Introduction, thanks, Enric, for initiating this. This is the area that I recently, I

mean, I would say maybe last six months, I've become interested. Well, thanks to COVID,

I cannot travel. So, well, this joint work with my colleague, your colleague, Nana Liu,

who is, she's really expert in quantum computing, and she got her PhD from Oxford.

She was selected by this MIT technical review for 10 innovators under the age of 35 in Asia Pacific

region. So she's somewhat like a land of the ABC from quantum computing.

All right. So why we want to do quantum computation?

Well, basically, we know the class computer has this more or less, which

everybody recognizes probably has reached its limit. And of course, there are other restrictions

with classical computation, like computations, bandwidth, energy cost, memory, and time.

All right. So actually, Feynman was the first one to propose that we should develop quantum computing

because he was curious about whether one can make a computer that just simulates quantum mechanics,

which we know for M-body Schrodinger equations is impossible due to the curves of dimensionality.

It's impossible to run it by a classical computer. So then he said maybe some computer that's based

on quantum mechanics principle can solve this M-body Schrodinger equation. All right. So

essentially what he conjectured is maybe a quantum system can be used to simulate other quantum

systems. And he was dreaming about some universal quantum simulators. He even suggested how to do it

by array of spin particles, which is actually later will be called qubits in the quantum computer

language. So this audience is probably a mathematic audience. I give a very quick, very elementary

review of what is quantum computing. So basically, a classical computer is based on these

binary things, either 0 or 1. That's classical computer. With a quantum computer, you compute

the quantum states. All right. So you input the quantum states, then it goes through the

computations, through so-called gates. And then you output the quantum states, and you have to

take the measurement. All right. Then you get this solution that's more like classical solution.

So that's the measurement. All right. So basically, in the quantum computing, we also got the bits.

So here are just the two simple bits. One is like a 0, and the other one is 1. These are basically

bases in a two-dimensional space. All right. So then the qubits are the linear combination

of these two bases, vectors. And here, the coefficient of linear combination is the complex

numbers. But this Psi is a vector, which has a norm 1. So it is a square plus b square. I mean,

the modulus should be the n to 1. You have to normalize. It's the unit vector. Then from 1 bit

to n bits, you just do this tensor product. And quantum computer basically just computes

based on the qubits. So what it gives you is so-called these qubits, these

quantum states. And trying to find out, this is famous, the Schrodinger cats, right?

So basically, you have this probability between 0 and 1. And a square, modulo a square and b square

are just probability of getting this 0 state and 1 state. And the measurement basically means that

you open this box to see whether this cat is alive or dead. That corresponds to the measurement.

OK. But before you take the measurement, you have all these quantum states.

So here, we're going to work on Hilbert space, which is in complex domain, but dimension capital

N. We usually assume capital N is 2 to the little n. When there's little n, it's usually the number

of qubits. So what one hope is that with the quantum computer, we can achieve so-called

exponential speed up. But that basically means you want the quantum algorithm. I mean, for a

problem of this size n, usually, the class of computers, the cost will be some polynomial

factor of n, n to some power, integer power. But for quantum algorithm, you hope the cost will be

poly little n. It's polynomial little n. So that which is like a log of capital N.

And if you can achieve that, we say you will get so-called exponential speed up. So that's the goal

of a quantum computation. And quantum computation starts with some quantum computation. It's

basically like linear algebra, linear operators. You multiply these quantum states by unitary

operators, unitary matrices. Here are some of the very elementary matrices that quantum computers

use. For example, here, this is the Pauli matrices, sigma x, sigma y, and sigma z. And this

Hadamard gate, this so-called c-naught gate, c means controlled, not. So this is a 2 by 2 matrix.