Hello, everyone. We're already a quarter
of the way through this class. So this
might be a good time to take stock of the
situation and see where we are and
re-orient ourselves. So the first thing
I'd like to do is say a little bit about
the organizing principles of this course
just to make sure that we're on the same
page. So, you may have noticed that this
class takes a certain approach to math
formulas which, which I'll call the Kanban
approach. I don't know if you, well may,
maybe you weren't even, some of you were
not even born then but you know, back in
about 1990. There was, there was a lot of
talk about how Japanese, the Japanese were
going to take over every aspect of, you
know they were taking over the
semiconductor industry and they just did
everything well. And one, one thing that
people talked about was that they used
this Kanban approach to manufacturing
which, which was translated as just in
time. So with small inventories and they
were extremely efficient about it. Okay so
what do we mean by Kanban approach to math
formalism? So what I mean is that, you
know when we've in, in this class, I've
tried to keep the mathematical formalism
down to the minimum. So, for example when
we, when talked about two qubits, you know
I said well, if you have the first qubit
in this state a0(0) + a1(1) and the second
qubit in b0(0) + b1(1) then the state of
the two qubits system together is a0b0(00)
+ a0b1(01) + so on. Well, if you want to
do this formally what, what you are doing
here is computing the tensor product of
these two qubit states, you know but
rather than load you with all this
mathematical formalism, what I want to do
is put off this formalism as far as
possible. The same is the case with
measurements where, later, we'll describe
quantum measurements in terms of, in terms
of the measurement operator. And the
measurement outcomes in terms of
eigenvalues and eigenvectors of this
operator but once, once again. It's, it's
actually useful to talk about as much of
this as possible, without introducing a
[laugh] the math, all the mathematical
notation. And that's what we are going to
do in this class. Now, as we go along as
you get comfortable with these concepts,
okay? We will actually talk about this
proper mathematical formalism with which
to discuss all these concepts. And so, I
hope that this is actually going to be
better for many of you, it's a friendlier
way of introducing the subject. And for
those of you who, who already know the
formalism and more prefer to see it, well
you'll see it as we go along. Now, one of
the reasons that I'm trying to introduce
the subject in this way, is that by
keeping it light on formalism, it also
highlights the counter-intuitive aspects.
The intuitive aspects of the subject and
once you look at things intuitively, you
actually realize how counter-intuitive
quantum mechanics is. I think this is
extremely important when you're doing
quantum computing, because quantum
computing actually as a subject. It
exploits some of the most counter
intuitive aspects of quantum mechanics.
And, so if you're going to understand
quantum computing at a deep level, you've
got to grapple with these aspects of
quantum mechanics. And you've got to
understand them at an intuitive level. And
so, it's only by confronting your
intuition with how strange quantum
mechanics is, how, how strange it's
behavior is. That you can start forming
this new level of intuition, okay. So I
hope that this, this, you know may be some
of you had already figured this out
implicitly but I hope this helps you
understand how this course is being
treated and organized. Okay, so the second
thing about, about taking stock is, well
at the beginning of the course we, we
conducted a survey to try to understand
your background and so this is probably a
good time to, to actually conduct another
survey to get a sense of your reaction to
the lectures, the homework, you know how
easy, how difficult you find it, whether
you actually attempt the optional
assignments and would you like to see more
of those. How do you feel about your
linear algebra preparation for this
course? Would you like to see a review of,
of, of the material and if so, which
topics would you like to see a review? And
also would you like to see some discussion
of, of current research topics related to
the lecture what's, what's being done in
the lectures. So for example in the last
lecture I talked about the Bell experiment
and I said actually, this Bell experiment
is , you know it's a subject of current
research current research in, in
cryptography , and what's called device
independent cryptography and in random
number generation. So there are, there are
several, you know it, it is actually an
active subject of, of research. And so
maybe I'll give you a little bit of a
hint, you know a little bit of a flavor of
this, of this current research. And then
maybe you can, in the survey, you can
comment about whether you'd like to see
more such minuet or whether you prefer to
see just the basic lecture topics, okay.
So let's, let me just remind you what the
Bell experiment setup is. The, you know
the apparatus was divided into two
spatially separated parts which we think
of as two boxes. Each of those got a
random bit as input x and y. Each of those
output a bit in b and what we wanted was.
That if, if x and y were both one then we
wanted the alphabets to be different. In
all of the three cases we wanted them to
be the same and what we said is that, if
the boxes are described by local hidden
variable theory so if, if, if there is
some sort of classical description of
what's going on in these boxes, then you
can succeed in this task with no more than
75 percent probability. On the other hand,
if the two boxes are allowed to share a
Bell State, if they share entanglement and
if they measure these qubits in suitable
basis. Then they can succeed with
probability as high as .85 cosine square
phi by eight and in fact, in each of the
four cases, they succeed with probability
exactly .85 in that, in that particular
protocol. Okay so, so now here's a
problem. Here's a, here's a question t hat
people have been trying to think about for
a long time which is how do you construct
a physical source of randomness? How do
you construct a device which outputs truly
random bits. So Intel last year announced
a chip on which it digitally produces
random bits for use in cryptography but
now you could ask the question. How do you
know that the chip is working correctly?
So, in other words, you could ask suppose
you have, you're given a black box which
produces as output what it claims is a
random string. How you can test it? If it,
if it outputs let's say a thousand bits,
how would you test that every one of the
two to the 1000 different 1000 bit strings
are equally likely, or whether it's close
to that. Well, classically it seem
impossible to do this. But it turns out
that you can use the Bell experiment to
certify that you have got real randomness.
And the way this works is, if you run the
Bell experiment and if, if the experiment
succeeds in it's goal with 85 percent
probability or very close to 85 percent
probability, then the only way it can
happen is if the output bits a and b, are
random bits. So for example you cloud, you
could choose to pick b and you could
output it and it would be guaranteed to be
a random bit if you could certify that you
succeeded with probability close to 85%.
So now the, you know okay, so starting
from this observation, there are schemes
that, that actually show how to, how to
create a random number generator so that
you can certify that the particular output
that you've been you, you've got must have
been sampled from a distribution which is
close to a uniform distribution on let's
say, 1000 bit strings. Okay so, if you
want, if you want to read more about how
this comes about, you know I've given you
pointers, references, to two different
papers which are, which have been posted
on the archive. The archive is actually a
good place for you to look for interesting
papers on the on, on, on quantum
computing. Okay so, you know that's an
example of, you know active research, very
close to somethi ng we've already talked
about in, in lecture.