Introduction to quantum computing and quantum information - PowerPoint Presentation

Slide1

Introduction to quantum computing and quantum information

Dung Nguyen

Chicago 19

th

JanuarySlide2

Content

Motivation

Quantum bit (qubit) vs Classical bit (bit)

Quantum Computation

Quantum Communication

ConclusionSlide3

Motivation

The end of Moore’s law scaling in silicon (because of quantum effects of particle at scale smaller than 7nm).

There are only three more generations of Intel processor based on current silicon technology.

There could be new improvements for silicon technology (3D stacking), but the issues of cooling and energy consuming are hard to pass

The new carbon nanotubes computer is questionable (easy to have defects that can ruin an entire chip)Slide4

Motivation

The classical computer can not simulate

efficiently

quantum systems (an unproven conjecture).

Quantum computer can simulate quantum systems, so there is a possible solution for strong correlated

correlated problems (high Tc superconductor) using quantum computation. Quantum computer can solve the hard problems of classical computing using computational power inherent in quantum mechanics. Quantum algorithms can solve some of the NP problems (but not NP-complete) like searching.

Slide5

Motivation

The security of quantum communication (we will talk about it)

Quantum computation can break the classical cryptography (finding prime factors) using reasonable time and energy consumption.Slide6

Quantum bit (Qbit) vs Classical bit (bit)

Classical bit and classical computer

Classical bit take value 0 or 1

We can consider logic gates as the function

Example of

 Slide7

Quantum bit (Qbit) vs Classical bit (bit)

Classical bit

Example of

 Slide8

Quantum bit (Qbit) vs Classical bit (bit)

Universal gates

Any

can be built up using NAND gates and Fan-out.

Example:

 Slide9

Quantum bit (Qubit) vs Classical bit (bit)

Stronger Universality statement (Strong Church-Turing thesis)

Any algorithmic process (in Turing’s definition)can be simulated efficiently using a probabilistic Turing machine (loosely speaking a computer which is built up by logic gates and can generate random numbers)

Every Turing machine has equal power (

polynomially

).

The idea behind is a Turing machine can simulate an other Turing machine within a polynomially bounded overhead in

time. Slide10

Quantum bit (Qubit) vs Classical bit (bit)

Quantum logic bit (qubit)

A single qubit can be described by a vector two dimensional complex Hilbert space

When we

mesuare

a qubits, we either get the result 0, with probability

,or the result 1, with the probability

. The normalization condition

, we can rewrite the

qbit

as

We can represent it by the Bloch sphere.

The single qubit can be represented by 2 real numbers

and

.

 Slide11

Quantum bit (Qubit) vs Classical bit (bit)

Because of quantum

decoherence

,

the quantum logic bit usually is not a single physical qubit.

The single physical qubit can be constructed by single atom, by photon polarizations, by spin direction of single electron, superconducting circuit (based on Josephson effect)

The orthonormal basis can be chosen arbitrary. We can chose the basis

and

 Slide12

Quantum bit (Qubit) vs Classical bit (bit)

How much information we can store in a single qubit if we measure it? There are an infinite number of points on the Bloch sphere, could we store infinite binary expansion of

or

?

NO! Because of behavior of qubit when observed, when we measure a qubit, it gives the value either 0 or 1. From a single measurement we only can have a single bit information.

If you want to have all information of

and

, you need to measure

infinite

identically qubits, which is impossible in general due to “no-cloning” theorem (we will talk about it latter).

 Slide13

Quantum bit (Qubit) vs Classical bit (bit)

How much information can be stored in a qubit, if we don’t measure it? There is something conceptually important here, we can do the computation without doing measurement in the middle step, we do the measurement at the end.

A qubit contain hidden quantum information, and this hidden information grows exponentially with the number of qubit. Slide14

Quantum bit (Qubit) vs Classical bit (bit)

Multiple qubits

If we have 2 qubits, we have four computational basis states

The state vector that describing the 2 qubit system is

For a 2 qubit system, we can measure subset of the qubits, if we measure the first qubit, and get the value 0 with probability

, and the post measurement state (collapsed state)

 Slide15

Quantum bit (Qubit) vs Classical bit (bit)

The 2 qubit system can be described with 4 complex number, however if we consider normalization and ignore arbitrary phase factor, we only need 6 real numbers.

More generally, we may consider a system of n qubits, we need a vector in

to represent a state (but in practice, we use

complex numbers ). For

n=300

, the dimension of our vector space is bigger than total number of atoms in the visible Universe. The hidden information is huge.

Could we use this hidden information to do computation (quantum computation)?

YES!

 Slide16

Quantum computation

Single qubit gate

is any

unitary transformation act on the vector space that represent a qubit, it can be considered as an evolution matrix of a time dependent Hamiltonian

We do have access to hidden information, not just

and

states.

Examples:

Quantum NOT gate

Phase gate

 Slide17

Quantum computation

Z gate

Hadamard

gate

Transformation of a qubit state under action of qubit logic gates

There are infinite many 2 by 2 unitary matrices, do we need infinitely many single qubit logic gate? NO!Slide18

Quantum computation

Universal single qubit gates

We know that any unitary transformation in 2 dimension can be decomposed in to Euler rotations

So with any unitary transformation, we can built it up from 4 gates? Not really, since

are real numbers, we still need infinite number of gate to built up exactly any unitary transformation.

However with in an approximation

|U|

w

here

is the transformation built from quantum gates.

We need only some certain special fixed value of

, and the price is we need more than 4 gates to built up

.

 Slide19

Quantum computation

Example for an arbitrary real number

, and a fixed irrational number

, we always can find a integer number m such that

Thus and

gate can be built up from m

gates.

This is just an example to show that we can have a set of universal single qubit gates. In practice, we may use different set of gates as the universal single qubit gates.

 Slide20

Quantum computation

Multiple qubit quantum gate

We have a special 2 qubit quantum gate is controlled-NOT or CNOT gate act on the basis of 2 qubit state

.

Or in matrix form

Where

is addition modulo 2.

 Slide21

Quantum bit (Qubit) vs Classical bit (bit)

Universal multiple qubits gates

We can prove that any multiple qubits transformation can be constructed from CNOT gates and single qubit gates.

For an approximation

, any multiple qubits gate can be built from CNOT gate,

Hadamard

gate and

gate.

Thus we have

CNOT gate,

Hadamard

gate and

gate are the set of universal gates for multiple qubits.

We have quantum Church-Turing thesis: Every quantum Turing machine has equal power (polynomially).

 Slide22

Quantum Computation

No-cloning theorem

We can not do quantum copy on a qubit

Proof:

If we can find an unitary operator C (clone operator) such that with any normalized qubit state

, we have

Thus we have

=

This can not be true for general state.

There is no quantum Fan-out.

 Slide23

Quantum Computation

Control quantum gate:

Consider U is any unitary matrix acting on

qubits. Define a controlled-U gate as a extension of controlled-NOT gate with an extra control qubit. If the control qubit is set to 0, then nothing happens to the

target qubits. If the control qubit is set to 1 then the gate U is applied to the target qubits

 Slide24

Quantum Computation

Can quantum computer do the classical computation?

YES!

We require the input and output are both classical signals (either

or

)

We can construct the classical gates using the quantum circuit (indeed we make the reversible version of the classical gate) named

Toffoli

gate

 Slide25

Quantum Computation

We just need to show that we can construct reversible version of NAND and Fan-out

Generate random number is the property that is inherited from quantum mechanics. Slide26

Quantum computation

Can quantum computer do better?

YES!

One of the most important example is quantum parallelism

We consider a function on classical bit

We can construct quantum gate

gate with the specific input

(We can easily show that

is unitary by writing down the matrix form in the canonical basis)

 Slide27

Quantum Computation

We see that the output state is

The different terms contain information about both

and

.

Generally, if we want to calculate the function of n bit

, we can make a quantum circuit including n+1 input and n+1 output qubits, such that the output state has the information of

with

take

value simultaneously

But what information we can get from that final state?

 Slide28

Quantum Computation

Let investigate it more

Using this quantum circuit, at the end we found that the final state

When

mesure

the first qubits, we can determine

, quantum circuit gives us a

global property

of function

with only one evaluation. The power is exponentially faster (in the number of bit) in comparison to classical computer.

 Slide29

Quantum Computation

Exploit the quantum parallelism, we can solve some of NP problems, we can finish the computation

polynomially

in time rather than exponentially in time.

Some of the most famous problem are quantum searching and prime factoring (Shor’s algorithm). The prime factoring is the key of classical cryptography

If we could do quantum computing now, the online trading is dead. Slide30

Quantum Communication

Using qubit, we can store and transfer information with save.

It doesn’t keep people from stealing our information. The key point is when people steal the information, we knew it.

Suppose I prepare a qubit in a state

or

, and I tell my friend that I store my qubit in a box and go out for lunch. I come back latter, and I know if he could steal my information or not.

To distinguish the qubit in the state

or

secretly, he need to do unitary transformation that preserve my qubit.

He will get his qubit state is

or

But U is unitary so

, thus if

,

=1

He can not steal my information without letting me know it.

 Slide31

Quantum Communication

We can use quantum entanglement to send the decode key for classical cryptography.

If Alice want to share with Bob a secret key that they can use latter to decode the information. However they worry about Eva, who can steal their key.

Alice prepare a

entanglement

state

And send one of qubit to Bob

Then Alice and Bob do the measurement on their qubit in Z and X, if they have 100% correlation, they know that they share an entanglement pair of qubit.

 Slide32

Quantum Communication

Since Eva can interfere with the system, the state of the three is

But the measurement in Z direction are correlated, thus the state should be

Furthermore, the measurement in X direction are correlated, thus the state should be

So they know that Eva’s state is uncorrelated with their entanglement pairs. They can use their shared entanglement pairs to generate random key latter. Slide33

Conclusion

Quantum computation is promising.

The idea of quantum gate as the driven of time dependent Hamiltonian gives us a hope that we can simulate a quantum system using quantum computer.

Quantum communication is really a big deal.

Slide34

Appendix

Creat

Bell states using

Hadamard

gate