Quantum Computation and Quantum Information – Lecture 3

Part 1 of CS406 – Research Directions in Computing

Nick Papanikolaou

Motivation

Quantum computers are built from wires and logic gates, just as classical computers are

The potential of such devices stems from the ability to manipulate superpositions of states

Quantum algorithms solve problems which are not known to be solvable classically!

Lecture 3 Topics

Quantum logic gates Simple quantum circuits Quantum teleportation as a circuit Deutsch’s quantum algorithm

Quantum vs. classical gates

The simplest boolean gate is NOT, with truth table:

Quantum gates have to be defined not only on the equivalents of 0 and 1, but on their superpositions too!

in out

0 1

1 0

Quantum NOT gate: Linearity

Suppose we define a quantum NOT gate as follows:

The action of the quantum NOT gate on a superposition must then be:

All quantum operations are linear

01NOT ,10NOT

01

1NOT0NOT10NOT

The NOT Gate as a Matrix

Because all quantum operations have to be linear, we can represent the action of a quantum gate by a matrix

The quantum NOT, or Pauli-X gate, is written:

01

10X

Quantum State Vectors

Remember that a quantum state is represented by a vector

Notation:

1

01 ,

0

10

10

Quantum NOT

We can express the NOT operation on a general qubit as matrix multiplication:

01

1010X

Other Single Qubit Gates

The Pauli-X gate works on only one qubit Other common single qubit gates are:

– Pauli-Z gate:

– Pauli-Y gate:

– Hadamard gate:

10

01Z

XZY

11

11

2

1NOTH

Z

Y

H

Summary of Simple Gates

X

Z

H

10

10

10

01

10

2

10

2

10

Y10 01 i

Reversibility Requirement

All quantum operations have to be reversible

Boolean operations are not necessarily so A reversible operation is always given by a

unitary matrix, i.e. one for which:

1)*( UUT

The Controlled NOT Gate

The CNOT gate is the standard two-qubit quantum gate

It is defined like this:

1011

1110

0101

0000

CNOT

CNOT

CNOT

CNOT

The Controlled NOT Gate (2)

CNOT is a generalisation of the classical XOR:

The CNOT gate is drawn like this:

ABABA , ,CNOT

“control qubit”

“target qubit”

A

B

A

AB

The Controlled NOT Gate (3)

The matrix corresponding to the CNOT gate is:

The CNOT together with the single qubit gates are universal for quantum computing

0100

1000

0010

0001

CNOT

Quantum Circuits

Using the conventions for control and target qubits, we can build interesting circuits

Example: A Qubit Swap Circuit

A

B A

B

Qubit Swap Circuit

A

B A

BA

BA BA

BBAA )(

ABAB )(

Features of Quantum Circuits

1. No loops are allowed; quantum circuits are acyclic

2. Fan-in is not allowed:

3. Fan-out is not allowed:

Generalised Control Gate

Any quantum gate U can be converted into a controlled gate:

U

One control qubit

n target qubits

If the control qubit is “high,” U is applied to the targets. CNOT is the Controlled-X gate!

Quantum Measurement

Measurement in a quantum circuit is drawn as:

M(classical bit representing outcome of measurement)

10 If then:M = 0 with prob. orM = 1 with prob.

2

2

A Qubit Cloning Circuit?

Using the XOR gate, it is possible to copy a classical bit:

x x

y xy

x

0

x

x

Can we build a quantum circuit that performs does this with qubits?

?

0

A Qubit Cloning Circuit? (2)

0

0

0

0

0

1 1

1

0

10 1100 entangled!!

OK here

A Qubit Cloning Circuit? (3)

11100100

101022

It is impossible to clone a qubit!

Also note that

unwanted terms!

The Bell State Circuit

Hx

y

x y Output

0 0

0

0

1

1

1 1

11002

1

11002

1

10012

1

10012

1

The Bell State Circuit By Example

H0

0

102

1

?

11002

1

10002

1

2

10000

2

10

CNOTCNOT

CNOTCNOT

Quantum Teleportation Circuit

H

XM2 ZM1

11002

1100

M1

M2

Quantum Teleportation Circuit (2)

H

XM2 ZM1

11001110002

11

M1

M2

Quantum Teleportation Circuit (3)

H

XM2 ZM1

01111010

01011000

2

12

M1

M2

Quantum Teleportation Circuit (4)

H

XM2 ZM1

00, 01, 10 or 11

M1

M2

Quantum Teleportation Circuit (5)

If Alice obtains

Then Bob’s qubit is in state

So Bob applies gate

obtaining

00 I

01 X

10 Z

11 Y = ZX

10

01

10

01

10

What have we achieved?

The teleportation process makes it possible to “reproduce” a qubit in a different location

But the original qubit is destroyed!

Next topic: Quantum Parallelism and Deutsch’s quantum algorithm

Quantum Parallelism

Quantum parallelism is that feature of quantum computers which makes it possible to evaluate a function f(x) on many different values of x simultaneously

We will look at an example of quantum parallelism now – how to compute f(0) and f(1) for some function f all in one go!

Quantum Circuits for Boolean Functions

It is known that, for any boolean function

it is possible to construct a quantum circuit Uf

that computes it Specifically, to each binary function f

corresponds a quantum circuit:

1,01,0: f

)(,,: xfyxyxU f

binary addition

Quantum Circuits for Boolean Functions (2)

What can this circuit Uf do? Example:

x x

y yf(x)

0

1

)0(1 ,0

01

10

f

U

U

f

f

fU

Quantum Circuits for Boolean Functions (3)

But what if the input is a superposition?

x x

y yf(x)1fU

102

1

2

)1(,1)0(,0

2

)1(0 ,1)0(0 ,0

2

1000

02

10

ff

ff

U

U

f

f

amazing! we’ve computed f(0) and f(1) at the same time!

Quantum Parallelism Summary

So, a superposition of inputs will give a superposition of outputs!

We can perform many computations simultaneously

This is what makes famous quantum algorithms, such as Shor’s algorithm for factoring, or Grover’s algorithm for searching

Simple q. algorithm: Deutsch’s algorithm

Deutsch’s Algorithm

David Deutsch: famous British physicist Deutsch’s algorithm allows us to compute,

in only one step, the value of

To do this classically, you would have to:1. compute f(0)2. compute f(1)3. add the two results

– Remember:

)1()0( ff

1,01,0: f

Circuit for Deutsch’s Algorithm

x x

y yf(x)

H

H

H0

1

010 2

10

2

101

fU

Circuit for Deutsch’s Algorithm (2)

x x

y yf(x)

H

H

H0

1

2

10

2

10 ),1()0( if

2

10

2

10 ),1()0( if

2

ff

ff

fU

Circuit for Deutsch’s Algorithm (3)

x x

y yf(x)

H

H

H0

1

2

101 ),1()0( if

2

100 ),1()0( if

3

ff

ff

2

10)1()0(

tosimplifies this

3

ff

...and so we have computed

)1()0( ff

fU

End of Lecture 3

Congratulations! If you are still awake, you have learned something about:– quantum gates (X, Y, Z, H, CNOT)– quantum circuits (swapping, no-cloning problem)– teleportation– quantum parallelism– and Deutsch’s algorithm