qubits, quantum registers and gates

IPQI-2010-Anu Venugopalan 1

qubits, quantum registers and gates

Anu Venugopalan

Guru Gobind Singh Indraprastha UniveristyDelhi

_______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-

2010) Institute of Physics (IOP), Bhubaneswar

January 2010


The Qubit______________________________________

0

2 2| | | | 1Normalization

Physical implementations - Photons, electron, spin, nuclear spin

1

‘Bit’ : fundamental concept of classical computation & info. - 0

or 1

‘Qubit’ : fundamental concept of quantum computation &

info 0 1

- can be thought of mathematical objects having some specific properties


The Qubit- computational basis______________________________________

10

General state of a qubit

and are complex coefficients that can

take any possible values and satisfy the

normalization condition:

122

Computational basis

1

01 ;

0

10

State space : C2

orthonormal basis


The Qubit- measurement in the computational basis

_________________________________

•In general the state of a qubit is a unit vector in a two dimensional complex vector space.

• Unlike a bit you cannot ‘examine’ a qubit to determine its quantum state

10


The Qubit- measurement in the computational basis

_________________________________

-

•A qubit can exist in a continuum of states between and until it is observed

•Measuring on the qubit:

10 measurement

0

1

with prob

with prob2

2

0 1


How much information does a qubit hold?_________________________________

10 Geometric representation in terms of a Bloch sphere:

1

2sin0

2cos

ieA point on a unit 3D sphere

There are an infinite number of points on the unit sphere, so that in principle one could store a large amount of information

But – from a single measurement one obtains only a single bit of information.

In the state of a qubit, Nature conceals a great deal of hidden information


multiple qubits - quantum registers_________________________________

-

More than one qubit…..The state space of a composite physical system is the tensor product of the state spaces of the component systems.

Example: for a two qubits, the state space is C2

C2=C4

computational basis for C2 : 1; 0

computational basis for C4 :

11;01;10; 00

11;10;01; 00alternate representation :


Two qubit register – computational basis_________________________________

11;10;01; 00computational basis :

The state vector for two qubits:212,1

10

10

21

20

2

11

10

1

state of qubit 1

state of qubit 2

11100100 111001002,1


Two qubit register – matrix representation

_________________________________

1

0

0

0

11;

0

1

0

0

10;

0

0

1

0

01;

0

0

0

1

00

computational basis :

21

202

120

2

11

101

110

1

10

10

state of qubit 1

state of qubit 2


Two qubit register – matrix representation

_________________________________

The state vector for two qubits:212,1

11100100 111001002,1

11

10

01

00

21

11

20

11

21

10

20

10


n-qubit register_________________________________

Two-qubit register- state space C2 ; 4 basis states

4 terms in the superposition for the state vector

11100100 111001002,1

C8 :three-qubit register : 23=8 terms in the superposition

C16 :four-qubit register : 24=16 terms in the superposition

Cn :n-qubit register : 2n terms in the superposition


n-qubit register_________________________________

n-qubit register- 2n basis states 2n terms in the superposition for the state vector

n............3,2,1 : a superposition specified by 2n amplitudes

e.g. for n=500 (a quantum register of 500 qubits), the number of terms in the superposition, i.e., 2500 , is larger than the number of atoms in the Universe!

A few hundered atoms can store an enormous amount of data - an exponential amount of classical info. in only a polynomial number of qubits because of the superposition.


Computing – gates – quantum analogs_________________________________

-

Quantum Mechanics as computation

Classical computer circuits consist of logic gates. The logic gates perform manipulations of the information, converting it from one form to another.

Quantum analogs of logic gates are Unitary operators which can be represented as matrices.

Unitary operators (quantum gates ) operate on qubits and quantum registers.


Computing –gates – quantum analogs_________________________________

example: single qubit quantum gates:

The NOT Gate

01ˆ ;10ˆ

ˆ01

10

XX

Xx

0

1

1

0ˆ ; 1

0

0

1ˆ XX

ˆ

X 0110ˆ X




The phase flip Gate

11ˆ ;00ˆ

ˆ10

01

ZZ

Zz

0

1

1

0ˆ ; 1

0

0

1ˆ ZZ

ˆ

X 0010ˆ Z




The Hadamard Gate

11

11

2

1ˆ

H

1

1

2

1

1

0ˆ ; 1

1

2

1

0

1ˆ HH

102

11ˆ

102

10ˆ

H

H

This gate is uniquely quantum-mechanical with no classical counterpart



example: single qubit quantum gates: The Hadamard Gate

11

11

2

1ˆ

H

102

110ˆ H


Computing –gates – quantum analogs_________________________________example: two-qubit quantum gates: The quantum controlled NOT gate

The classical

C-NOT gate

0 1 1

1 0 1

1 1 0

0 0 0

cba

The Quantum C-NOT gate

(reversible)

a

b

a

ba

control qubit

target

qubit

1011 ;1110

0101 ;0000

action


Quantum Gates – the C-NOT gate_________________________________Quantum C-NOT gate is reversible- it corresponds to a Unitary operator,

a

b

a

ba 1011 ˆ ;1110ˆ

0101 ˆ ;0000ˆ

CNCN

CNCN

UU

UU

CNU

matrix representation

0100

1000

0010

0001

ˆCNU


Quantum Gates – the swap circuit_________________________________

Three quantum C-NOT gates

a

b

b

a

abbabbbba

baababaaba

,)(,,

)(,,,

CNU CNU

CNU


A quantum circuit for producing Bell states

_________________________________

2

110000

2

100101

2

110010

2

100111

xyHx

y

These are very useful states


The No-Cloning Theorem_________________________________

Copying/cloning a single classical bit

Use a C- NOT gate

x

0

x

y

x

xy

x

x

Two bits of the same input x

The C-NOT operationCan we have a similar quantum circuit that can clone/copy a qubit?



A quantum C-NOT gate to clone/copy a qubit?

X= 10

0

010

Input state

Action of C-NOT

inputCNU ˆ

1100 output state



Can a quantum C-NOT gate clone/copy a qubit?

X= 10

0

010

Input state

1100 output state

If the circuit had cloned the input state

x as in the case of the classical circuit,

the output state should be

XX



X= 10

0

1000 Input state:

1100 output state:

If the circuit had cloned the input state x as in the

case of the classical circuit, the output state should

be

XX 11100100

101022



1000 Input state:

1100

output state:

Output state expected of a cloning machine

11100100 22 clearly

11100100 22

1100

We have not managed to clone the state



1100 11100100 22

We have not managed to clone the state

0,1or 1,0 if

11or 00RHS = LHS

Cloning happens only if the input state is either or 0 1

- These are like classical bits!



Cloning happens only if the input state is either or 0 1

Only orthogonal states (classical bits) can be cloned

It is impossible to clone an unknown quantum state like an

input state of the qubit = 10

The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state.



The no cloning theorem was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields. The theorem follows from the fact that all quantum operations must be unitary linear transformation on the state

•W.K. Wootters and W.H. Zurek, A Single Quantum Cannot

be Cloned, Nature 299 (1982), pp. 802–803.

•D. Dieks, Communication by EPR devices, Physics Letters

A, vol. 92(6) (1982), pp. 271–272.

•V. Buzek and M. Hillery, Quantum cloning, Physics World

14 (11) (2001), pp. 25–29.


Some consequences of the no-cloning theorem

___________________________________•The no cloning theorem prevents us from using classical error correction techniques on quantum states.

•the no cloning theorem is a vital ingredient in quantum cryptography, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key.

•Fundamentally, the no-cloning theorem protects the uncertainty principle in quantum mechanics

•More fundamentally, the no cloning theorem prevents superluminal communication via quantum entanglement.

qubits, quantum registers and gates

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