Exploitation of Quantum Mechanics

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Exploitation of Quantum Mechanics

• What we now need to discuss
• Ways to use Q.M. to our advantage such as
  quantum
• algorithms, teleportation, and cryptography
• Ways to make quantum information robust
• How do we do this in physical systems?Note
  Todays quantum computers have at most 7 qubits.

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Sufficient Conditions for Quantum Computing
Set of sufficient criteria for quantum
information processing are Having a Hilbert
space Having quantum control Set up a
fiducial initial state 1 and 2 qubit
operations Measuring the qubits Having low
noise
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Devices for Quantum Information Processing

• Ion Traps
• Cavity QED
• Atom Traps
• Superconducting Josephson Junctions
• Nuclear Magnetic Resonance
• Solid State
• Electron Floating on Helium
• Electron Trapped by Surface Acoustic Waves
• Spintronics
• Quantum Dots
• Quantum Optics

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Quantum Information Processing using Ion Traps

• NIST (Wineland)
• Ann Arbour (Monroe)
• Innsbruck (Blatt)
• Oxford (Steane)
• LANL (Hughes)
• Munich (Walther)
• IBM (DeVoe)
• See Cirac Zoller Physical Review Letters,1995

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Plusses and Minuses of Ion Traps
Plusses-demonstration of manipulations of up to
4 qubits -decoherence time operation
time -possible combination with cavities
photon Minuses -uncontrolled heating, limits
of operations -increased complexity of cooling
with of qubits -difficulty to address different
qubits independently -linear trap provides no
parallel operations
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Cavity QED

• Groups
• Paris (Haroche)
• Achievements
• -quantum Rabi oscillations
• -entanglement knitting
• -quantum memory
• -EPR atom pairs
• -single photon QND detection
• -Quantum phase gates
• Caltech (Kimble)
• Georgia Tech (Chapman)

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Superconducting Devices
Groups Delft (Mooij) Stony Brook
(Friedman) Paris (Devoret) (Nakamura)
NATURE, VOL 398, 29 APRIL 1999 NATURE, VOL 406,
6 JULY 2000
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Benchmarking Quantum Devices Goal

• Must demonstrate the ability to
• control the system
• be scalable
• be device independent
• Application to NMR
• -create a cat state and decode it into a
  pseudo-pure state
• -similar to fault tolerant operations
• -create a n-coherence state

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Examples of Discrete Quantum Systems

• Energy levels in atom
• Electron Spin
• Photon Polarization
• Magnetic flux
• Angular momentum
• Sound vibration in a lattice

We describe these situations using the
mathematics of vector spaces!
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Vector Spaces

• Closure (add 2 vectors, get a vector)
• Has a Zero ( for every )
• Scalar Multiplication ( is also a vector)
• Inverse (for every there exists a so
  that )
• Associative

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The Vector Space
For a discrete quantum system with n possible
states, we will be interested in the space
n-tuples of complex numbers
Closure
Scalar Multiplication
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Quantum Notation
Complex conjugate of z
Vector (a ket) -- this will represent a possible
state of the discrete quantum system
Vector dual to (a bra)
Inner product of two vectors
Tensor product of two vectors
A matrix -- this will represent an operator
which can modify a quantum state
Inner product of
and
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Bases and Linear Independence
Spanning set a set of vectors such that any
vector in the space can be written as a
linear combination of vectors in the set
for any
Linear independence a set of vectors is
linearly independent if there is no linear
combination of them which adds to zero
non-trivially
Basis a linearly independent spanning set
(always exists!)
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Linear Operators
Physical operations on quantum states are
represented by linear operators which act on the
states
Linear operator An operator which maps one
vector space into another that is linear in its
arguments is called a linear operator
Basis for W
Basis for V
Linear operators matrices (matrix elements
determined by specifying action on a basis)
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Pauli Matrices
A useful set of matrices which acts on a
2-dimensional vector space are the Pauli matrices
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Inner Products
Inner Product A method for combining two vectors
which yields a complex number
that obeys the following rules

• is linear in its 2nd argument

Example

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More on Inner Products
Hilbert Space the inner product space of a
quantum system
Orthogonality and are orthogonal if
Norm
Unit is the unit vector parallel to
Orthonormal basis a basis set
where
Gram-Schmidt Orthogonalization an algorithmic
procedure for finding an orthonormal basis
from a given basis
(inner product of 2 vectors is equal to inner
product of the matrix reps of the 2 vectors)
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Outer Products
Let be a vector in the vector space W
Let be a vector in the vector space V
Outer product is the outer product of
and It is a linear map from V into W
defined by
Completeness relation Let be a basis for
V. It is easy to show that
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Eigenvalues and Eigenvectors
Eigenvalue
obtain by finding all roots to the equation
Eigenvector
Diagonalizable A matrix is diagonalizable
if it can be written as

e.g.
orthonormal basis
Degeneracy when two (or more) eigenvalues are
equal In this case the eigenspace is larger
than one dimension
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Hermitian Operators
Adjoint is the adjoint of if
for all vectors ,
in the vector space V
Properties
Hermiticity is Hermitian if
Projects any vector into a k-dimensional subspace
e.g.,
Normal is Normal if
Can show Normal Diagonalizable
(spectral decomposition)
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Unitary and Positive Operators
Unitary is unitary if
where and are any two distinct
orthonormal bases for the vector space V, such
that
can write
Note
(preserves inner product)
Positive is positive if
for every in V
(no negative eigenvalues!)
If for every in V
is positive definite
(all positive eigenvalues!)
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Tensor Products
A tensor product is a larger vector space formed
from two smaller ones simply by combining
elements from each in all possible ways that
preserve both linearity and scalar multiplication
If V is a vector space of dimension n W is a
vector space of dimension m then V W is a vector
space of dimension mn
e.g.,
are elements of V V
and so is
qualitatively new feature entangled states!
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More on Tensor Products
scalar multiplication
linearity
tensor product of operators
e.g.
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Functions of Operators
Can define the function of an operator from its
power series
e.g.
For normal operators, can go beyond this using
their spectral decomposition
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Trace and Commutator
Trace
(sum over the diagonal elements)
Commutator
Anti-commutator
Simultaneous Diagonalization Two Hermitian
operators and are diagonalizable in the same
basis if and only if
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Polar Decomposition
For any linear operator acting on a vector
space we can write
(left polar decomposition)
where is a unitary matrix -- it is unique if
has an inverse
Alternatively
(right polar decomposition)
Singular-value decomposition For all square
matrices, can write
where is a diagonal matrix