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.

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 (-ħ/2) state: Spin-up (+ħ/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

If the basis states for a qubit are written and , then the state vector for the qubit is

where and are complex numbers with

Basis States

Instead of and 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:

Basis states (2)

For example, we could use the basis to describe the state of a qubit:

Now:

orthogonality:

Systems of many qubits

If we know the individual states of the electrons in the system below:

1 2 3

... 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:

This is the “outer product” of vectors Note that this is different from the inner

product

Systems of many qubits (3)

We have

By convention, we write as

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

you’re 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

... you will never be able to obtain the values

of and . A measurement has to be made with respect

to a particular basis.

Quantum Measurement (3)

If you measure with respect to the basis:– if the answer will be with probability 100%– if the answer will be with probability 100%– in all other cases (e.g. ), the result will be

probabilistic.

After measurement, the value of will change permanently to the result obtained.

Quantum Measurement (4)

If you measure with respect to a different basis, things are worse!

Measuring with respect to will give one of the results and with particular probabilities.

Also, the value of will change permanently to the result obtained.

Quantum Measurement, Formally

Formally, when you measure

with respect to you will get:– result with probability |

– result with probability | 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:

the first measurement will reduce to one of these smaller states

prob.2

prob.2

Measuring many qubits (2)

The second measurement will reduce to one of the four states

22

2

||||

||

22

2

||||

||

22

2

||||

||

22

2

||||

||

Measuring many qubits (3)

By multiplying the branches in the overall tree, we can obtain the probability of each result. So for the state

two consecutive measurements will give– result with probability |

– result with probability |

– result with probability |

– result with probability |

Entanglement

There exist states of many-qubit systems that cannot be broken down into a tensor product

E.g.: there do not exist for which

These are termed entangled states.

The Bell states

For a two-qubit system, the four possible entangled states are named Bell states:

100 11

21

00 112

101 10

21

01 102

Measuring Entangled States

After measuring an entangled pair for the first time, the outcome of the second measurement is known 100%

0.5

0.5

1

1 1

00 112

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, σ

x, σ

y, σ

z)

– the Controlled Not (CNot)

The Hadamard gate

The Hadamard gate acts on one qubit, and places it in a superposition of and :

102

21

102

20

H

H

The Pauli gates

The Pauli gates act on one qubit, as follows:

– phase shift, σz:

σz

– bit flip, σx:

σx

– phase shift and bit flip, σy:

σy

– identity, I, does not change the input

The Controlled Not Gate

The CNot gate acts on two qubits:

CNot( ) =

CNot( ) =

CNot( ) =

CNot( ) =

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 σ

x

Remember: σx

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)

PersonAction

performedState

Bob H

Alice σx

Bob H

10 1

2

11 0

2

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

23 2 3 2 3

10 0 1 1

2

1 1 10 1

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.

1 2 1CNot , ; H

Quantum Teleportation

Depending on the outcome of Alice’s measurement, Bob applies a Pauli operator to particle 3, “reincarnating” the original qubit

If outcome=00, Bob uses operator I If outcome=01, Bob uses operator σ

x

If outcome=11, Bob uses operator σy

If outcome=10, Bob uses operator σz

Bob’s 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!