ipqi-2010-anu venugopalan 1 quantum mechanics for quantum information & computation anu...
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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
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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
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Moore’s law [www.intel.com]
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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]
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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
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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)
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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
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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
IPQI-2010-Anu Venugopalan
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
IPQI-2010-Anu Venugopalan
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 |||||
IPQI-2010-Anu Venugopalan
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
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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
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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
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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
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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
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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
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Linear Algebra & vector spaces___________________________________
0
1;
1
021 vv
1v
22112
1 vavavaa
av
iii
Example: For a Qubit: Vector Space C2
and span the Vector space C21v
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Linear Algebra & vector spaces___________________________________
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
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Linear Algebra & vector spaces___________________________________
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
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Linear Algebra & vector spaces___________________________________
• 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
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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
 : V W, which is linear in its inputs
Î: Identity
operator
Ô: Zero
Operator
Once the action of a linear operator  on a basis is specified, the action of  is completely determined on all inputs
ii
iii
i vAavaA ˆˆ
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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
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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:
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Linear operators and matrices - some properties____________________________________
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
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Linear operators and matrices - some properties____________________________________
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
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Linear operators and matrices - some properties____________________________________
iiAi
i ˆ
0ˆˆdet)( IAc
vvvA ˆ
Eigenvalues and
eigenvectors
Diagonal Representation
i An orthonormal set of eigenvectors for  with corresponding eigenvalues i
examplediagonal representation for z
110010
01
z
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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.
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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
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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_________________________________________
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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
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Unitary operators & Matrices- examples___________________________________
Unitary operators acting on qubits
102
11ˆ ;10
2
10ˆ
11
11
2
1ˆ
HH
H
The Quantum Hadamard Gate
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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
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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
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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
IPQI-2010-Anu Venugopalan
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
|
IPQI-2010-Anu Venugopalan
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
IPQI-2010-Anu Venugopalan
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