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Quantum Computation and Quantum Information

Lecture 2

- Part 1 of CS406 Research Directions in Computing

Dr. Rajagopal Nagarajan Assistant Nick

Papanikolaou

Lecture 2 Topics

- Physical systems on the atomic scale
- State vectors and basis states Qubits
- Systems of many qubits
- Quantum Measurement
- Entanglement
- Quantum gates
- Quantum coin-flipping and teleportation

Quantum physics and Nature

- There exists a vast array of minute objects on

the atomic scale electrons, protons, neutrons,

photons, quarks, neutrinos, - Quantum mechanics is a system of laws that

describes the behaviour of such objects - With computer chips getting smaller and smaller,

by 2020 we will store 1 bit of data on objects of

that size!

Quantum physics and Nature (2)

- Atom-sized objects behave in unusual ways their

state is generally unknown at any given time,

and changes if you try to observe it! - Several properties of these systems can be

manipulated and measured.

(No Transcript)

Qubits

- A qubit is any quantum system with exactly two

degrees of freedom we use them to represent

binary 0 and 1 - Hydrogen atom
- Spin-1/2 electron

Ground state

Excited state

Spin-down (-h/2) state

Spin-up (h/2) state

Qubits (2)

- In general, the state of a qubit is a

combination, or superposition, of two basis

states - The rest state and the excited state are the

basis states of the hydrogen atom - The spin-up and spin-down states are basis states

for the spin-1/2 particle

The State Vector

- The state of a quantum system is described by a

state vector, written yñ - If the basis states for a qubit are written 0ñ

and 1ñ, then the state vector for the qubit is - yñ a 0ñ b 1ñ
- where a and b are complex numbers with
- a2 b2 1

Basis States

- Instead of 0ñ and 1ñ we can use any other basis

states, as long as we can distinguish clearly

between the two. - Mathematically, basis states must be given by

orthogonal vectors.

The inner product of the two vectors must be

0 á0 1ñ 0

Basis states (2)

- For example, we could use the basis ñ, -ñ

to describe the state of a qubit

1ñ

Now yñ g ñ d -ñ orthogonality á

-ñ 0

-ñ

ñ

0ñ

Systems of many qubits

- If we know the individual states of the electrons

in the system below

y1ñ 0 0ñ 1 1ñ 1ñ y2ñ 1 0ñ 0 1ñ

0ñ y3ñ 0 0ñ 1 1ñ 1ñ

- ... then what is the overall state of the

three-particle system?

Systems of many qubits (2)

- The state of a composite quantum system, when all

the component states are known, is their tensor

product - yñ y1ñ Ä y2ñ Ä y3ñ
- This is the outer product of vectors
- Note that this is different from the inner

product áf½cñ

Systems of many qubits (3)

- We have
- yñ y1ñ Ä y2ñ Ä y3ñ
- (0 0ñ 1 1ñ) Ä (1 0ñ 0 1ñ) Ä (0 0ñ

1 1ñ) - 1ñ Ä 0ñ Ä 1ñ
- By convention, we write 1ñ Ä 0ñ Ä 1ñ as

101ñ

Quantum Measurement

- To extract any information out of a quantum

system, you have to perform a physical

measurement - By measuring a quantum system
- you automatically change its state, the very

state youre trying to measure - you obtain, in general, a random result, which

may be different from the original state

Quantum Measurement (2)

- When you try to measure a qubit
- yñ a 0ñ b 1ñ
- ... you will never be able to obtain the values

of a and b. - A measurement has to be made with respect to a

particular basis.

Quantum Measurement (3)

- If you measure with respect to the 0ñ, 1ñ

basis - if yñ 0ñ the answer will be 0ñ with

probability 100 - if yñ 1ñ the answer will be 1ñ with

probability 100 - in all other cases (e.g. ab0.5), the result

will be probabilistic. - After measurement, the value of yñ will change

permanently to the result obtained.

Quantum Measurement (4)

- If you measure with respect to a different basis,

things are worse! - Measuring yñ a 0ñ b 1ñ with respect to

ñ, -ñ will give one of the results ñ and

-ñ with particular probabilities. - Also, the value of yñ will change permanently to

the result obtained.

Quantum Measurement, Formally

- Formally, when you measure
- yñ a 0ñ b 1ñ
- with respect to 0ñ, 1ñ you will get
- result 0ñ with probability a2
- result 1ñ with probability b2
- If you use a different measurement basis, the

result will be one of the basis states, with

different probabilities

Measuring many qubits

- We want to know the possible outcomes of

measuring the two qubit state - yñ (a 0ñ b 1ñ) Ä (g 0ñ d 1ñ)
- ag 00ñ ad 01ñ bg 10ñ bd 11ñ

prob. ag2 ad2

prob. bg2 bd2

the first measurement will reduce yñ to one of

these smaller states

Measuring many qubits (2)

- The second measurement will reduce yñ to one of

the four states 00ñ, 01ñ, 10ñ, 11ñ.

ag 00ñ ad 01ñ

bg 10ñ bd 11ñ

00ñ

01ñ

10ñ

11ñ

Measuring many qubits (3)

- By multiplying the branches in the overall tree,

we can obtain the probability of each result. So

for the state - yñ ag 00ñ ad 01ñ bg 10ñ bd 11ñ
- two consecutive measurements will give
- result 00ñ with probability ag2
- result 01ñ with probability ad2
- result 10ñ with probability bg2
- result 11ñ with probability bd2

Entanglement

- There exist states of many-qubit systems that

cannot be broken down into a tensor product - E.g. there do not exist a, b, g, d for which
- m 00ñ n 11ñ (a 0ñ b 1ñ) Ä (g 0ñ d

1ñ) - These are termed entangled states.

The Bell states

- For a two-qubit system, the four possible

entangled states are named Bell states

Measuring Entangled States

- After measuring an entangled pair for the first

time, the outcome of the second measurement is

known 100

1

0ñ

0ñ

0.5

1

1ñ

1ñ

0.5

Review

- Thus far we have seen
- how qubits are represented
- how many qubits can be combined together
- what happens when you measure one or more qubits
- where entangled pairs come from, and what happens

when you measure them - Now we will take a look at quantum gates

Quantum gates

- As in classical computing, a gate is an operation

on a unit of data, here a qubit - A quantum gate is represented by a matrix that

may be applied to a state vector - We will talk about this in more detail next time

for now we will look at some examples of commonly

used quantum gates - the Hadamard gate (H)
- the Pauli gates (I, sx, sy, sz)
- the Controlled Not (CNot)

The Hadamard gate

- The Hadamard gate acts on one qubit, and places

it in a superposition of 0ñ and 1ñ

The Pauli gates

- The Pauli gates act on one qubit, as follows
- phase shift, sz
- sz(a 0ñ b 1ñ) a 0ñ - b 1ñ
- bit flip, sx
- sx(a 0ñ b 1ñ) a 1ñ b 0ñ
- phase shift and bit flip, sy
- sy(a 0ñ b 1ñ) a 1ñ - b 0ñ
- identity, I, does not change the input

The Controlled Not Gate

- The CNot gate acts on two qubits
- CNot( 00ñ ) 00ñ
- CNot( 01ñ ) 01ñ
- CNot( 10ñ ) 11ñ
- CNot( 11ñ ) 10ñ

Quantum Coin Flipping

- Quantum coin flipping is based on the following

game - Alice places a coin, head upwards in a box.
- Alice and Bob then take turns to optionally turn

the coin over (without looking at it). - At the end of the game, the box is opened and and

Bob wins if the coin is head upwards. - In the quantum version of the game, the coin is a

quantum state

Quantum Coin Flipping (2)

- Assume that Alice can only perform a flipping

operation, i.e. gate sx - Remember sx(a 0ñ b 1ñ) a 1ñ b 0ñ
- There is a strategy that allows Bob to win

always he must perform Hadamard operations. - Thus Bob places the state of the coin in a

superposition of heads and tails!

Quantum Coin Flipping (3)

Person Action performed State

0ñ

Bob H

Alice sx

Bob H 0ñ

The No-cloning principle

- It has been proved by Wootters and Zurek that it

is impossible to clone, or duplicate, an unknown

quantum state. - However, it is possible to recreate a quantum

state in a different physical location through

the process of quantum teleportation.

Quantum Teleportation The Basics

- If Alice and Bob each have a single particle from

an entangled pair, then - It is possible for Alice to teleport a qubit to

Bob, using only a classical channel - The state of the original qubit will be destroyed
- How?
- Using the properties of entangled particles

Quantum Teleportation

- Alice wants to teleport particle 1 to Bob
- Two particles, 2 and 3, are prepared in an

entangled state - Particle 2 is given to Alice, particle 3 is given

to Bob

Quantum Teleportation (2)

- In order to teleport particle 1, Alice now

entangles it with her particle using the CNot and

Hadamard gates - Thus, particle 1 is disassembled and combined

with the entangled pair - Alice measures particles 1 and 2, producing a

classical outcome 00, 01, 10 or 11.

Quantum Teleportation

- Depending on the outcome of Alices measurement,

Bob applies a Pauli operator to particle 3,

reincarnating the original qubit - If outcome00, Bob uses operator I
- If outcome01, Bob uses operator sx
- If outcome11, Bob uses operator sy
- If outcome10, Bob uses operator sz
- Bobs measurement produces the original state of

particle 1.

Quantum Teleportation (Summary)

- The basic idea is that Alice and Bob can perform

a sequence of operations on their qubits to

move the quantum state of a particle from one

location to another - The actual operations are more involved than we

have presented here see the standard texts on

quantum computing for details - Recommended S. Lomonaco, A Rosetta Stone for

Quantum Computation see www

Review

- Quantum gates allow us to manipulate quantum

states without measuring them - Quantum states cannot be cloned
- Teleportation allows a quantum state to be

recreated by exchanging only 2 bits of classical

information - Quantum coin flipping is more fun than classical

coin flipping!