quantum mechanics for quantum information & computation

IPQI-2010-Anu Venugopalan 1

Quantum Mechanics for Quantum Information & Computation

Anu Venugopalan

Guru Gobind Singh Indraprastha UniveristyDelhi

_______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-

2010) Institute of Physics (IOP), Bhubaneswar

January 2010


Computer technology in the last fifty years- dramatic miniaturization

Faster and smaller –

- the memory capacity of a chip approximately doubles every 18 months – clock speeds and transistor density are rising exponentially...what is their ultimate fate????

Real computers are physical systems


Moore’s law [www.intel.com]


The future of computer technology

If Moore’s law is extrapolated, by the year 2020 the basic memory component of the chip would be of the size of an atom – what will be space, time and energy considerations at these scales (heat dissipation…)?

At such scales, the laws of quantum physics would come into play - the laws of quantum physics are very different from the laws of classical physics - everything would change!

[“There’s plenty of room at the bottom”Richard P. Feynman (1969)

Feynman explored the idea of data bits the size of a single atom, and discussed the possibility of building devices an atom or a molecule at a time (bottom-up approach) - nanotechnology]


Quantum Mechanics_______________________________

• At the turn of the last century, there were several experimental observations which could not be explained by the established laws of classical physics and called for a radically different way of thinking

• This led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature


Some key events/observations that led to the development of quantum mechanics…

___________________________________• Black body radiation spectrum (Planck, 1901)

• Photoelectric effect (Einstein, 1905)

• Model of the atom (Rutherford, 1911)

• Quantum Theory of Spectra (Bohr, 1913)

• Scattering of photons off electrons (Compton, 1922)

• Exclusion Principle (Pauli, 1922)

• Matter Waves (de Broglie 1925)

• Experimental test of matter waves (Davisson and Germer, 1927)


Quantum Mechanics___________________________________

Matter and radiation have a dual nature – of both wave and particle

The matter wave associated with a particle has a de Broglie wavelength given by

The wave corresponding to a quantum system is described by a wave function or state vector

p

h


Quantum Mechanics___________________________________

Quantum Mechanics is the most accurate and

complete description of the physical world

– It also forms a basis for the understanding of

quantum information

IPQI-2010-Anu Venugopalan

Quantum Mechanics

_______________________________________________________

Quantum Mechanics – most successful working theory of Nature……..

The price to be paid for this powerful tool is that some of the

predictions that Quantum Mechanics makes are highly counterintuitive and compel us to reshape our classical (‘common sense’) notions.........

Schrödinger Equation

Linear superposition principle

|| Hdt

di

Linear

Deterministic

Unitary evolution


Some conceptual problems in QM: quantum measurement, entanglements, nonlocality___________________________________

Quantum Measurement

Basic postulates of quantum measurement

Measurement on yields eigenvalue with probability

Measurement culminates in a collapse or reduction of to one of the eigenstates,

‘non unitary’ process….

|| Hdt

di iiaA |,:ˆ ii

c ||

| ia 2|| ic

|}{| i


Some conceptual problems in QM: quantum measurement, entanglements, nonlocality_________________________________________

Macroscopic Superpositions

linear superposition principle

Schrödinger's Cat

Such states are almost never seen for classical (‘macro’) objects in our familiar physical world….but the ‘macro’ is finally made up of the ‘micro’…so, where is the boundary??

|| Hdt

di

deadalivecatatom |||||


Conceptual problems of QM: quantum measurement, entanglements, nonlocality___________________________________

Quantum entanglements – a uniquely quantum mechanical phenomenon associated with composite systems

A B

BAAB cc ||||||| 222111

}|||{|2

1| BABAEPR


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


Quantum Mechanics & Linear Algebra___________________________________

Linear Algebra: The study of vector spaces and of linear operations on those vector spaces.

Basic objects of Linear algebra Vector spaces

C nThe space of ‘n-tuples’ of complex numbers, (z1, z2, z3,………zn)

Elements of vector spaces vectors


Quantum mechanics & Linear Algebra___________________________________

nz

zz

.

.2

1

Vector : column matrix

The standard quantum mechanical representation for a vector in a vector space :

: ‘Ket’ Dirac notation

The state of a closed quantum system is described by such a ‘state vector’ described on a ‘state space’

V


Quantum mechanics & Linear Algebra_____________________________________________

Associated to any quantum system is a complex vector space known as state space.

A qubit, has a two-dimensional state space C2.

0 1

The state of a closed quantum system is a unit vector in state space.

Most physical systems

often have finite

dimensional state spaces

0 1 2 1

0

1

2

1

0 1 2 ... 1

:

d

d

d

‘Qudit’

Cd


Linear Algebra & vector spaces___________________________________

vnvvvv .......,, 331

i

ii vav

•Vector space V, closed under scalar multiplication &

addition

•Spanning set: A set of vectors in V :

such that any vector in the space V can be

expressed as a linear combination:

Example: For a Qubit: Vector Space C2



0

1;

1

021 vv

1v

22112

1 vavavaa

av

iii

Example: For a Qubit: Vector Space C2

and span the Vector space C21v



1

1;

1

121 ww

1w

221

121

2

1

22w

aaw

aa

a

av

A particular vector space could have many spanning sets.

Example: For C2

and also span the Vector space

C2

2w



nvvvv .......,, 331

0 with ......, 21 in aaaa

A set of non zero vectors, are

linearly dependent if there exists a set of complex

numbers

for at least one value of i such

that 0....................2211 nn vavava

A set of nonzero vectors is linearly independent if

they are not linearly dependent in the above sense



• Any two sets of linearly independent vectors that

span a vector space V have the same number of

elements

•A linearly independent spanning set is called a

basis set

•The number of elements in the basis set is equal to

the dimension of the vector space V

•For a qubit, V : C 2 ;

0 and 1 are the

computational basis states


Linear operators & Matrices________________________________

1

01;

0

10 Computational Basis for a Qubit

A linear operator between vector spaces V and W is defines as any function Â

Â : V W, which is linear in its inputs

Î: Identity

operator

Ô: Zero

Operator

Once the action of a linear operator Â on a basis is specified, the action of Â is completely determined on all inputs

ii

iii

i vAavaA ˆˆ


Linear operators & Matrices__________________________________

0

0

i

iy

01

10x

10

010 I

Linear operators and Matrix representations are equivalent Examples: Four extremely useful matrices that operate on elements in C 2

10

01z The Pauli Matrices


Linear operators and matrices - some properties____________________________________

*,, wvwv

wvwv ,

0, vv

Inner product - A vector space equipped with an inner product is called an inner product space- e.g. “Hilbert Space”

runit vecto afor 1 vvvNorm:



runit vecto afor 1 vvv

v

vNorm:

Normalized form for any non-zero vector:

A set of vectors with index i is orthonormal if each vector is a unit vector and distinct vectors are orthogonal

ijji The Gram-Schmidt orthonormalization procedure



wvvvvwvvw

vw

'''

:

:

:

w

v

vwOuter Product

vector in inner product space V

vector in inner product space

W

A linear operator from V to W Iii

i

completenessrelation



iiAi

i ˆ

0ˆˆdet)( IAc

vvvA ˆ

Eigenvalues and

eigenvectors

Diagonal Representation

i An orthonormal set of eigenvectors for Â with corresponding eigenvalues i

examplediagonal representation for z

110010

01

z


The Postulates of Quantum Mechanics____________________________________

Quantum mechanics is a mathematical framework for the development of physical theories. The postulates of quantum mechanics connect the physical world to the mathematical formalism

Postulate 1: Associated with any isolated physical system is a complex vector space with inner product, known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space

A qubit, has a two-dimensional state space: C2.


The Postulates of Quantum Mechanics____________________________________

U'

Evolution - How does the state, , of a quantum system change with time? Postulate 2: The evolution of a closed quantum system is described by a Unitary transformation

A matrix/operator U is said to be Unitary if

IUU

Unitary operators preserve normalization /inner products


a bA

c d

Hermitian conjugation; taking the adjoint

† * TA A

* *

* *

a c

b d

A is said to be unitary if † †AA A A I We usually write unitary matrices as U.

†

Example:

0 1 0 1 1 0XX

1 0 1 0 0 1I

The Postulates of Quantum Mechanics - Unitary operators/Matrices_________________________________________


Linear operators & Matrices –operations on a Qubit (examples)

___________________________________

01ˆ ;10ˆ ;ˆ01

10

XXXx

11Z ;00Z ;ˆ10

01

Zz

The Pauli Matrices- Unitary operators on qubits - Gates

01ˆ ;10ˆ ;ˆ0

0iYiYY

i

iy

NOT Gate

Phase flip

Gate


Unitary operators & Matrices- examples___________________________________

Unitary operators acting on qubits

102

11ˆ ;10

2

10ˆ

11

11

2

1ˆ

HH

H

The Quantum Hadamard Gate


The Postulates of Quantum Mechanics

____________________________________Quantum Measurement

• The outcome of the measurement cannot be determined with certainty but only probabilistically

• Soon after the measurement, the state of the system changes (collapses) to an eigenstate of the operator corresponding to measured observable


The Postulates of Quantum Mechanics

____________________________________Quantum Measurement

Postulate 3:. Unlike classical systems, when we

measure a quantum system, our action ends up

disturbing the system and changing its state. The act

of quantum measurements are described by a

collection of measurement operators which act on the

state space of the system being measure


Measuring a qubit_____________________________________

0 1 Quantum mechanics DOES NOT allow us to determine and .

We can, however, read out limited inf ormation about and . If we measure in the computational basis, i.e.,

and 22(0) ; (1)P P

Measurement the system, leaving it in a state 0 or 1 determi

unavoidably disned by the outc

turbsome.

0 1


More general measurements____________________________________

iiaA |,:ˆ

iic ||

|

Observable A (to be measured) corresponds to operator A

A has a set of eigenvectors with corresponding

eigenvalues

To measure on the system whose state vector is

one expresses in terms of the eigenvectorsA

|


More general measurements____________________________________

iic ||

|ia

2|| ic

i|

1.The measurement on state yields only

one of the eigenvalues, with probability

2.The measurement culminates with the state

collapsing to one of the eigenstates,

The process is non unitary


Quantum Classical transition in a quantum

measurement

Several interpretations of quantum mechanics seek

to explain this transition and a resolution to this

apparent nonunitary collapse in a quantum

measurement.

The collapse of the wavefunction following measurement

The quantum measurement

paradox/foundations of quantum mechanics

quantum mechanics for quantum information & computation

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