MASOUD MOHSENI: Good
afternoon, everyone.
It's my pleasure to
introduce Seth Lloyd, who's
our first speaker in
Quantum AI series in 2014.
Seth Lloyd is considered
as one of the founding
fathers of quantum
information science
and he's done
major contributions
to a broad range of any of those
in quantum computing, quantum
control, quantum simulation,
transport, illumination
sensing, and quantum
communication.
So he's been in many
groundbreaking research
over the last 25 years.
And so he's actually a
theoretical physicist
who happen to be a professor of
mechanical engineering at MIT
and would like to consider
himself as a quantum mechanic.
SETH LLOYD: Your
Quanta are broke.
We fixed them.
MASOUD MOHSENI: I happen
to have the opportunity
to work with him
over the last six,
seven years while I
was at Harvard and MIT.
We worked on a number
of interesting projects
such as developing material
of noise-assisted quantum
transport with applications
to biological systems.
And also more recently,
developing new quantum
algorithms with applications
to machine learning.
This is actually the topic that
he's going to discuss today.
And we are very happy that
he accepted our invitation
to come here and
present his results.
SETH LLOYD: Great.
Thank you very much, Masoud.
MASOUD MOHSENI: Seth Lloyd.
[APPLAUSE]
SETH LLOYD: Yeah,
it's a good idea
to clap now, because
after I'm done
who knows what's
going to happen.
So thank you very much
for bringing me here,
Masoud was a postdoctoral
fellow with me.
And then he was
working so well, I
hired him as a
research scientist.
And everything was doing great
and then Google hired him away.
And actually it's OK,
because I miss Masoud telling
me what to do, so now he
can tell you what to do.
And my advice is do it, because
it always seemed to work out.
All right, so it's great
pleasure to be here.
I lived in Santa Monica
like 20 years ago
and used to come
to the Rose Cafe
all the time which
is around the corner.
I don't think this
building was here then.
But I have a question.
So I know this is a
very general audience.
So I'm going to talk about
quantum mechanics and machine
learning algorithms.
But my first
question is who here
has studied quantum mechanics
at some point in their lives?
Who here has not study
quantum mechanics?
All right, and the
remainder of people
who are people who have and have
not studied quantum mechanics.
And those are the
real-- those are
the people who know
it intuitively.
OK, good.
So let me just start off
with a really brief course
in quantum computing.
So the exam will
be easy, I swear.
So the thing to remember
about quantum mechanics
is that quantum
mechanics is weird.
This is a technical term.
My brother who's
sitting right here
told me he once saw James
Brown at a concert and somebody
said, so James, what
you're going to play next?
And James said, I don't
know, but whatever
it is it's gotta be funky!
And quantum mechanics
is the funky science
because it's
strange, it's weird,
it goes against your intuitions.
And it's hard to
understand what's going.
So then if I say
things that sounds
strange and weird and against
your intuitions, that's good.
Because as Niels Bohr
once said, anybody
who can contemplate quantum
mechanics without getting dizzy
hasn't properly understood it.
Except he said in Danish,
which was more impressive.
So the key thing about
quantum mechanics, here's
a key feature.
If I have something
like a particle-- so
this is a particle.
By the way, I apologize.
I don't do PowerPoint unless
I have signed a government
grant that requires me
to deliver PowerPoint.
And the reason is that
PowerPoint is a form of Satan
and I object to the
government requiring
me to perform satanic rituals.
But that's the way it is.
Life is too short
for bullet points.
OK, so key features
about quantum mechanics.
If I have a particle
such as an electron-- now
we're normally
accustomed to thinking
of particles like little tiny
basketballs or something.
They can sit in one
place at a time.
But a particle such
as electron, if you
get to things that
are that small,
particles can be in
two places at once.
Now I know this sounds
kind of strange.
So how can a particle be
in two places at once?
Well the thing is in
quantum mechanics,
a particle has a wave
that's associated with it.
So particle over here has a
wave associated with over there.
Particle over there
has a wave that's
associated with it over there.
The wave tells you what's
the probability of finding
a particle in a particular spot.
But quantum mechanics has a
feature that waves add up,
just like waves on
the beach out there.
So if this is an OK wave
and if this is an OK wave,
then this is also an OK wave.
So it's OK to have a
wave for the electron
where the electron is both
here and there simultaneously.
Now I know this
sounds weird and it
does contradict our
intuitions about what
things do microscopically.
Because I'm from Boston, I can't
talk about basketball here.
So let's talk about soccer.
Even if Lionel Messi
makes the soccer ball look
like it's two places at
once, it isn't really.
However, quantum
mechanically things
are like this all the time.
Everybody OK with this?
You shouldn't be OK.
This is wrong.
It's like absolutely wrong.
Einstein hated this crap.
He said this is awful.
I hate this.
My intuition tells
me this is wrong,
and he never believed
in quantum mechanics.
Well, he was wrong.
I mean, he was wrong.
So this kind of wave particle
duality, is what it's called,
when you translate
this into a situation,
some device is performing
computation like a computer.
Well, in an ordinary
computer, a zero
is stored by having a whole
bunch of electrons here,
capacitor uncharged.
And one is stored by having
a whole bunch of electrons
over here, capacitor charged.
And if you go down to a
single electron transistor,
you could have one
electron over here as zero.
And one electron
over here is one.
So electron over here is one.
And the same electron being
here and there at the same time
is zero and one
at the same time.
So if you look at quantum
mechanics and logic, or just
bits, quantum bits, or Q bits
as they're whimsically called.
I mean, I don't know,
when I was a kid
a Q bit was this distance
right here from your elbow
to your finger.
I still think of
that as a Q bit.
But a quantum bit is now
a Q bit spelled with a Q.
So a Q bit, quantum bit, or a
Q bit-- spelled differently,
too-- can be zero and
one at the same time.
So that's the primary
feature of quantum mechanics.
And quantum computers
operate by trying
to take advantage of this.
For many years being a
professor of quantum mechanical
engineering, I used to
say I was at a wedding
and I was seated with-- this
is at our cousin's wedding.
She's a cultural
anthropologist and I
was seated at this table of all
these cultural anthropologists.
They asked me what I did
and I said, oh, I take atoms
and I exploit their intrinsic
information processing power
to make them compute.
And they're like, oh my god,
that's the most politically
incorrect thing I've ever heard.
So Barbara Ehrenreich
was at this table.
She's a wonderful
social commentator.
And she actually has a degree
in chemistry, as it turns out.
So she got me to
explain it better
and finally they
were OK, they figured
I wasn't really exploiting
elementary particles too badly.
And afterwards
her companion, who
was the older African
American activist
from Greensboro, North Carolina,
he put his arm or my shoulder
and said, "Seth, I
suggest the next time
you describe what
you do that you
say you empower
atoms to compute."
And it was a transformational
moment in my life
because I went just like
that, in a nanosecond,
from being an exploiter of
atoms to an empowerer of atoms.
So how can we empower
quantum bits, Q bits, atoms
and electrons, that
we can manipulate
at this subatomic scale-- atomic
and subatomic scale-- how can
we empower them to compute?
And if we can do so, how can
we exploit-- I mean, sorry,
how can we use this feature that
quantum bits can effectively
read or register zero and one at
the same time to do things that
ordinary classical
computers can't.
So, any questions so far?
Yes.
AUDIENCE: What's the
difference between [INAUDIBLE]?
SETH LLOYD: What's the real
advantage of it, I'm sorry?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: I'm not
exactly sure I understand.
I mean, you accept
immediately that it
can be zero and one
at the same time.
So you obviously lived
with us for a long time.
Let me just say more
precisely how this can be so,
because actually there's a
little bit of mathematics
has to take place now.
So how many people have a
background in computer science?
All right.
How many people don't have a
background in computer science.
All right, how many people
do and--no, never mind.
OK.
All right, there we go.
Yeah, me too.
So let me give you a little
bit of the mathematics of how
this works and
then actually this
will help with your question.
So the way that the
mathematics works
is that actually
this state, zero,
suppose we have something that
can have two states, a quantum
bit.
It corresponds to a vector in
a two-dimensional vector space.
And one corresponds to
an orthogonal vector
in a two-dimensional
vector space.
And a general Q bit,
so a generic Q bit,
corresponds to something which
is a generic complex vector
in this two-dimensional
complex vector space.
This is just mathematically
what they correspond to.
And the way that this works--
and so this of course,
this is just equal to alpha
times this plus beta times
this.
And often there's a
quantum mechanical notation
right here where this
vector is called zero.
This means this
funny little thing,
this brackety-like thing,
means that whatever's inside
it is quantum mechanical.
That's what it means.
It's called a Dirac bracket.
That's this notation.
But I'm just defining
it to mean this vector.
And it just means it's a
quantum mechanical dookickey,
another technical term.
So this is equal to alpha
times zero plus beta times one.
So this means these
alphas in the beta
represent basically the
relative height of these.
And because they're complex
numbers, there's a phase.
And you demand that the modulus
squared of alpha squared
plus a modulus squared of
beta squared is equal to one,
because alpha squared is equal
to probability of finding--
if I make a measurement on this
cube and then I say, electron,
are you over here?
Are you zero?
Or are you over there?
Are you one?
Then this modulus
squared of alpha
is the probability of getting
zero and data squared is
the probability of getting one.
And when you add them up,
they have to add up to one.
And why should this be true?
Nobody knows, OK?
This is just the way it is.
You've got to suck it up.
No, in fact, it's the
square of this amplitude,
this complex number
that's a probability.
This is footnoted in the paper
written by Max Born in 1929.
He says, oh, look,
this amplitude
is proportional to
the probability.
It says that in the paper.
But then in the
footnote, it says,
"Under more
consideration, I realized
that the probability is
the amplitude squared."
Note added in proof.
It's the most famous footnote
in physics, actually.
OK, so nobody knows why
this is the case actually.
If I had longer time,
I could give you
some arguments for
why it might be,
but the real answer
is nobody knows.
So are people OK with this?
This is what a quantum bit is
in the formulation of quantum
mechanics.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: As OK as you'll
ever be, that's right.
The great thing about this is
that computer scientists know
not only about algorithms
and computation than I do,
but they know a whole bunch
of stuff about linear algebra.
Because what is
computer science besides
glorified linear algebra?
Or at least much
computer science.
And actually this fact
that linear algebra
is very, very, very
important in computer science
is what I'm now
going to exploit.
Oh sorry, not exploit.
I'm not supposed
to exploit things.
This is what empowers
quantum systems
to be able to do
interesting things
that classical systems can't do.
Because quantum
mechanics at it's
very guts, at the very
bottom of quantum mechanics,
it's about vectors.
It's about vectors in complex
Hilbert spaces they're called,
because there's actually
a norm for this,
but it is in complex
vector spaces.
And quantum computers
when they're
flipping bits and
moving things around
are actually performing linear
operations on these vectors.
So if you look at the
mathematical formalism
of quantum mechanics
to people who
are taking quantum
mechanics know,
it's all about linear algebra,
matrices, eigenvectors,
eigenvalues, blabitty,
blah, blah, blah.
Which is what our family
uses for et cetera.
It's OK if you
don't write a paper.
All right, so that's
what we're going to use.
This in fact is this feature
that quantum mechanics
is about vectors, matrices,
linear operations-- the way
that we're going to construct
quantum machine learning
algorithms is to map ordinary
machine learning algorithms
when they have to do with
manipulating vectors, figuring
out inner products, performing
linear transformations,
inverting matrices,
stuff like that.
We're going to map all
those transformations
on the things we can do at
the quantum mechanical level.
And that's the basic
intuition for why
this whole method works.
And I should say that the
reason I got involved in this
is that Masoud's
and my colleagues,
Patrick Rebentrost--
he's a post-doc at MIT
right now, he was a graduate
student of Harvard--
he came to office a
year ago in November
and he said, Seth, we should
work on quantum machine
learning.
And I said, machine learning?
What's that?
He said look, it's like
all about this manipulation
of vectors and things like that.
I said, oh, I see.
Anyway, so the dumbass
intuition is that lots
of machine learning tasks about
linear manipulations of vectors
in large vector spaces.
And quantum mechanics is all
about vectors and large vector
spaces.
And so maybe you can do
these manipulations quantum
mechanically.
And that dumb ass
intuition turns out
to be correct in a
large number of cases.
And also not correct in
other cases and that's
very interesting.
And so one of the things
I'd like to explore here
since we have a group of people
who I bet know I hell of a lot
more about machine
learning than I do,
is maybe we can cook up
some examples of things
where this works, might work,
that we haven't thought of.
And Masoud and
[INAUDIBLE] and these guys
are working on trying to figure
out things that have worked
and also to figure out
places where it doesn't work.
Because failure, though
less fun than success,
is often more illuminating.
Everybody ready for--
good question time.
Because-- yes?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Complex numbers.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: So the alpha and
beta, the way they show up,
is that they're moduli
squared our probabilities.
So you never actually measure
their complex features
directly.
The way that you actually see
their kind of complex features
are in famous things like
the double split experiment
where you send these waves,
which are complex waves.
This is now a single
electron, right?
It corresponds to a wave
that spreads out here.
It shows up on a screen and then
you get interference patterns
between these different waves
coming through these slits.
And so these
interference patterns
come from adding up
these complex amplitudes.
But you only end
up having access
to experimental
information that's like,
did the electron show
up in someplace here?
So actually your question
is a very good one
because though often you
want to have information
about these complex amplitudes,
what you're going to do
is you need to massage
your problem in order
to actually extract the
information that you need.
And that's often a difficult
and complex thing to do.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: It will
be a bit like that.
Yeah, absolutely, it
will be like that.
In fact, We'll start with
this mapping of these states
onto these kinds of
onto kinds of problems
that we're interested
in in machine
learning just right now.
Unless there are more
questions at this point.
OK, good.
Oh, yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Sure, yeah.
You could.
In fact, for
elementary particles,
the number of internal states
depends upon those spin.
So a massive spin one
electron, for instance,
could be spinning
up or spinning down.
Or actually you
could just imagine
to have three
different places, let's
us imagine it like
that since that's
way you were describing it.
Sure.
You could have a ternary
system or quaternary system.
So, yeah, it often in the
case of quantum computing--
as in the case of
classical-- it's
easier to manipulate things that
just have two different states.
But there's nothing preventing
you from doing that.
And in fact, in
quantum mechanics
we often refer to
qudits, where you
have D different
possible states.
And this is a very useful way
for manipulating information
on that scale.
So suppose I have a
qubit and it can be
and it can have two different
states, like zero or one,
and this is one electron.
Then I could have
another electron
that could also be
in different states.
Like this could be in
two different states,
use a third electron.
I have n different electrons.
So each one of
these belongs in c2,
this two-dimensional
complex space.
I put them together,
it's in actually
what's called the
tensor products space.
And if you don't know the tensor
product space is, don't worry.
It's just what we have
when we put together
a whole bunch of copies of c2.
It's in c2 to the n,
which is basically
a2 to the n dimensional space.
So there are 2 to the n
different possible space here.
And so the key feature
here-- and this
is the key place where all our
quantum speedups will come--
is that if I have n
qubits, they live in this 2
to the n dimensional space.
So when we're going to
manipulate our vectors,
the nice thing is if we
have some if we have data
and our data is in the
form of, let's say,
some gigantic vector which is a
2 to the n dimensional vector.
Now, the classical
representation of this data
takes 2 to the end bits,
or these could be words.
So 2 to the n times
32 for 32 bit words.
But the thing is that
our quantum state,
we can have a quantum
state of n qubits,
and this has 2 to the n
different components in it.
So I can write
this as I can make
a quantum version
of this vector.
And this is going to be equal
to the sum over I. x of i,
this state i, where
i is like i is
an n bit-- this is
an n bit number.
So this is an important
place to be annoyed
or ask questions
or get confused.
This is sort of like
a wedding, but instead
of having one place where you
have to speak now or forever
hold your peace, there are
like 10 places along the way.
So if the bride
and groom actually
end up getting married here
it's going to be a miracle.
So there are n bit
numbers, this is n qubits,
but there are 2 to the
n different components.
So because these quantum
mechanical objects are vectors,
I can have a small
number of quantum bits
that represents a
huge honking vector.
People are looking at
me closely like I'm
trying to pull the
wool over their eyes.
This is just a feature
of quantum mechanics
that the states are
vectors, I put together
n two-dimensional vectors.
They live in a 2 to the n
dimensional vector space.
And so a state of n quantum
bits is a vector on a 2
to the n dimensional space.
And so, this mapping
is very important
because this is the source of
all of our quantum speedups.
We're going to be manipulating
these huge honking vectors.
Things that would take a very,
very large amount of power
classically if these were all
just numbers in a computer.
So these were just like 2 to the
n real numbers in a computer.
It would take me a
long time to do things
like invert matrices
multiplied by this number
or taking our products
of two of these vectors.
But quantum
mechanically, it's going
to be basically
exponentially faster
for all of these operations.
That will be the
source of our speedup.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Yeah, this
is a very good point.
So here, if these
are complex numbers,
we're actually demanding for
it to be a legitimate vector
that the square of these
numbers sum to one.
So there's one constraint,
so it's actually
2 to the n minus 1 vector.
So you're absolutely right.
You're exactly right.
But it only reduces
the constraint.
It only constraints
it by one constraint
to make sure that all
these squares add up to one
so that these modulus squared
are a legitimate probability.
And that's actually
a very good point.
having.
Mentioned it, I
will proceed to try
to ignore it for the rest of
the talk because, of course,
we're often interested in
the norm of this vector.
So we actually have
to have techniques
for dealing with these norms,
moving the norms around
and manipulating them.
But actually the
norms all end up
turning out to be probability.
So we end up
estimating these norms
by when we make measurements, we
find the probability of success
will be like one over
the norm of the vector.
So that's more than
I wanted to say,
but since you brought
up a rather deep point,
I thought I'd address it.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Exactly.
Exactly.
That's exactly right.
And in fact, we have to
take a lot of trouble.
So I hope everybody's
hearing these questions.
These are all extremely
important questions.
The idea here is that
quantum computers
work because you're representing
all these bits simultaneously.
The information
you're manipulating
are that of these amplitudes,
these complex amplitudes here.
And classical computers simply
don't have that in them.
They could have
one of these, they
could have a list of all
these components right here.
So yeah, but you said
it better than I,
so the answer to
your question is yes.
Since you expressed
it so eloquently.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Yeah,
let me tell you
what we need to do this, OK?
Because one of the reasons
I'm making this claim
that if we can manipulate
these relatively small vectors
quantum mechanically,
we should be
able to find out very
interesting features
of these vectors.
And if we have large
collections of vectors,
we ought to be able to do
machine learning on them.
So suppose our data
set consists of 2
to the m vectors of this form--
so it's some huge honking
set of data-- and we want to
perform manipulations of those.
And basically what I'm going
to argue is that we can.
And I will probably
not have time
to sketch out-- I'm relying on
you to tell me how much time I
have, Masoud, because
I don't have a watch.
Oh, I see.
It's a digital clock.
I see.
No wonder.
No, I see it now.
I can see the clock just fine.
So everything everybody
said is correct right now.
So let me say how you
do these kinds of things
and let me tell you the
necessary ingredients
we need to make this happen.
So the first component--
well, zero is just the idea
that quantum mechanics
allows manipulation
of vectors in c to the 2 to
the n using only n qubits.
That's the basic features,
I'm just putting out there.
So what do we need?
First of all, if we have
classical data of this form-- I
guess I could just
write it as x--
we need to create this vector x.
So that we have this
gigantic list of bits
and we need to
create this vector.
So we need to be able to do
this mapping from this large set
into some kind of
quantum mechanical state.
Now this actually sounds like
it might be very hard to do.
But actually, merely to
map a very large amount
of information to a
quantum mechanical state
is a rather easy thing to do.
And I'll do it right now.
So here I have a CD.
It's got like a couple
billion, 10 billion bits on it.
Something like that.
It consists of a very large
number, 10 billion very small,
either mirrors for a one, or
a lack of a mirror for a zero.
And the way you normally read
it is that you just have a laser
focused at one point.
It spins very rapidly.
You look at the
reflections of this point.
However, if I want to map
this gigantic vector of bits
onto a quantum state,
of some quantum system,
I can do that very simply.
I take a single photon, a
single particle of light.
I send it through a
lens, it spreads it out.
It's in a circle that
just encompasses the CD.
It bounces off of it.
There's one photon.
In the case where
there's a mirror,
the photon gets
reflected into the mode
of the electromagnetic
field that
corresponds to this
little tiny mirror.
And there's a case
where there's no mirror,
the photon is absorbed.
And the state of the single
photon after it's bounced off
this CD is now a
quantum superposition
of being in all the different
modes where there's a one,
and not being in all
the different modes
where there's a zero.
So it's actually not so
hard to create a quantum
state that has this funny form.
I mean this one photon just
created this quantum state.
Basically if these x's
are zeroes and ones,
then I end up with
a superposition
of photon in all the states
where they were ones.
So it's not hard to do.
What's hard is to get
the information in a form
where you can manipulate it.
OK?
I know this sounds like
this like a cheap trick,
but it's really true.
I mean, if you actually go
to quantum optics people
and then say, OK,
describe to me the state
of a single photon going
off, they'll write something
with a whole bunch of
annihilation creation operators
that will be that state.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: That's an
extremely good point.
And actually the analog with
holography is a good one.
There's one thing that's
actually quite different.
So, in fact, our
goal here is not
to take our
classical information
and to expanded it
into a quantum state
where there's more information.
It's to take our
classical information
and to compress it
into a quantum state
where there's way
less information.
Actually, if you have
n qubits and you just
want to store
classical bits, you
can only store n classical its.
And our data set had
2 to the n bits in it.
So in fact these
vectors that we're
creating, when I create this
bounce of single photon off
of this, the single
photon in itself
does not possess
10 billion bits.
In fact, if I go and
measure it, I can hold it
and say, is it in
this mode or it's not?
It's only going to possess,
like basically, a bit.
So my quantum state is
much more compressed
than the classical state.
That's the first thing.
And then your second question,
which is well what about noise?
That's a very important
question because as we all know,
quantum systems are
much more susceptible
to noise than classical systems.
Systems that involves
waves like analog systems
are more susceptible to
noise than digital systems.
So the question is is this
susceptible to noise or not?
And I'll address that in a bit.
Sorry, it's a little
while down the road.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Right.
In fact, in just in the same way
that if you were to read this
out one photon at a time with
a conventional laser, right?
You could read it out one
photon, one pixel at a time,
one mirror at a time.
If you expanded on
all the photons,
if you took 10 billion photons
and balanced it off of this,
then you'd have enough
information of those photons
to recreate what happened here.
Of course, it might
be a little hard.
You'd have to have a pretty
fancy optical system together.
But that's just so.
So there's a story about this.
So now, so we need to create x.
Creating this is OK, and we
have to put it in digital form.
That is to say qubit form.
And for this, the
device that does this
is what's called a quantum
random access memory.
And because this is Google,
I have to tell the story.
All right, so the
spring of 2004,
I was at a dinner in Monterey
associated with an early TED
Conference and it was called
the Billionaire's Banquet.
And you might ask me what I
was doing at a dinner called
the Billionaire's Banquet
and I was basically
the guy who jumped out of
a cake wearing a bikini.
That was basically
my role there.
And there I met the founders
of your glorious organization,
Sergery Brin and Larry Page.
And it turned out they
were extremely interested
in quantum computing,
because they had actually
been graduate students at
Stanford with Ike Chuang who's
now a professor at MIT who was
one of the main people quantum
computing.
And so we spent a long time
talking to this banquet,
and at the end of it we agreed
that my postdocs and I would
come up with
something that would
be called quantum
internet search.
We didn't know what
it was because it's
much better to come
up with a name first.
And of course, the reason
is to have something
which could be trademarked
as Quoogle, right,
for quantum internet search.
And it turned out to
be very hard to come up
with something that
was good for this.
But we realized that--
after like three years,
OK-- we realized that
if you could build
a quantum random access
memory, which is a device that
basically takes this
information and it codes it
in a digital form-- a
quantum digital form-- then
you can do the following thing.
If you included a database
in quantum mechanical form,
then someone could
access the database.
So to access Quoogle, you
would ask Quoogle a question,
and you would get
the answer back.
And when you're done, you
know for absolute sure,
guaranteed by the
laws of physics
that Quoogle doesn't
know the question.
This seems impossible,
because actually,
classically to do
this, you actually
have to recode your
entire database
for the person who's
making the private query.
So Google would have to recode
their entire database according
to someone who gives
them a private code,
and they're not
going to do that.
That's a bit expensive.
But quantum
mechanically, it's easy.
It actually has to do with the
so-called no cloning problem.
The solution is when Quoogle
sends back the answer,
they also send
back the question.
And in quantum mechanics
there's something
called the no cloning theorem.
Which says, if you don't
know a quantum state,
you can't make a copy of it.
So if Quoogle sends
back the question,
then the person who asked the
question in the first place
can check to see if the same
question is the same question,
and Google didn't
keep a copy, and so
this is guaranteed to be secure.
So provably secure given
the laws of physics.
OK, fine, three years passed.
We patented it because I'd
had a terrible experience
because my graduate student,
Bill Kaminsky and I in 2001
and 2002, wrote a
series of papers
about how you build an
adiabatic quantum computer using
superconducting
systems with base
qubits and tri-layer
niobium lithography
with inductive couplings.
And we didn't
patent it because we
did a back of the
envelope calculation.
We realized that once you've got
more than a few hundred qubits,
it wouldn't be
adiabatic anymore.
It wouldn't work.
So we didn't patent it.
Then D-Wave spent $100 million
building a superconducting
adiabatic quantum computer
with tri-layer and niobium
lithography with flux qubits
and inductive couplings,
and we felt pretty dumb.
So we patented this, quantum
random access memory, and then
at some other boutique kind
of conference in Napa Valley,
I met Sergey and Larry again.
And in the hot tub at
like 2:00 in the morning,
I said OK, you guys
just bought YouTube.
We've got this
great thing for you.
No, we worked really
hard, it took three years,
it's yours if you want it.
For the first access to it, you
don't want it to become Quahoo.
And they're like wow, this
is like really super cool.
So after the next day, they
came around and they came to me
and they looked
really sorrowful.
And they said, Seth, we
talked with our business guy.
We just agreed that our
corporate philosophy
is to know everything
about everybody
and we just couldn't possibly
invest in a technology that
involved not knowing
about people.
I said look, you're
not getting this right.
There's the European Court
of Justice is like suing you.
You've got all these problems.
And so I'm like, look,
let's do it this way.
You give me $10 million,
and I won't do anything.
So I'm not creating a
competitor for Google
because I'm not doing
anything, so that's fine.
Meanwhile, you tell the
European Court of Justice look,
we care so much about
people's privacy.
We've invested $10 million
in this brand new technology.
It'll absolutely
guarantee people's privacy
with internet search.
And then the European Court
of Justice will be happy.
European Court of
Justice is happy.
You're happy.
I'm really happy.
But they didn't see it that way.
It was very sad.
Anyway, so these
quRAMs can now be
used for these
kinds of processes.
So it's a very interesting
question of how you build them.
If you look at our
patent, you can find out
you could build them
with optical methods,
you could build them using
superconducting circuits.
I know that actually
there's a effort
here to build things that are
much more ambitious and more
interesting than a quRAM.
But for this, it's just
quantum random access memory
which allows us to
create these vectors.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Yeah that's
a very good question.
And this actually shows
insight into something
that somebody
asked earlier about
it's often very hard
in a quantum computer
to figure out how to
get the information out.
So what we have to
do is not only show
that you can do the
kinds of manipulations
we need to do to do things
like clustering algorithms
to build support vector
machines, that kind of thing,
or the stuff that I'm
doing with Paolo here
to do quantum versions of
topological big data analysis.
We have to show that
having done that analysis,
that the information in the
quantum states is at the form
we need to actually get
the information out.
So let me actually
address that in what I'm
going to talk about right now.
So having constructed
these datas,
and now using,
basically, techniques
from quantum computation and
quantum information processing,
exploiting-- I mean, empowering,
empowering those atoms,
the superconducting
circuits in those
photons to do these
manipulations,
we need to construct things
such as, for example,
things like interproducts.
x dot y overlaps
between vectors.
That's easy.
This is takes time order n.
So this is logarithmic in
the size of the vector.
So that's the first step
you might want to do.
We might want to do
things as we might
want to compare some vector
x with the mean value
of some other vectors.
This is a basic step in the
clustering algorithm, k-means.
And k-means, your job is to
take a whole bunch of vectors
and divide them
up into clusters.
And the idea is you assign
the standard algorithm-- which
by the way, happens to be
called Lloyd's algorithm--
but it was made by
Stuart Lloyd in the 1950s
and I think he's any relation
to us so far as I know.
Anyway, so you have
a bunch of vectors.
The way that Lloyd's
algorithm works
is you take a bunch
of seeds-- guesses
for the centers of the clusters.
You assign each vector
to the nearest cluster.
And then, when
everything's done,
you calculate the mean
values of the clusters.
And then you just do it again.
You reassign to
the nearest mean.
And then you keep on
going until it converges.
So that's the kind
of thing you'd
want to do is calculate
the distance of a vector
to a cluster of a vector.
And this also
takes time order n.
So logarithmic in the
dimension of the Hilbert space
and logarithmic in the number
of vectors in the Hilbert space.
So j equals 1 to m.
This goes s order n, log of m.
So these are all--
if you actually
compare these with
the exact versions
of the classical ones--
you may notice something.
That these things,
calculating this exactly
for classical vectors
takes time to the n.
Because you've got a sum of
all 2 to the n components.
Approximating it by
various sneaky techniques
by Monte Carlo or sampling
can take time order n,
but not always.
So there's very
interesting questions
about how much better we're
doing then good heuristic
algorithms.
But if I'm comparing
this algorithm
that we're making to just the
straight up classical versions
of these algorithms, the quantum
versions of these algorithms
are exponentially faster
than the classical ones.
The straight up classical ones.
I'm not saying that they're not
excellent heuristic clustering
algorithms, of course there are
excellent heuristic clustering
algorithms.
Just saying that these quantum
algorithms that we have
are exponentially faster than
the corresponding classical
ones.
OK?
So, for instance, we can
do things like k-means.
If you do the quantization
of this Lloyd's algorithm,
then you end up with something
like k n log m, which in order
to do the k-means Lloyd's
algorithm versus the quantum--
this is quantum-- versus k
times 2 to the n times m.
This is the classical,
k squared, sorry,
for the classical.
So when you actually start
quantizing these algorithms,
what you find is
that what you get
is assignments of
vectors to clusters
that are a lot faster
than you had previously.
And now this starts to attempt
to answer these questions that
have been raised about
how you measure stuff.
What the output of this
algorithm is is it says,
OK, it gives you a
quantum superposition
of a list of vectors
that are assigned
to a particular cluster.
So of course it could be
very, very, very large
number of those vectors and if
you want to know all of them,
you'd have to do the
algorithm many, many times.
But if all you want to do is
sample representative members
of this cluster,
then these algorithms
are exponentially faster
than the classical versions.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: I'm not sure.
So what this is
actually, what you
calculate quantum
mechanically is this.
Right?
This is the true Dirac bracket.
This is called a bra, not an
article of women's clothing.
And this is a cat.
So this together is a bracket.
This is why this notation
is what it's supposed to be.
So this is the inner product
between these vectors.
So this is equal to
norm of x, norm of y.
Sorry, this is equal to x dot
y over norm of x, norm of y.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: But
that's hard to do.
I mean this raises a whole bunch
of interesting, open questions
actually.
So for instance, just
evaluating x dot y,
x dot y is often
very easy to do.
You can do this
with order n steps
quite accurately by Monte Carlo.
But if you take something like
x with some transformation of y,
like before you transform
of this vector y,
then there's no known way
and it seems very unlikely
you could do this with
classical Monte Carlo.
Where this is very easy to
do quantum mechanically.
Indeed, what can you do
quantum mechanically?
So here are the tools.
So what did I describe there?
I described this quantum version
of Lloyd's algorithm, quantum
clustering algorithm.
Let's look at the other tools
that-- [COUGHS] excuse me,
that we actually have.
Sorry, it's too warm here
so it's making me cough.
It was five degrees
when I left Boston.
[COUGHS]
So we take a vector
x and we can map it--
we can do the fourier transform.
This corresponds to
taking a quantum vector
and doing the quantum
fourier transform.
So the classical time
for this takes time order
n 2 to the n for the classical
fast fourier transform.
Even for the fastest fourier
transform in the West.
And this takes time
order n squared.
So things that involves
fourier transforms
are exponentially faster
quantum mechanically.
I mean, these are, you
take a 2 to the n vector,
you have 2 to the n by 2
to the n sparse matrix.
That's not easy to do.
Here's another thing
that's quite useful.
If I take x goes to a inverse
x where a is a sparse matrix,
this is order again n
times 2 to the n classical.
And then the quantum
version of this,
once again, this
is actually just,
once again, order
n squared, maybe
n cubed, depending
on how accurate
you want to be
quantum mechanical.
So matrix inversion
for sparse matrices
is also exponentially
faster than
in a quantum mechanical system
then in a classical version.
In fact, our third algorithm--
this is our first algorithm
right here, this clustering
algorithm we came up.
Well, I'll call her
our second algorithm,
even though it's
our third algorithm.
So this actually, if you look
at these kinds of manipulations
like this, if I have a bunch
of data points right here,
and suppose I want to create
a support vector machine.
So what a support
vector machine does,
support vector
machine identifies
the maximum marginal
hyperplanes that
separate these two
data points right here.
So if I have this
set of data points
it's a big set of vectors.
I have this set of data points
it's a big set of vectors.
You can basically construct
support vector machines
by doing linear manipulations
in matrix inversion
on the so-called covariance
matrix of this data.
And you can actually
show that constructing
the covariance matrix and
performing the requisite matrix
inversion, solving
the linear equations,
you have to find these
kernel operators-- blabbidy
blah, blah, blah, as they say.
And you find that this
quantum support vector machine
has exactly the same
kind of features.
And then of course, the kind of
question that you want to ask
is here's another data point.
Does it belong over here or
does it belong over there?
So you can show that
constructing these hyperplanes,
finding these vectors
from the data,
and then comparing this
data point to find out,
is it closer to this side or
was it closer to that side?
That can all, again, be
done in time order n where
the classical versions all
take time order 2 to the n.
And the reason is,
again, just to remind you
of why this is,
it's because you can
do these kinds of simple linear
transformations, like quantum
fourier transforms
or matrix inversions,
solving sets of
linear equations,
on very high dimensional vector
spaces is actually quite easy
quantum mechanically.
It's because these
transformations,
it's like, what is
this photon doing
when it bounces off this CD?
A photon is a quantum
mechanical object
moving through a humongous
dimensional vector space--
the vector space of all
the propagational modes
of the electromagnetic field.
It bounces off of here and
quantum mechanics automatically
does this linear transformation.
In fact, just sending a single
photon through a diffraction
grating is exactly performing
a fourier transform
on the mode labels
of the photon,
for people who know
optics kind of stuff.
So in fact, these kinds
of transformations,
like doing fourier
transforms, there
are many quantum
mechanical systems
that do things that
look like they would
be very hard from a classical
computational perspective.
And they just do
them automatically
because that's
just what they do.
They're quantum
things and performing
linear transformations on
quantum things like fourier
transforms is what happens.
So I'm almost out of time.
When do you want me to stop?
It's 1:58 right now.
I'm going to advertise one more
nice thing because since Paolo
Zanardi is here from
USC and this is the work
that we've been doing recently
that-- so was there a question?
Or was that just an echo?
All right.
So this is a very fun topic.
Topological big data analysis.
You've got a whole
bunch of data points.
There in some funny shape.
But there's a hole
in the middle.
A hole is a topological feature.
How the hell do
you find the hole?
OK.
And this goes under the
name of persistent homology.
Who here has heard
of this stuff,
these topological methods?
They were developed by
Carlson at that UC San Diego,
though I found out
from-- I'm sorry?
He's now at Stanford.
He was at UC San Diego.
But I found out from
Michael Friedman,
who was this field medalist who
actually does quantum computing
stuff, that Michael
Friedman developed
a whole method in a secret
DARPA program, or ARPA program,
in the 1960s.
And he showed me his
recently declassified notes.
Anyway, so this is
actually really cool.
So what is persistent homology?
You think, wow,
topological features.
How do we actually find
those kinds of things?
So the way persistent
homology works
is you basically
tessellate, or triangulate,
if you like, this kind
of systems right here.
You create what's called
a simplicial complex.
So you create a
topology, an ocean
of things that are
close to each other.
Topology.
And then you create
the simplicial complex.
I'm not going to draw
all the simplicies
because there's too many.
You see where I'm
going with this, right?
So now the great thing is that
now the next step-- and this
is what I all learned from
Paolo here, right here--
the next step in this
homology is each simplex gets
mapped to a vector in a gigantic
complex dimensional vector
space.
Complex vector space
of high dimension.
And then finding holes,
voids, connected components,
et cetera.
This is just linear algebra now.
So it was something that looks
like it might have nothing
whatsoever to do with things
we quantized that is performing
linear algebra operations
on large dimensional vector
spaces, excuse me.
If you actually go into what's
called algebraic topology,
it turns out that the way
people do algebraic topology is
to say, we can't deal with
these topological spaces.
Let's map everything
into vectors
on some gigantic,
dimensional vector space.
And let us do linear
operations on that.
It's a great field.
But it's a great field
that actually happens
to be perfectly
adapted for making
these quantum
mechanical algorithms
and the quantum
mechanical algorithms.
As you might not be surprised
out of this if this is an m,
the classical logarithms.
Classical goes 2
to the m where m
is the dimension of this vector
space and the quantum, go s m.
So, these quantum
algorithms, again,
give exponential speedup.
So Masoud is looking
edgy right there.
I know what happens.
That means he's
about to beat me up.
I'll get you a
beer later, Masoud.
Don't worry.
So let me just then conclude.
So what happened?
So that was a short course
in quantum mechanics.
Everybody passed.
So I was supposed to be here
a few days earlier but then I
had to give all these
graduates oral exams.
The graduate students just go to
the dentist themselves, I feel.
So I'm tired of failing people.
So everybody passed.
So quantum mechanics
has this weird feature
that things we think of as
particles correspond to waves.
They can be in two
places, electrons
can be in two places at once.
Mathematically, this means
that the mathematical structure
of quantum mechanics
is that the states
of quantum mechanical
systems, these waves
are, in fact, vectors in high
dimensional vector spaces.
And the kind of
transformations that
happened when things
like particles of light
bounce off of CDs are just
linear transformations
on these high dimensional
vector spaces.
Quantum computing is
the effort to exploit--
I mean, or empower-- quantum
systems to transform,
allow these linear
transformations
to perform the kinds of
things that we ourselves
would like to do for
manipulating data
in this quantum mechanical form.
And if you have a quantum random
access memory which allows you
to map data onto big data
into big quantum data.
I know for a fact,
being in Academia,
that in order to get
a grant these days,
you either have to have the
word big data in your title
or graphing.
So what we're aiming for here
is graphing-based quantum
random access memory for
quantum big data analysis.
It's a winner, I can just
smell the funds coming in.
So you have a quantum
random access memory
that maps your classical
data onto a quantum state.
This actually turns out
to be something that's
a lot easier technically than
full-blown quantum computing,
initial experiments, and proof
of principles have been done.
It's been patented because
I learned my damn lesson
with D-Wave the
first time around.
Then we have to make
sure that we can actually
do the manipulations
that we need to do.
The various linear
transformations,
evaluating inner products and
distances, looking at means
and clustering.
These turn out to be all
accessible in a quantum
computer and compared with
their classical versions,
they are exponentially faster.
Then you go to more fancy stuff.
You find you can do
fourier transforms,
you find that you can
do matrix inversion,
you can solve systems
of linear equations.
You can do things like
find kernels of operators
in high dimensional
spaces, which
is the essence of actually
doing this persistent homology.
And so what we found
is that once you have
this first step of
this dumbass intuition
that Patrick Rebentrost
came up with is like,
whoa, lots of
machine learning is
about manipulating large numbers
of vectors in high dimension
vector spaces.
Gee, quantum mechanics is about
manipulating large numbers
of vectors in high
dimensional vector spaces.
Wouldn't it be fun if we
could bring them together?
That's dumbass intuition
turns out to be correct.
And so now we're working with a
whole variety people, and here,
of course, with a few
folks at Google research,
to try to make this vision
of actually doing big data
analysis in intrinsically
quantum mechanical ways that
are potentially
much more efficient
than the classical ways
to make that a reality.
Thank you.
MASOUD MOHSENI: Thanks, Seth,
for that nice introduction
to quantum mechanics and
quantum machine learning.
So are there any
other questions?
SETH LLOYD: Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Yeah, that's a
very interesting question.
So quantum mechanics on itself
seems to be exactly linear.
At a fundamental level,
all the experiments
indicate the quantum mechanics
is linear to one part in 10
to the 18th or even better.
However, you can actually--
the word linear is
used in a variety of ways.
So you could certainly
have nonlinear interactions
in quantum mechanics.
The interaction of light with
matter is nonlinear typically.
So like absorption
of light of matter
is nonlinear, that's
a nonlinear reaction.
But the waves-- the
quantum mechanical waves--
still superpose in
a linear fashion.
So in some systems,
as in, for instance,
Bose-Einstein condensates,
you can actually
emulate what would be a
fundamental non-linearity
by manipulatively using
these existing nonlinear
interactions.
But quantum mechanics
is basically,
the way that we know it, it
is linear, exactly linear,
and so we're stuck
with that linearity
at this fundamental level.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: And in
the number of vectors.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Well
normally to compare,
normal k-means
goes as m squared.
Where m is the
number of vectors.
And ours goes as log m.
I'm sorry, when I take
m and take it to log m,
I call that an
exponential speedup.
That's what I mean.
Because saying a
logarithmic speedup
doesn't sound-- I mean, it's
true that taking m to log m,
going, wow, it's a
logarithmic speedup,
nobody gets very
excited about that.
But of course, taking m to log
m is an exponential speedup,
so I call it that.
It sounds more exciting.
It's also true.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Yeah,
you can, always.
We tried lots of stuff.
There are many things we
haven't been able to do.
So for instance,
this is things that I
want to talk with
people about here.
So we can solve linear
equations and even
polynomial or
nonlinear equations
with equality
constraints, because then
a Lagrangian method
turns us into another set
of linear equations
that we can solve.
But it's hard.
We can't do that with
inequality constraints.
And these show up all the
time in machine learning.
You have some convex space.
You're trying to identify the
interior of the convex space.
Like the support vector is
lying on the exterior of this
and you solve some equation
with inequality constraints.
And then you're stuck with
Karush-Kuhn-Tucker equations.
And who knows what
the hell to do then?
And we don't know what to do.
We've tried really hard,
all kinds of methods.
Basically what we've
been doing right now
is we've been doing the stuff we
know how to do well, like solve
linear equations, do
fourier transforms,
identify kernels
and eigenspaces.
Or we can diagnalize
matrices and find
eigenvectors and eigenvalues.
That's something you could do
very effectively on a quantum
computer.
But there's some
things we can't do.
And so, we certainly--
this is not a magic bullet.
Even if you could build a
large scale quantum computer,
this wouldn't be a magic bullet
to solve all machine learning
problems by no means.
Was there another
question over here?
Yes.
AUDIENCE: [INAUDIBLE].
SETH LLOYD: Oh, a product.
Oh, I don't have a product
except with a lot of phonons.
I am not a company.
But in terms of building
quantum computers-- so,
basically what's
happening right now
is that there are
small scale quantum
computers with a few
tens of quantum bits.
We're hoping to have like,
within the next five years
or so, there are people
like John Martinez who
say they'll build you a quantum
computer with 100 quantum bits.
I don't know if it's true.
But it's quite possible.
And I think actually
the way to go
here is one of the philosophies
behind this approach
is the people in
quantum computing--
which is a field that's been
around maybe for almost 20
years as a very young field--
have been talking about trying
to build very general purpose
quantum computers that
act in the same way that
regular digital computers do
but operate in a fully
quantum mechanical fashion.
But I think one of the
philosophies that I pursue--
and certainly the people
here at Google are pursuing--
is to try to construct.
Because there's
this intrinsic power
in the way that quantum
mechanical systems behave.
Can we actually
construct devices
that-- I was just about
to say exploit again, it's
like, really hard.
Like when your professor
of engineering at MIT.
Not to say we exploit the
properties of this materials.
Can we find ways of using this
ability of quantum systems
intrinsically to
perform operations
in high dimensional
vector spaces
without actually having to
go through this whole entire
digital formalism?
The fact that
diffraction grating
makes a fourier
transform automatically.
Can we use that kind
of thing, actually
construct devices that would
do this kind of manipulations
without having to go the route
of making a large scale quantum
computer.
That's what we're aiming for.
And there I think that
there's actually--
I think there's very good
possibilities for that.
Yeah?
AUDIENCE: [INAUDIBLE].
SETH LLOYD: No, actually,
you do see that.
In fact, I just last
Thursday and Friday
was at a conference at
[INAUDIBLE] Cambridge.
It was the first conference
on integrated quantum nano
photonics.
So one of the big developments
in just switching theory
right now, or practice,
is the development
of integrated photonics.
So you etch waveguides
and silicon,
you can put millions of
switches on a silicon wafer.
And so for many
years, there have
been lots of experiments
in quantum computing
that did quantum computation
using interaction
of light and
polarization with atoms.
And all these
experiments look the same
because all optical
experiments look the same.
There's a gigantic optical
table and all these lasers
off on the side and a
million mirrors there.
And then light beams
are bouncing around all
over the place and have
no idea what's going on.
Every single optical
experiment looks like that.
So a remarkable thing is now
happening, driven actually
by the desire of places
like IBM to build
these on chip-switching
arrays is
these gigantic interferometers
with interactions between light
and matter are now
being miniaturized
into individual wafers.
So a 3 centimeter--
well, they're
slightly bigger than that.
So 30 centimeter silicon wafer
contains an interferometer
that would've taken five
optical cables previously.
So I think there's a
lot of hope for that.
The real problems are
getting strong interactions
between light and matter.
Superconducting systems are
very good way to go there.
In fact, there's a strong
analog between transmissions
of microwaves in
superconducting microwave line
and analogs of photons.
In fact, it's so strong
because they're the same thing.
They're microwave photons.
But instead of atoms you can
use these little superconducting
quantum bits that now interact
much, much, much more strongly
with these microwave photons
than ordinary light interacts
with atoms.
So there's a lot of great
quantum technologies out there.
And, as usual, a lot of
the progress in this field
is being made by the combination
of industrial push creating
new technologies and
fabrication technologies that
allow us to do things
we couldn't before.
And then pull from application
on both classical and quantum
application.
But of course, what you're
saying is completely correct.
And I think there's a lot
of hope in the near future
to have much more
powerful devices that
use these techniques.
MASOUD MOHSENI: Thanks
again, Seth, for coming.
SETH LLOYD: Thank you.
[APPLAUSE]
