if the light was -- I mean, the light is a wave, so principle
superposition is just the fact that we add the amplitudes for the light
and because we add amplitudes, not probabilities, we can have this
interference pattern on the screen. So the idea is you send the light
here and here, and then the light can go either through this pass or
this pass. And then depending on the position of the screen, you'll
see a sequence of these dark and bright spots. And so that's the
principle of superposition because you have to add the amplitudes for
electromagnetic field and then square the amplitudes to get the
intensity on the screen.
So a long time ago, in 1982, Richard Feynman, very famous American
physicist, asked the following question: Can we explore the God-given
resource, the source of quantum mechanics for something useful, for
quantum computation, for example.
So if you do that, then you immediately realize that because each of
the -- each state of the system, if it's a two-level system, can be in
superposition than if you have N such two-level systems, then the
information that is encoded in these two -- in such a system scales is
two to the n. So it scales exponentially with the size -- with the
size of the system. And that's actually something we are trying to
explore in quantum computing. We're trying to explore the fact that we
can store much more information in quantum realm than in classical
realm where you can just have zero and once.
So what is the potential of quantum information processing? Actually,
there are many. And, for example, if you want solve a system and you
want to -- you would like to understand, say, properties of particular
system, like a chemical reaction that involves some quantum mechanics,
then we can simulate that on a -- on this quantum computer so we can
basically solve any quantum system by just programming it into our
quantum computer and running the simulation.
Another application is factorization of large integer numbers. And
that basically triggered this interesting quantum computing because
this particular operation lead to -- this particular algorithm leads to
exponential speedup. So our security will be in danger if we have a
quantum computer, or maybe we can invent something more complicated.
Quantum encryption.
Another thing which is fairly useful is quantum search algorithm. And
so basically a search of an entry in a database, but here we have
algebraic speedup rather than exponential, but it's also very useful.
And finally quantum cryptography. So quantum physics allows you to
send information securely and the security is guaranteed by the
physics. And so these are just a few potential applications. People
are working and discovering more and more algorithms, but what I would
like to talk today is actually how to implement, how to build a quantum
computer. What are the systems that are suitable to do this task.
And Krista here is doing research in quantum algorithms, so if you are
interested, you might want to talk to her and learn some more details
about this.
So in classical computer, which I have a bit which is a basic logical
element. It has zero and one. So we need to find something similar in
quantum computer. And it's called qubit or quantum bit. Essentially
qubit is a quantum two-level system, so these are the two levels. And
by controlling the couplings, so very often two-level system as we
present it by spin, so you can have essential [indiscernible] algebra.
And if you control all the couplings, if you control all the couples
with a spin, then essentially you can program the state you want to
get.
So a qubit is a controllable two-level system. You can drag this
vector by applying some gate, by applying some gate voltages and
signals and changing these coefficients, you can drag this vector and
put anywhere on a bloch here you want. So basically, you can change
all the coefficients here and here. So that's an idea. So you would
like to have a controllable two-level system.
Well, these are the potential realizations of these controllable
quantum two-level systems, and I would say they come in two flavors.
So the first one is based on microscopic degrees of freedom. So like a
nuclear spin in MRI, this is a microscopic degree of freedom. Or
position of an atom in particular location, that's also a microscopic
degree of freedom.
And another type of qubit are -- another type of qubit is placed on
collective degrees of freedom. And these are superconducting qubits
and actually also topological qubits that I'll talk about today.
So the idea is there is that a lot of electrons, they behave in a
coherent way so that it's like an orchestra. You listen to orchestra
and you see that everybody is following same, you know, pattern. So
the same thing happens here to the extent that you can describe ten to
the nine electrons by a single variable. So that's what happens in
superconductors, and in topological qubits, we would like to, you know,
we would like to have that but even go beyond that paradigm.
So one of the problems in quantum computing, I mean, these approaches
fairly developed now and people have created these two-level systems.
They manipulate them. They couple them and try to play with them. But
there's a fundamental problem. And the fundamental problem is called
decoherence.
So if you have a microscopic system and you want to couple to this
microscopic system and control its position on a bloch sphere, it turns
out there are other couplings which are unavoidable. And we cannot
just filter out useful couplings and, say, from unuseful couplings.
So these unavoidable, say, harmful couplings lead to errors in our
quantum computation. And this process is called decoherence.
So that's why we talk to experimentalists, because quantum phenomena
occur in the lab, and in the lab, you really have to make your hands
dirty, and you know.
So the process of quantum decoherence, something that people know from
the lab experiments and they measure it, so they start in some state
and then they see that this state will decay to a ground state, for
example, if you start in the excited state with time. So your
information can be preserved for some time.
And we would like to actually -- we would like to try to correct these
errors, but it turns out that, unlike in classical computers where
error correction is very developed and fairly successful, in quantum
computers, it is very hard to do. And the reason it is very hard to
do, because if you want correct more and more, you need to introduce
more and more resources.
So there are some approaches where people established what should be
the error rate, and this error rate is ten to the minus five. So it's
fairly -- fairly small. And so far, we are trying to achieve this
error rate but it's very hard.
>>:
[Indiscernible]?
Roman Lutchyn:
Yeah. So you have a decay time, and you have a speed
of particular operation. So if you divide these two, that will be your
error rate.
>>:
[Indiscernible].
Roman Lutchyn:
Excuse me?
>>:
[Indiscernible] can people do today?
Roman Lutchyn:
It depends on particular system. So I think we are
reaching ten to the minus four in certain qubits, but the thing is,
there is something you think explored here. This is error rate for
some models. And if we scale the system, this actually decay rate
might also scale with the sizeable system, so this area needs to be
explored more, especially when you have more and more qubits together.
Because there is also a collective effect.
But what we are trying to do at Station Q in Santa Barbara, we are
trying to solve this problem in a different way. We're trying to find
a system which will be immune to these errors at the hardware level.
So rather than having to correct for these errors, we would like to
build a hardware which is fault tolerance.
And the idea for this approach actually goes back to somebody who spent
a lot of time here in Redmond, Alexi Kitaev, who is now professor at
Caltech, and he said the following thing. He said, well, let's try to
encode our information in varying nonlocal degrees of freedom. And the
errors that we usually have, they come from some local fluctuations,
local changes in the environment. But if you have -- if information is
stored locally, then it will be insensitive to these local
perturbation.
So you can think about this in using a topologist's view. And
topologists, they say, well, these three objects are actually
identical. They all have just a single hole. So as long as we don't
do anything drastic to the system, as long as we don't destroy this
whole, the information which is encoded in this geometry will be
protected. You can bite a piece of this donut and the information is
still going to be preserved. And so that's the idea.
You'd like to find a system which will be insensitive to these local
perturbations because we encode the information in a completely
different way.
So when -- so usually we are talking about topological phases of matter
because phases of matter were experimentally observable quantities can
be insensitive to local perturbations.
And one of these -- one of the well-known examples in condensed matter
physics is quantum whole state. Quantum whole state is a state of a
semiconductor in a strong magnetic field.
So these two discoveries actually were awarded Nobel Prizes in '85 and
'98 and there's some difference between them, but I don't want to go
into details as far as -- this is concerned -- this is the topological
phase of matter.
So if you take an integer quantum whole state, which is a state, say,
in lower magnetic field, and you apply electric field to the system,
then there will be a current flowing in the perpendicular direction.
And then if you just measure this current and find whole conductivity
of the system, this is essentially the coefficient proportional to the
electric field applied, what you'll find, that this coefficient will be
proportional to fundamental constants, E squared over H times the
integer. And this integer is accurate to ten to the minus nine. So
you take a sample, different geometry, the sample can have different
disorder (inaudible), different parameters, but still this quantity is
protected to ten to the minus nine.
In fact, this whole conductivity is a standard of resistance now.
>>: And again, you're talking about the lifetime of this integer, not
about -- I mean, the integer is accurate to one, not to ten minus nine.
Roman Lutchyn:
No, no. You measure something and something is close
to one, but how close is it to one? Right? So I changed parameters in
the system, and it still stays in one, whereas in some other system,
nontopological, you change something, this coefficient will change
completely. It's going to be zero. Or it's going to be ten.
So the point is that this inter, this number close to one
[indiscernible] ten to the minus nine. And it doesn't matter what I do
to the system.
>>: Another way to say it is if it was a billion, it wouldn't be a
billion and one or ->>:
I see.
It's always [indiscernible].
>>:
It's an integer, but it's -- yeah.
>>:
Yeah.
It's accurate [indiscernible].
Roman Lutchyn:
Yes.
>>: So I'll ask a question [indiscernible]. There are the software
approaches to where correction [indiscernible] quantum computing. What
do you propose [indiscernible] hardware approach for that
[indiscernible] uses replications, right? And since this is what
happened, the state is not captured by a single particle. Instead, it
is captured by a lot of repetitions. So what is the fundamental
difference here? Why not [indiscernible] in software and just have so
many repetitions?
Roman Lutchyn:
Well, I mean, if you refer to collective nature of
the degrees of freedom which we are encoding information as repetition,
then in this sense, yeah, there's some similarity.
But you have to realize that you know, there is just a single parameter
that characterizes a system which consists of, you know, ten to the ten
electrons. And we don't have to control each of the electrons. It
just happens that the nature does it in a particular way.
Now, if I have a large system, then the number of electrons can be ten
to the 23 or, you know, even larger. And I don't have to control each
of them. But I can characterize the state just by a single parameter.
So it's different from error correction where you control different
elements separately, right? So there are similarities, and there are
differences. Okay. Did I answer your question?
>>:
Probably need to study more physics.
>>:
Mm-hmm.
Roman Lutchyn:
So that's an example of a topological phase of
matter. It's a phase of matter which the state of which is not very
sensitive to local perturbations. And what you want to try to find is
a state which is similar but has additional properties. It's even more
sophisticated state of matter, and it's called non-Abelian
topological -- this phase is called non-Abelian topological phase.
So once you find such phase, and I'll give you some examples where we
predict these phases can occur, we also have to be able to manipulate
certain excitations in these systems. They're called quasiparticles or
anyons.
So these are the candidate systems. And in today's talk, I will spend
some time on superconductors. And I will show you that superconductors
are very interesting systems which can actually be engineered. So in
some sense we are trying to engineer the state by combining various
properties of different materials.
So I mentioned the world non-Abelian, and this word non-Abelian is at
the heart of the topological quantum computing proposal. So let me
explain why this is so.
When, as a freshman in the graduate school, I took quantum mechanics, I
was taught that there are bosons and fermions. There are two
particles: bosons and fermions. But that's true only in
three-dimensional world. And the reason it's true in three dimensions
comes from this argument. So we know that if you take a
[indiscernible] function of the system and try to exchange two
particles, then depending whether they have bosonic or fermionic
statistics, the wave function will pick up a plus or minus sign. And
the reason there are just these two possibilities comes to the, you
know, essential to the reality of a braid group in three dimensions.
So if I exchange these two particles, and then I will exchange these
two particles twice, so essential I return to the same point. So I did
some operation on the system. But I can deform this loop. I can
essentially take advantage of this degree of freedom outside of the
board, and essentially deform it to a zero, to a point. Which means I
have to return to essentially the same thing, which means that there
operation of exchanging T, T squared, should be actual to one.
And if T squared is equal to one, the only possibility is that T is
equal to plus minus one. And that's why you have only bosons and
fermions.
But if you take two dimensions, in two dimensions, we have more
possibilities because we don't have this degree of freedom. We don't
have -- if you can imagine -- if you constrain our particles to the
plane, we don't have this degree of freedom. And so these exchanges
are actually nontrivial. So you cannot deform it to -- this argument
doesn't hold.
And that was the prediction made by two Norweigian scientists, Leinaas
and Myrheim, and also Frank Wilczek from the 1980s. And it has been
realized later that such particles exist in these two dimensional
system, quantum hole systems, and there are -- there are situations
there where when you exchange, you pick up the phase which can be
anything. That's why these particles in two dimensions are called
anyons, which means any phase.
>>:
[Indiscernible] non-Abelian statistics on this [indiscernible].
Roman Lutchyn:
Should be non-Abelian statistics.
[Laughter].
Non-Abelian statistics is -- happens when you have the generalcy in the
many body ground state. So let's imagine we have many states which
have the same energy, and then such exchanges that I'm talking about
here can induce certain transformations between these many body ground
states.
So then, I cannot just describe this operation as just a phase. It can
be described by matrix. That's what -- something that we are trying to
exploit here. We are trying to exploit the fact that when you have an
non-Abelian statistics, then exchanging different particles between
each other can be explained in terms of different matrices. And
matrices do not commute, so we can make various combinations and encode
information in these states.
>>:
[Indiscernible].
Roman Lutchyn:
Yeah.
So we have a --
>>:
[Indiscernible]?
Roman Lutchyn:
Yeah. You have a state which has many particles.
You know, X, N, so X one, X2 N. I'm writing [indiscernible] function
which explains a state. And then I can take two particles from this
state and try to exchange them together. So I can exchange X one and X
three, and that will be described by a matrix M, or I can exchange X2
and X three. And that's going to be explained by matrix M.
>>:
[Indiscernible].
Roman Lutchyn:
Because in three dimensions, as I explained, the
exchange is very trivial. The braid group is trivial. So two
exchanges give you just one. And so they -- the square root of one can
be either plus or minus one.
>>: So there is an implicit [indiscernible] similarity behind all
this. You were referring to additional degree of freedom where it
improved to a point. But if you puncture the plane, you cannot prove
[indiscernible].
Roman Lutchyn:
>>:
What is the similarity that -- what do you puncture in both cases?
Roman Lutchyn:
somewhere.
>>:
Well, you have a particle, physical particle sitting
Right.
Roman Lutchyn:
>>:
Yes.
And if you can go around this particle --
I see.
Roman Lutchyn:
-- then you can avoid the similarity. But if you
don't have a degree of freedom, it's like having a plane with defect
which has a line, and you cannot just go around the line. So that's
the difference. That's the way to think about it.
And the reason we are trying to avoid -- we can avoid the coherence
here because these manipulations never involve excited state of the
system. So in the previous proposal where we had quantum bits, but say
conventional quantum bits, they always involved excited and ground
state. So in some superposition between zero and one. And we know
that, well, excited states, they decay. So they relax to a ground
state. So lifetime is limit.
>>: [Indiscernible].
excited state --
I can [indiscernible].
Doesn't have to be
Roman Lutchyn:
Well, that's not igan (phonetic) state of the still.
The superposition is not an igan state of the system.
>>:
[Indiscernible].
Two position doesn't have to be [indiscernible].
Roman Lutchyn:
Well, zero and one refers to ground state and excited
state, right? And you're trying -- and you're saying it's
superposition?
>>:
[Indiscernible].
Roman Lutchyn:
Well --
>>: I mean [indiscernible].
that [indiscernible].
I'm just confused as to why you're saying
Roman Lutchyn:
Right. So okay, maybe I'll just draw something here.
So this is a two-level system. Two-level system has a ground state and
excited state.
>>: [Indiscernible] what was all that?
side by side too.
Roman Lutchyn:
>>:
Yeah, right.
I mean, I can write zero, one,
Then it's a degenerative two-level system?
If I have a superposition.
Roman Lutchyn:
So if you have a degenerative two-level system, it's
kind of similar what I'm saying, but [indiscernible]. So the reason it
doesn't work if you have a degenerate two-level system which is kind of
local, because you add some small perturbation, and that's the -- this
degeneracy is going to be splitted.
So this degeneracy is accidental, and you change something
[indiscernible] and the degeneracy is gone.
>>:
For example, very small magnetic field from somewhere.
Roman Lutchyn:
>>:
Exactly.
Spin up and spin down.
Roman Lutchyn:
So this is not the robust degeneracy, but you
actually getting to the point they'll explain in the next slide. So
you would like to create a degenerate system which is robust. Which is
robust to these small perturbations. And you're trying to use these
degeneracies, the robust degeneracies to encode the information.
Okay. So that's the basic idea. That's why if you have a state which
is non-Abelian state, then you do manipulations in the ground state,
doesn't invoke any excited state and the information is going to be
robust. And this robustness will be guaranteed by the quantum
mechanics and by physics.
So it turns out that the particles that the state that satisfies these
properties is the state which carries Majorana fermions. So that's why
my topic today was quantum computing was Majorana fermions. If you
find Majorana fermions, then you'll find the state which will satisfy
these properties.
So what is Majorana fermion anyway? We know about fermions. Now what
is Majorana fermion? What's so special about this Majorana fermion?
It's an interesting object which was predicted a long time ago, in 1937
by [indiscernible] Majorana, who asked the question in the context of
high energy physics whether the -- whether Dirac equation which
describes essentially spin one-half -- corresponds to spin one-half
particles, has to be necessarily complex. And then he showed that by
choosing a particular representation, you can make equation real and
therefore there should be a real solution of this equation. And if
there is a real solution of the equation, then essentially the particle
will be the same as its own antiparticle.
And since then, there has been a lot of surge in particle physics, high
energy physics, super [indiscernible] which is also related to high
energy physics and also solid state systems.
But as of now, we don't have a convincing evidence for the existence,
and I would like to present one road to overcome this problem. And
I'll present complete proposal where we can find these Majoranas?
>>: I have a quick question. So take a Dirac equation and split it
[indiscernible]. So it must be something more [indiscernible] on that.
The first question. The second question is why [indiscernible]?
Roman Lutchyn:
Well, the [indiscernible] emulations, which are
formulaic. And now, if you really split Dirac equation two, two, I
mean, you can always split complex number into real parts. But the
point is that in these Majorana presentations, there is no complex
part. There's just real part. So if you count degrees of freedom ->>:
[Indiscernible].
Roman Lutchyn:
It's a real equation. A real equation has a real
solution. So if you have real solution, then it means that the complex
part is gone.
But I will also get to this in the next slide.
>>:
[Indiscernible].
Actually in this slide.
>>:
Yes [indiscernible].
Roman Lutchyn:
No, no, you can always -- they can have complex
solutions, but if you have a real equation, you can always find a basis
where there's a real solution.
>>:
Okay.
Roman Lutchyn:
If you have a complex equation, there is no basis
where you find a real solution. So really, this Majorana fermion is a
real -- you can think about it as a real part of Dirac fermion. And if
I want -- if I want to construct a Dirac fermion, I have to take this
to [indiscernible]. So our intuition actually comes from having Dirac
fermions. You know, in [indiscernible], we describe everything by
Dirac fermion.
And this Dirac fermion that I create from two Majoranas is very special
because it's not a composite object, and these two Majoranas as
depicted in this picture can be far away from each other. So
essentially, we created -- if you have Majoranas, you'll have a fermion
will be, you know, [indiscernible] and local. And this locality helps
us to make a protected systems against decoherence. Yes?
>>: I have a very basic question because I'm not a physicist.
state here is represented by the relative position of the two
particles? Is that --
So the
Roman Lutchyn:
The state is represented here by the fact whether
this -- it's occupied or unoccupied by a particular Dirac fermion. So
I have -- I created artificially this Dirac fermion. Okay. And now I
want to enumerate my Hilbert space, my states. And I can enumerate by
either saying, well, this mold is unoccupied, then it's zero. Or it's
occupied, which is this -- which is denoted by this, then it's kind of
one. And so my logical qubit is a qubit which is either occupied or
unoccupied by this Dirac fermion.
And because of that, if I have many of these particles, then from 2n
Majoranas, I can roughly create N qubits.
And so that's the idea. You have these Majoranas on the plane, and you
know, you have a Hilbert space that they correspond to, and this
Hilbert space will be degenerate because these Majoranas, they are
sitting at zero energy. So there's a degeneracy involved.
>>:
So can you explain in which sense this is [indiscernible]?
Roman Lutchyn:
Well it's a complex, right, object. So each of these
two are real, but now if I conjugate this, I'll pick up a minus sign.
So definition of Dirac fermion is that C and C conjugated are not equal
to each other. And Majorana fermion, they're equal to each other. So
it's essentially a half a degree of premium. So Majoranas, one
Majorana is a half of Dirac fermion. It's only real part of Dirac
fermion.
So now I want to illustrate why exchanging these particles will
actually lead to something nontrivial and why this happens in the
system.
So suppose now I make -- I make the situation a little bit more
complicated. I add another Dirac fermion, which is localized between A
three and A four. And then I can say, well my Hilbert state, my
Hilbert space can be encoded in either this mode occupied or unoccupied
or this mode occupied and unoccupied.
And there are four states which essential enumerate this Hilbert space.
And now I can do something very nontrivial because I can take a half of
my fermion, which is encoded here, and exchange with another half of
the fermion, which is encoded here.
So normally, in other systems, you would take one electron and exchange
with another electron. But here, because this electron is very
nonlocal, it's actually a composite object, you can take a half of a
electron in exchange for another half of electron. And that's why
some -- you know, we are doing something nontrivial. And if you just
check the math, you will find that you know, the state you will end up
will be different. And that's basically a simple quantum operation -quantum gate operation.
>>:
[Indiscernible]?
Roman Lutchyn:
>>:
Red and blue?
[Indiscernible].
Roman Lutchyn:
Oh, oh, there is no significance.
know, just labeling which ones we are braiding.
>>:
It's just, you
So [indiscernible]?
Roman Lutchyn:
Yeah. So this is the -- this is the excitement in
the community because of these ideas. And the reason there is so much
excitement, because we found how to borrow a trick from superconductor
and use this trick to find a systems where Majorana fermions can be
detected. And the idea is the following: So I'm going to give us some
sort of introduction for superconductors, about you if you guys are
familiar with superconductors, I can just skip it. So I'll -- should
I --
>>:
Do it.
Roman Lutchyn:
Okay. [Laughter]. So superconductors are actually
interesting -- is very -- superconductivity is very interesting state
of matter. It's a state of the matter where at low temperatures, once
you lower the temperature, the state doesn't have any resistance. So
if I measure -- if I take mercury and I lower the temperature and
measure the resistance of mercury as a function of temperature, then at
some point, resistance is essentially vanishing. And that was
discovered by Kamerlingh Onnes in 1911. And soon after discovery, two
years after discovery, he was awarded Nobel Prize.
And the theory of superconductivity remained unsolved until 1957 where
these three theorists, Bardeen, Cooper, and Schrieffer, came up with
the explanation why this happens, why you have zero resistant state at
low temperature. And the basic idea is that electrons in the metal,
they kind of glue to each other and they form Cooper pairs. From two
electrons, you can create the boson, quote, which is called Cooper
pair. And bosons and fermions have different -- very different
properties. At low temperatures, bosons want to occupy zero momentum
state. They want to condense and sit in the same level, whereas
fermions, they don't want to do that.
And so -- and that actually is an explanation for the zero resistance.
Bosons can have zero resistance because there is coherence between
them.
But there's another property which is important in the theory of BCS.
The theory of BCS has the state -- the Hamiltonian, the effect of
Hamiltonian that is described by the BCS theory mixes states with
different particle number. So the particle number is not conserve.
And that's actually key to having Majoranas because if the particle
number is not conserved in Hamiltonian -- and I wanted to say that this
theory of the superconductivity is so developed that we can even
explain this cool stuff. We have a magnet on top of superconductors,
this magnet can be levitated because of the Meissner effect.
So we know superconductivity pretty well now.
[indiscernible] superconductivity.
At least conventional
But I was going to say that this fact that superintendent conducting
Hamiltonian doesn't conserve the particle number is actually key to the
following of the ration. In order to diagonalize Hamiltonian which
doesn't conserve particle number, I have to have a superposition
between sort of particles and holes or electron creation and
[indiscernible] operators.
And the superposition has some amplitudes, UNV. And if I want to
create a Majorana fermion, all I immediate to do in the superconductor
is to find the state where these two amplitudes are actually the same.
Well, up to conjugation. Then you can just check because
[indiscernible] that you know, if U is equal to the star, then you will
get the particle will be self-conjugated. So that is a Majorana
fermion.
So where do you find these states [indiscernible]? Well, where do we
find these states with such properties? Well, we find them whenever we
have an equal superposition of a particle in a hole. And where do we
find such superposition? Essentially somewhere between -- somewhere
between the empty and occupied band. So somewhere here. So we have to
look at zero energy. And if you find the state of zero energy, such
state actually, by construction, will have these properties. And the
reason that it has by construction these properties actually is
something fundamental in the theory of superconductivity.
There is certain symmetry in the serial superconductivity which is
called particle-hole symmetry. And the symmetry tells you that if you
have a solution which corresponds to energy E, then there should be
another solution which corresponds to energy of minus E.
Now, if I have a solution with zero energy, you can just check that
this symmetry guarantees that U should be equal to the star, and
therefore, the state that is sitting at zero energy will be
self-conjugated. Yes?
>>: [Indiscernible].
one-half.
I thought the rule is Fermions [indiscernible]
Roman Lutchyn:
Yes, that's right. This is a model for now, but I'll
give you a physical realization later. I'll show you how one can
enslave the spin.
So basically, we are guaranteed to have the zero energy state if, as
you correctly pointed out, we have spinless Fermions.
Now, the zero energy state is not going to be accidental like in the
other example. It's going to be protected by the symmetry. Because
see, the symmetry tells us that we should have states that come in
pairs, E and minus E.
Now if I tried to -- if I add some
I try to lift the state from zero,
partner. And since it's not going
violate particle hole symmetries.
allowed.
perturbation to the Hamiltonian and
then it's not going to have a
to have a partner, that should
And so this -- these changes are not
So unless we do something drastic to the system and we completely
destroy it, weak perturbations do not live to degeneracy. So it is
robust. Okay.
Now, I'll give you some examples of spinless for now, but then I will
show how one can engineer the spinless Hamiltonians from spinful
Fermions.
So why do we have these zero energy states annual where do we have
these zero energy states? So there is a simple model which one can
write, and it's a superconductivity model where the older parameter is
proportional to momentum. So this coefficient depends on momentum and
is proportional to P.
And within this simple model, what I wanted to illustrate here, without
any fine-tuning, you have these zero energy states. So let's imagine
that we have a situation where there's chemical potential, sort of
varying in space, like this. Then whenever chemical potential is
positive, we have a metal. That one becomes negative, we have an
insulator, because there is no fermion surface, so the Fermions cannot
populate. There is no -- the dense state is zero.
So if you assume this configuration, actually this configuration models
one dimensional system [indiscernible] these points.
Now I'll do another simplification, and I'll say that, well, this
chemical potential changes very slowly in space. Okay. Then I can
drop essentially higher order derivatives and what I end up with is
something well known in high energy physics. It's called one plus one
Dirac equation, which we are solving for zero energy.
One plus one Dirac equation is very well studied and in particular, it
is well known that whenever the mass of this Dirac equation which here
plays the role of chemical potential, changes a sine, you'll have these
states. You have some bound states.
In other words, you don't need to know high energy physics. You can
just say I have this equation. I want to find the normalizable
solution of this differential equation, and what you will find that
there are normalizable solutions at the ends.
So bottom line, if I have this spinless [indiscernible] superconductor
at the ends of one dimensional system, I'll have zero energy solution.
And these zero energy states will be robust against small
perturbations. So I don't need to fine tune any parameters like delta
or new or anything. As long as I have a [indiscernible] spinless
[indiscernible] superconductor, I will have these states.
So that's idea. We need to somehow engineer a spinless superconductor.
How can we do that when all electrons have spin?
So there are two ingredients that we will use, and first ingredient is
spin-or bit interaction. So spin-or bit interaction is an interaction
which couples spin to momentum. And if you can somehow enslave spin to
the momentum, we can get rid of this half of a degree of freedom that
is redundant, and we can make the system effectively spinless.
And the second thing is superconducting proximity effect. So we want
to induce -- rather than looking for these materials in nature, you
want to have the superconducting state on demand. So then you'll use
some other superconductor and make -- and create some interfaces, and
those interfaces, we will have the necessary states.
So I'm be talking today about superconductor, semiconductor
heterostructures. And these heterostructures will have essentially the
required properties.
So first, let me just spend a few minutes on spin-or bit coupling.
Actually, spin-or bit coupling in condensed [indiscernible] systems
appears quite generically. It appears because at the interface between
different materials, you might have an electric field. A very strong
electric field, in fact.
And then, if the motion of electrons is, say, constrained to this
plane, then if I make a, you know, a [indiscernible] transformation and
go to a arrest frame of this electron, this electric field will induce
a magnetic field in this rest frame will be pointing up. And if the
electron has a spin, obviously it interacts with magnetic fields, so
therefore, I have a coupling between the spin and electric field.
And this effect is called -- I mean, this mechanism is called Rashba
mechanism, and this mechanism is -- can be written in terms of -- in
this form. So I will use this in the rest of my talk.
And what is important for this Rashba spin-or bit coupling that it
actually does the job because now you can see that the spins are
enslaved to the momentum which live on different fermion surfaces. So
the spins, they precess around the fermion surfaces either in clockwise
or counterclockwise fashion.
The second ingredient that I will need is the proximity effect.
what is proximity effect?
So
Well, you have a semiconductor, and you put it very close to
superconductor. So electrons in the semiconductor will tunnel into
superconductor. And for some short time, they will feel these
correlations, superconducting correlations. So they will inherit these
properties from the superconductor. And effectively, it's a way to
create a Hamiltonian which doesn't conserve particle number because
what I can do, I can send in an electron and then measure the -- and
this electron will be reflected as a hole, and as a result of this, the
Cooper pair, the two electrons will be emitted in this superconductor.
So if I want to describe this kind of special scattering that appears
at this interface, I ever to add a term in the Hamiltonian which
describes this appearance of two electrons from the semiconductor. So
that's effectively a term which allows us to break particle number
conservation.
Okay. So that's the proposal. So suppose we first take this
one-dimensional system which has the Rashba spin-or bit coupling. And
we also have some magnetic field along the longitudinal direction. And
so what happens in the geometry that first the -- this Rashba coupling
separates the spectrum into two probable lows, but then this additional
term essentially opens up a gap in the spectrum at zero momentum.
So at the end of the day, if the chemical potential is somewhere here,
you will have a single fermion surface. So it's going to be -- if you
have a single fermion surface, which is kind of tricky because I
started from a spinful system, but in the basis that diagonalizes this
Hamiltonian, I will have spinless particles. I'll have single specie
of particles that live on this single fermion surface. And so that's
how we get around the fermion doubling. You spin degeneracy.
The nanowires actually that are useful for our purposes are indium
arsenide and indium antimonide nanowires. And for technical reasons,
they are very favorable. They have very large couplings, very large
couplings to magnetic field, to Rashba also -- Rashba [indiscernible]
coupling, and they have very good contacts with metals.
So now this is a rather technical slide, but I want to present the
schematics, why this state, this complicated Hamiltonian is related to
this spinless [indiscernible] superconductor. So you have -- so now
let's imagine that we have a wire that is lying on top of the
superconductor and then you'll have these additional terms to use the
proximity effect.
Then let's do the following trick. Let's first diagonalize this part
and then we write the other part in the basis that is diagonal where
the single particle part is diagonal.
So what happens when the chemical potential is here, that essentially
this band is unoccupied. And I can project the system. I can just
keep the lowest band.
And if I do that, all these terms that are here, I will throw away side
plus component. So they are gone.
And now what I'm left with is essentially single component for single
species. Now index minus is kind of a spin index. And I have a P wave
superconductor. Okay.
So this construction allows us to map this spinful model to a spinless
P-wave superconductor, and I showed you that spinless P-wave
superconductors have zero molds at the ends. So therefore, this model
should have zero molds at the ends as well.
And you can see that these zero -- that these zero molds can be only if
the chemical potential is somewhere in the gap. If the chemical
potential is somewhere here, then this construction that I showed you
doesn't work.
So what we filed, we find that this system can generate Majorana
fermions on demand. Okay. So I changed some parameter in the system,
and I can go from a state which has these molds to a state which
doesn't have these molds. Okay. And you can keep all the details,
keep the full Hamiltonian and check these ideas numerically. So that's
what you find. So this is the plot of the density of states is a
function of the length of this wire. So this is a length. And the
function of energy.
And it's just numerical [indiscernible] of the Hamiltonian and then
computing density of states. And you can see that you have zero energy
states at the end where the magnetic field is in the right regime. And
in the opposite regime, you just don't have anything. So they
disappear.
>>:
Technical question.
Roman Lutchyn:
>>:
What do you use [indiscernible] Hamiltonian?
How we did the --
[Indiscernible].
Roman Lutchyn:
Oh, it's mathematical. Mathematic [indiscernible].
You can do many things with mathematics these days?
>>:
Oh, yeah.
Painful, but yes.
Roman Lutchyn:
Okay. All right. So that's basically the system
that we'd like to use to build our topological quantum computer.
Well, one can also say, well, what if you're theory is missing some
important ingredients and how do you test your theory? Well, there are
simple experiments that one can do to test this theory. I mean, the
fact that you have zero energy states in one phase and you don't have
it in the other phase should be seen in the experiments.
And the simplest experiment is to try some kind of a tunnelling or
canning tunnelling microscopy and try to tunnel at the end of this
wire. If you have a state of zero energy, then the tunnelling should
be allowed. If you don't have it, then the tunnelling should be
forbidden.
And what you should find -- actually there's a concrete prediction that
if the state is in topological phase, then this tunnelling conductor,
the IDV, should be equal to something very universal. Two E squared
over H.
And in the other limit, it should be exactly zero.
So there are other experiments which also have interesting signatures
of these Majorana fermions in the system, but I just wanted to maybe
talk more about quantum computation in this audience. I would be happy
to explain more about experiments if somebody is interested.
So suppose we have this system.
we do with it?
Suppose we have this phase.
What can
Well, by the way, this is how the current experimental situation -this is where the current experimental situation. Actually, we have
these nanowires and we are doing the experiments as we speak. And you
know, we'll keep you posted?
>>:
I think they're [indiscernible].
Roman Lutchyn:
>>:
Yeah.
Right.
[Indiscernible].
Roman Lutchyn:
>>:
Okay.
[Laughter].
Right.
[Indiscernible].
Roman Lutchyn:
So these are the gates that you can apply to change
chemical potential in the system. And this is the wire lying on top.
And beneath, you have a superconductor.
>>:
Just a regular S-wave.
Roman Lutchyn:
>>:
Regular S-wave superconductor?
[Indiscernible].
Roman Lutchyn:
So how can we use these systems for quantum
computation. So there isn't -- there is a very interesting effect. So
I mentioned that if I change the chemical potential somewhere, I can go
through the phase transition, right? If I change the chemical
potential I can go from a state which doesn't support Majorana to a
state that supports them. Right.
So suppose I change the chemical potential only in half of the system.
[Indiscernible] instead of essentially I move this Majorana particle
somewhere here. And essential, that allows me to exchange -- to move
these particles and exchange them around because if I create now a T
junction, so I can move this particle here, then move this other
particle here then move it back, and so on, I can do the exchange
that -- I can do the braiding operation that I mentioned before. So
these wires essentially allow you to do braiding operations. A network
of these wires allow you to exchange the particles. Because it's
essentially, you know, a two dimensional network of one dimensional
wires.
And one can also measure the state of the system by coupling these
wires. So this is the wire and this is a set of gates. By coupling
the system to a squid, so this is the squid. And you have C Joseph and
junctions. And it turns out that because the state is either occupied
by an electron or unoccupied, these [indiscernible] junctions, these
squids are sensitive to -- and they can distinguish different states
with different fermion number. So I just measure some characteristic
of the squid like a splitting energy and I will be able to distinguish
the difference between the state occupied and unoccupied. So I can do
the redial of the state.
>>: So what are those tunnel barriers?
surfaces [indiscernible]?
Are they just a fictional
Roman Lutchyn:
No. You can get them for free actually. So if you
have aluminum and expose the aluminum in, you know, to the air, it will
create a natural layer of oxide. And so this oxide is a barrier. And
so if you basically take them out from the vacuum and then stick them
together after that, and you have a junction.
>>: Usually don't put them in the air [indiscernible] some chamber or
[indiscernible] basically.
Roman Lutchyn:
>>:
The basic idea is that they use oxides.
[Indiscernible].
They could have done it here.
Roman Lutchyn:
So another thing, what you can do, you can actually
create a device which is called topological quantum bus. So I
mentioned that the information that is encoded in these topological
degrees of freedom is very different than informations that encoded in
conventional qubits where the user freedom are microscopic. So if you
want to couple, say, conventional quantum architecture and topological
quantum [indiscernible], we need to find a bus which can do that.
And these are the ideas that allow you do that. Again, it helps that
these are superconducting systems and here's, for example, a proposal
to couple a semiconductor double-dot qubit where essentially the state
is encoded by the presence of electron or absence on this dot. So
that's one proposal.
Another proposal is to couple this wire network with so-called transmon
qubit, which is just a generalization of a charge qubit in
superconductors. But what is interesting, that one can then couple
these superconductors to photons. And photons, one can teleport, they
can live for over large distances.
So here's the example. You have a strip line resonator which it's
basically a cavity with very good mirrors. And you can control the
number of photons in this cavity by lowering the temperature. And
essentially this photon couples to this transmon qubit, which is
basically the qubit which consists of these two superconducting kind of
islands, large islands. And then transmon qubit couples to topological
qubit.
So bottom line, we also can use advantage of these conventional qubits
to do the logical operations which are very fast in those qubits and
then teleport information into topological space and store the
information there.
So I'll leave with you this final slide where you envision some kind of
topological quantum computer. It's basically a computer which is in
the [indiscernible] but it's also connected to some classical machine
and you can just dial a particular task that you want the quantum
computer to do and it will do the task and spit out the result and
communicate it to a classical computer.
So with this, I will end.
[Applause].
And thank you very much for your attention.
Yes?
>>: You mentioned speed. Is that -- so the topological computer tends
to be slower than the [indiscernible] ways of doing -Roman Lutchyn:
Well, you have to exchange the particles which tends
to be slower. Also ->>:
Are there other disadvantages that you're aware of?
Roman Lutchyn:
Also, one gate that we need to teleport from
somewhere from conventional computers, we know how to make many gates
with topological, but there is one gate that is missing, and we would
like to teleport in gate from the conventional quantum computers.
>>:
Yes, conventional quantum computers as opposed to --
Roman Lutchyn:
Yeah, yeah. So it's -- I understand. Quantum
computing is already unconventional, but you know, for the people in
the field, it's kind of conventional and topological is unconventional.
>>: Sort of as a follow-up, is there a limit that one-dimensionality
of it? Does that limit what you can do with it? Or can you make a
grid of those wires?
Roman Lutchyn:
You can make a grid. There's no problem. You can
make actually various networks and study the properties of these ->>:
There's no problem with them over laying each other?
Roman Lutchyn:
>>:
You can --
No.
-- [indiscernible] interaction between them?
>>: Yeah, the key is [indiscernible]s and surfaces and so on. And if
you can manufacturer those in quantity, then you have got a lot of
opportunity. That's all.
Roman Lutchyn:
>>:
How long before you expect to get results on this?
Roman Lutchyn:
>>:
There was another question?
Well, it's not the right question to me.
Well, there's an amplitude, right?
Roman Lutchyn:
Yeah. [Laughter]. We hope soon. I mean, they start
fairly recently. They actually started maybe in the spring of this
year. So it's hard to predict, but they already have some good
results. I mean, they've tested the parameters that we used for
example for some estimates of these nanowires and they find that
they're in the same ballpark. So now they have to couple them to
superconductors and characterize that part.
>>:
And then to adjust the --
Roman Lutchyn:
Chemical potential, tune into the right regime.
>>: Is this one of the experiments that it's been two years analyzing
the data afterwards?
Roman Lutchyn:
No, no. They will -- this is not a high energy
experiment. There you just measure something that comes out and
analyze it next day.
>>:
[Indiscernible] three, four, five, or six times to be sure.
Roman Lutchyn:
>>:
Okay.
[Indiscernible].
Yeah.
>>: What is the device you are using to try and observe the emergence
of the [indiscernible]?
Roman Lutchyn:
So [indiscernible] experiment, for example? So you
just have some normal metal tip which is in tunnel contact with the end
of the wire. And you will tunnel and see where the electron actually
has some density of states to go into or not.
>>:
It's hard to see [indiscernible].
Roman Lutchyn:
No, no, but it's very easy to see the IDV because
you're just measuring electric signal. And this electric signal will
be either zero or something finite. In fact, not just finite, but
something that is two E squared over H. Two times fundamental
constants. So these are very concrete prediction.
This is though zero temperature results. So if you have low
temperatures sufficiently low temperature, which is doable.
>>: This process has such high application in cryptology. Could it
be, say, that NSA already have that and they just won't tell anyone?
[Laughter].
Roman Lutchyn:
Well, if NAS has it, then I don't know of it. Okay?
But the rest companies that are very interested in these devices, like
Lockheed Martin, for example. So maybe they might have it. I don't
know. Yes?
>>: So there's some double [indiscernible] experiments that they're
looking for [indiscernible].
Roman Lutchyn:
Yes. In high energy physics, people are looking for
Majorana fermion in double (inaudible) decay.
>>: So if they find that, that confirms that [indiscernible] can exist
but it doesn't necessarily mean that [indiscernible].
Roman Lutchyn:
Exactly. So here, in condensed matter, what we are
predicting is actually more than just Majorana. We are predicting a
state which has [indiscernible] statistics and has degeneracy. It just
happens to be that these particles at zero energy, they satisfy
Majorana condition. So once you find the Majorana in solid state, then
it automatically should have these properties. But in high energy, it
doesn't have to be the case.
>>: Majoranas don't have a lot of measurement nicety for the
[indiscernible] crowd any more than [indiscernible] do as a matter of
fact. I mean, they just don't have a lot of properties that you can
measure in a [indiscernible].
Roman Lutchyn:
But this double barrel decay, I think that's the
experiment that they tried to do and as far as I know, the data is
inconclusive because there's not enough statistics. And they're trying
to ->>:
[Indiscernible].
I mean it doesn't happen that often.
Roman Lutchyn:
Right. They're trying to reproduce this experiment,
but that costs a lot of money. Costs millions of dollars, if not more?
>>: Here it's kind of interesting because the Majorana's trapped in
the end of that wire. We know where it is, if it is.
Roman Lutchyn:
Right.
>>: And so the tunnelling those where to look, I guess, is the way to
say it, right?
Roman Lutchyn:
Yeah.
>>: Get to decide whether there's a Majorana there or just the end of
these pairs. You know.
Roman Lutchyn:
Yeah.
Exactly.
>>: What does that mean physically when you said the changed the
chemical energy to try and kind of change the endpoint of the wire?
that -Roman Lutchyn:
Yeah.
>>: Chemical potential. I don't know why it's called chemical
potential, but it's . . .
>>:
What are you doing to the wire?
Roman Lutchyn:
So see, I can have some gate on top.
change the chemical potential in the ->>:
Are you adding electrons?
Roman Lutchyn:
>>:
And I can
What are you doing -- what is --
It's a capacity coupling.
Okay.
>>: Yeah, what you're doing is you're changing the level where the
states are occupied.
Roman Lutchyn:
Exactly.
Is
>>:
Right.
So [indiscernible] or not, you're trying to get --
Roman Lutchyn:
Trying to tune that level.
>>: You're trying to get this thing at zero energy by jacking that
thing up so that it's a permitted thing. It gets protected once you're
there because of this [indiscernible] ->>: [Indiscernible]. This is an electrical change in the material,
not like a physical ->>: Yeah, but we are talking yesterday about population changes, and I
asked Roman, I said, so what are all the ways we can change the
chemical potential?
Roman Lutchyn:
Well, let's don't go there.
>>:
No, but I mean, you know.
>>:
There's lots of ways.
Roman Lutchyn:
[Laughter].
Yes.
>>: [Indiscernible] many
suspect [indiscernible].
to do it. But the NSA -Maryland, [indiscernible]
people [indiscernible] quantum mechanics. I
Now, the Russians and Chinese might be able
if a hundred [indiscernible] people go
be noticed.
Roman Lutchyn:
Yeah. They actually created the institute, joint
quantum institute, so they already have a lot of people there.
>>:
Okay.
>>: And some of the people in Maryland know where [indiscernible] is
too.
Roman Lutchyn:
>>:
Okay.
Let's thank Roman again.
[Applause.]