Quantum Computing Part 1: Quantum Mechanics Overview

1
Quantum Computing Part 1 Quantum Mechanics
Overview
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Technological limits

• Gordon Moores law
• The observation made in 1965 by Gordon Moore,
  co-founder of Intel, that the number of
  transistors per square inch on integrated
  circuits had doubled every year since the
  integrated circuit was invented.
• Moore predicted that this trend would continue
  for the foreseeable future.
• In subsequent years, the pace slowed down a bit,
  but data density has doubled approximately every
  18 months, and this is the current definition of
  Moore's Law, which Moore himself has blessed.
• Most experts, including Moore himself, expect
  Moore's Law to hold for at least another two
  decades.

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Technological limits

• But all good things may come to an end
• We are limited in our ability to increase
• the density and
• the speed of a computing engine.
• Reliability will also be affected
• to increase the speed we need increasingly
  smaller circuits (light needs 1 ns to travel 30
  cm in vacuum)
• smaller circuits ? systems consisting only of a
  few particles subject to Heissenberg uncertainty

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Energy/operation

• If there is a minimum amount of energy dissipated
  to perform an elementary operation, then to
  increase the speed, thus the number of operations
  performed each second, we require a liner
  increase of the amount of energy dissipated by
  the device.
• The computer technology vintage year 2000
  requires some 3 x 10-18 Joules per elementary
  operation.
• Even if this limit is reduced say 100-fold we
  shall see a 10 (ten) times increase in the amount
  of power needed by devices operating at a speed
  103 times larger than the sped of today's devices.

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Power dissipation, circuit density, and speed

• In 1992 Ralph Merkle from Xerox PARC calculated
  that a
• 1 GHz computer operating at room temperature,
  with 1018 gates packed in a volume of about 1 cm3
  would dissipate 3 MW of power.
• A small city with 1,000 homes each using 3 KW
  would require the same amount of power
• A 500 MW nuclear reactor could only power some
  166 such circuits.

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Talking about the heat

• The heat produced by a super dense computing
  engine is proportional with the number of
  elementary computing circuits, thus, with the
  volume of the engine.
• The heat dissipated grows as the cube of the
  radius of the device.
• To prevent the destruction of the engine we have
  to remove the heat through a surface surrounding
  the device. Henceforth,
• our ability to remove heat increases as the
  square of the radius
• while the amount of heat increases with the cube
  of the size of the computing engine.

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Quantum Quantum mechanics

• Quantum is a Latin word meaning some quantity.
• In physics it is used with the same meaning as
  the word discrete in mathematics, i.e., some
  quantity or variable that can take only sharply
  defined values as opposed to a continuously
  varying quantity.
• Quantum mechanics is a mathematical model of the
  physical world

I'll bet any quantum mechanic in the service
would give the rest of his life to fool around
with this gadget. - Chief Engineer Quinn,
Forbidden Planet , 1956
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Heissenberg uncertainty principle

• Heisenberg uncertainty principle says we cannot
  determine both the position and the momentum of a
  quantum particle with arbitrary precision.
• In his Nobel prize lecture on December 11, 1954
  Max Born says about this fundamental principle of
  Quantum Mechanics
• ... It shows that not only the determinism of
  classical physics must be abandoned, but also the
  naive concept of reality which looked upon atomic
  particles as if they were very small grains of
  sand. At every instant a grain of sand has a
  definite position and velocity. This is not the
  case with an electron. If the position is
  determined with increasing accuracy, the
  possibility of ascertaining its velocity becomes
  less and vice versa.''

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Quantum mechanics displays some weird features
that cannot be duplicated by any classical system.
Quantum Weirdness

• Can Weirdness be useful?
• Quantum Computing
• Quantum Encryption
• Quantum teleportation
• Quantum Global Positioning

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Electromagnetic Weirdness

• In 1867 Maxwell proposed a theory unifying
  electricity, magnetism, and optics.

The theory was weird! Light was propagated as a
wave in a vacuum - no underlying medium.
Only when Marconi invented the radio transmitter
and receiver did the weirdness become accepted
as normal, or even obvious.
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Quantum Weirdness
some remarkable quotes

• The quantum is the greatest mystery we've got.
  Never in my life was I more up a tree than today.
  -John Wheeler
• If someone says that he can think about quantum
  physics without becoming dizzy, that shows only
  that he has not understood anything whatever
  about it. - Niels Bohr
• A picture may seem extraordinarily strange to you
  and after some time not only does it not seem
  strange but it is impossible to find what there
  was in it that was strange. - GERTRUDE STEIN
  concerning modern art.

John Wheeler and friends
Niels Bohr and big money.
Gertrude Stein by Picasso
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Classical versus Quantum Experiments

• Classical Experiments
• Experiment with bullets
• Experiment with waves
• Quantum Experiments
• Two slits Experiment with electrons
• Stern-Gerlach Experiment

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Experiment with bullets
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
H1 is open H2 is closed
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
H1 is closed H2 is open
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
H1 is open H2 is open
(b)
Figure 1 Experiment with bullets
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Experiment with Waves
H1 is closed H2 is closed
wave source
H1
H2
wall
(a)
Figure 2 Experiments with waves
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Experiment with Waves
H1 is open H2 is closed
wave source
H1
H2
wall
(b)
(a)
Figure 2 Experiments with waves
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Experiment with Waves
H1 is closed H2 is open
wave source
H1
H2
I2(x)
wall
(b)
(a)
Figure 2 Experiments with waves
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Experiment with Waves
H1 is open H2 is open
wave source
H1
H2
wall
This is a result of interference
(a)
Figure 2 Experiments with waves
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Two Slit Experiment
Are electrons particles or waves?
Figure 3 Two slit experiment
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Two Slit Experiment
Figure 3 Two slit experiment
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Two Slit Experiment With Observation
Now we add light source
H1
H2
? Decoherence
wall
(a)
Figure 4 Two slit experiment with observation
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Stern-Gerlach Experiment
1922 - a beam of silver atoms directed through an
inhomogeneous magnetic field would be forced into
two beams. Consistent with the possession of an
intrinsic angular momentum and a magnetic moment
by individual electrons.
Stern-Gerlach experiment with spin-1/2 particles
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Conclusions From the Experiments

• Limitations of classical mechanics
• Particles demonstrate wavelike behavior
• Effect of observations cannot be ignored
• Evolution and measurement must be distinguished

Can we use these phenomena practically? Quantum
computing and information
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Quantum Weirdnesses

• 1. Complementarity Every particle is a wave, and
  every wave is a particle. It just depends on how
  you measure it.

Leonard Mandel
classical particle
classical wave
quantum photon
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Quantum Weirdnesses

• 2. Superposition If Agt represents one state of
  the system, and Bgt represents another, then
• aAgt bBgt is also a state of the system.
• Examples
• heregt theregt
• live catgt dead catgt
• .

Schrödinger Cat State
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Quantum Weirdnesses

• 3. Entanglement The quantum state of a two
  particle system in which the states of the
  individual particles are inseparable even though
  the two particles are separated.

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A revolutionary approach to computing and
communication

• We need to consider a revolutionary rather than
  an evolutionary approach to computing.
• Quantum theory does not play only a supporting
  role by prescribing the limitations of physical
  systems used for computing and communication.
• Quantum properties such as
• uncertainty,
• interference, and
• entanglement
• form the foundation of a new brand of theory,
  the quantum information theory where
  computational and communication processes rest
  upon fundamental physics.

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Milestones in quantum physics

• 1900 - Max Plank presents the black body
  radiation theory the quantum theory is born.
• 1905 - Albert Einstein develops the theory of the
  photoelectric effect.
• 1911 - Ernest Rutherford develops the planetary
  model of the atom.
• 1913 - Niels Bohr develops the quantum model of
  the hydrogen atom.
• 1923 - Louis de Broglie relates the momentum of a
  particle with the wavelength
• 1925 - Werner Heisenberg formulates the matrix
  quantum mechanics.
• 1926 - Erwin Schrodinger proposes the equation
  for the dynamics of the wave function.
• 1926 - Erwin Schrodinger and Paul Dirac show the
  equivalence of Heisenberg's matrix formulation
  and Dirac's algebraic one with Schrodinger's wave
  function.
• 1926 - Paul Dirac and, independently, Max Born,
  Werner Heisenberg, and Pasqual Jordan obtain a
  complete formulation of quantum dynamics.
• 1926 - John von Newmann introduces Hilbert spaces
  to quantum mechanics.
• 1927 - Werner Heisenberg formulates the
  uncertainty principle.

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Milestones in computing and information theory

• 1936 - Alan Turing dreams up the Universal
  Turing Machine, UTM.
• 1936 - Alonzo Church publishes a paper asserting
  that every function which can be regarded as
  computable can be computed by an universal
  computing machine''. Church Thesis.
• 1946 - A report co-authored by John von Neumann
  outlines the von Neumann architecture.
• 1948 - Claude Shannon publishes A Mathematical
  Theory of Communication.
• 1961 - Rolf Landauer decrees that computation is
  physical and studies heat generation.
• 1973 - Charles Bennet studies the logical
  reversibility of computations.
• 1981 - Richard Feynman suggests that physical
  systems including quantum systems can be
  simulated exactly with quantum computers.
• 1982 - Peter Beniof develops quantum mechanical
  models of Turing machines.
• 1984 - Charles Bennet and Gilles Brassard
  introduce quantum cryptography.
• 1985 - David Deutsch reinterprets the
  Church-Turing conjecture.
• 1993 - Bennet, Brassard, Crepeau, Josza, Peres,
  Wooters discover quantum teleportation.
• 1994 - Peter Shor develops a clever algorithm for
  factoring large numbers.

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Deterministic versus probabilistic photon behavior
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The puzzling nature of light

• If we start decreasing the intensity of the
  incident light we observe the granular nature of
  light.
• Imagine that we send a single photon.
• Then either detector D1 or detector D2 will
  record the arrival of a photon.
• If we repeat the experiment involving a single
  photon over and over again we observe that each
  one of the two detectors records a number of
  events.
• Could there be hidden information, which controls
  the behavior of a photon?
• Does a photon carry a gene and one with a
  transmit'' gene continues and reaches detector
  D2 and another with a reflect'' gene ends up at
  D1?

Each detector detects single photons. Why? What
is a hidden information that controls this
In an attempt to solve this puzzle we design this
setup
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The puzzling nature of light (contd)

• Consider now a cascade of beam splitters.
• As before, we send a single photon and repeat the
  experiment many times and count the number of
  events registered by each detector.
• According to our theory we expect the first beam
  splitter to decide the fate of an incoming
  photon
• the photon is either reflected by the first beam
  splitter or transmitted by all of them.
• Thus, only the first and last detectors in the
  chain are expected to register an equal number of
  events.
• Amazingly enough, the experiment shows that all
  the detectors have a chance to register an event.

Why all these detectors detect light?
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Polarization experiment

• Polarization of light electric field vector is
  confined to a single direction. This is a
  quantum-mechanical property of photons.
• Light source Assume each photon has a random
  polarization
• Filter A Horizontally polarized

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Polarization experiment

• Polarization of light electric field vector is
  confined to a single direction. This is a
  quantum-mechanical property of photons.
• Light source Assume each photon has a random
  polarization
• Filter A Horizontally polarized
• Filter B Vertically polarized

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Polarization experiment

• Polarization of light electric field vector is
  confined to a single direction. This is a
  quantum-mechanical property of photons.
• Light source Assume each photon has a random
  polarization
• Filter A Horizontally polarized
• Filter B Vertically polarized
• Filter C polarized at 45 degrees

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Polarization experiment explanation

• Photons polarization state can be modeled by a
  unit vector pointing in the appropriate direction
• Let the basis vectors be ? and ?
• Any polarization can be expressed as
• where a and b are complex numbers such that

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• Measurement postulate of quantum mechanics
• Any device measuring a quantum system has an
  associated orthonormal basis with respect to
  which the measurement takes place.
• Measurement of a state transforms the state into
  one of these basis vectors.
• The probability that the state is measured as
  basis vector u is the norm of the amplitude of
  the component of the original state in the
  direction of u.

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• Example
• Let be the polarization
  state of a photon.
• Then this state will be measured as ? with
  probability a2
• and as ? with probability b2 .
• These are indeed probabilities, since
  .
• The measurement will also change the state to the
  result
• of the measurement.
• Its original value cannot be determined.

41

• Back to the polarization experiment
• Original light source produces photons with
  random polarizations
• Thus 50 are measured as horizontal, and let
  through. (Why?)
• Those photons are now in the horizontal state,
  
• None are let through by the vertical filter.

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• Now let the 45o filter be put behind the
  horizontal filter. The 45o filter measures
  photon polarization with respect to a new basis
• Photons in state ? will be measured as
  with probability 1/2. So half will get through.
  
• Likewise, filter B will let half of these
  through.
• Total photons getting through all three filters
  (1/2)3 1/8.

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Quantum Mechanics
Quantum mechanics vs. Classical
Mechanics
Formulated to explain the behavior of microscopic
systems.
Formulated to explain the behavior of macroscopic
objects.
Newtons second law Integrate twice ? x(t).
Two constants of integration ? two additional
pieces of information required to uniquely define
the state of the system (e.g. xo and vo). The
state of the system is defined by FORCES,
POSITIONS, VELOCITIES Knowledge of the initial
state of the system can predict future states
precisely.
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Quantum Mechanics

• Designed to describe the behavior observed for
  microscopic particles and systems.
• Behavior
• discrete rather than continuous energy levels
• photons, electrons and nuclei exhibit both wave
  and particle nature.
• History
• 1925 Werner Heisenberg, Max Born and Pascual
  Jordon introduced matrix-based mathematical
  formalism to describe observed quantum mechanical
  phenomena
• 1926 Erwin Schrödinger introduced a
  differential equation and its solution that
  equivalently describes equation observed quantum
  mechanical phenomena

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The postulates of quantum mechanics (QM)

• Postulate I
• For any possible state of a system, there is a
  function y of the coordinates of the parts of the
  system and time that completely describes the
  system.

Y Is called a wave function. For two particles
system,
The wave function square Y2 is proportional to
probability. Since Y may be complex, we are
interested in YY, where Y is the complex
conjugate (i ? -i) of Y. The quantity YYdt is
proportional to the probability of finding the
particles of the system in the volume element, dt
dxdydz.
that is the probability of finding the particle
in the universe is 1 ? normalization condition.
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The postulates of quantum mechanics (QM)

• Orthogonality of two wave functions

Example sinq and cosq are orthogonal functions.
Fourier series expansion sin(nq) and cos(nq)
orthogonal functions
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The Wavefunction
Postulate I For any possible state of a system,
there is a function y of the coordinates of the
parts of the system and time that completely
describes the system

• Y(t) f ig
• f and g are real functions of coordinates and
  time
• An abstract, complex quantity but related to
  physically measurable quantities
• State is dependent on coordinates (spatial and
  spin) and time
• The time-dependent Schrödinger equation
• A single integration with respect to time is
  required to obtain Y(t), so that only one
  constant of integration is required to predict
  future states of the system.

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The Wavefunction
How can we describe the state of a quantum
mechanical system such as nuclear spins? Complex
wavefunction Y(t) Y(t,t) description of
all knowable information about the state of the
system What if we want to know if the system
is in a given state Y(t) at time t? The
probability that the system is in the state given
by Y(t) at time t is P Y(t)Y(t) Y2 For
example, for a single particle at time t', the
wavefunction is Y(x,y,z,t'), and the probability
that time t', the particle is in a given volume
of space (dxdydz) is given by Y(x,y,z,t')2
dxdydz
Time dependence
Spatial and spin coordinates (independent of time)
probability density
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Complex Conjugate
Complex Conjugate Y Y f ig Y f
ig (replace i with i) YY (f ig) (f ig)
f2 ifg ifg (i2)g f2 g2
real, non-negative (as P should be!)
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Normalization Condition
Since the system must exist in some state at
time, t, if we integrate over all coordinates of
the system (t represents the generalized
coordinates, which may include spatial
coordinates and spin state), the probability
density is 1. Normalization condition ?
Y(t)Y(t)dt 1 i.e. the probability of finding
the particle somewhere in space is one. The
Schrödinger equation describes the evolution in
time of a given system
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The Schrödinger Equation
constant, often omitted
Hamiltonian FORCES
The Hamiltonian, H, represents the forces acting
on the system, which can be time-dependent or
time-independent.
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The time-independent Schrödinger equation
If the Hamiltonian is time-independent, then the
Schrödinger equation can be solved by separation
of variables. Y(t) y(t) f(t) The Hamiltonian
acts only on the generalized coordinate part of
Y(t), y(t), since H is independent of time (i.e.
f(t) acts as a constant). Choosing units so
that h 1 Multiplying both sides by ? Y(t)
dt
time indep.
time- dep.
1
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The time-independent Schrödinger equation
Noting that the wave function is normalized and
multiplying by i To simplify things, lets
define So that Now the goal is to solve
this differential equation.
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The time-independent Schrödinger equation
The solution is This can be seen by
differentiating with respect to t Now we can
write
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The time-independent Schrödinger equation
Differentiating with respect to
time Multiplying both sides by i Comparing
to the original form of the Schrödinger equation,
we get Which is the time-independent
Schrödinger equation. The time dependence can be
thought of as a phase factor that cancels when
the probability distribution is calculated
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The time-independent Schrödinger equation
We have just shown that H is an operator E is
an energy ? is the wave function
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Probabilistic nature of QM

• If Y is a solution to the Schrödinger equation,
  then so is cY (c arbitrary constant), and Sci?i
  is also a solution, where each ?i is a possible
  state of the system.
• Y isnt really a physical wave. It is an
  abstract mathematical entity that yields
  information about the state of the system.
• Y gives information on the probabilities for
  possible outcomes of measurements of the systems
  physical properties.

Quantum mechanics says a lot, but does not
really bring us any closer to the secrets of the
Old One. I, at any rate, am convinced that He
does not throw dice. Albert Einstein
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Operators

• Postulate II
• With every physical observable q there is
  associated an operator Q, which when operating
  upon the wavefunction associated with a definite
  value of that observable will yield that value
  times the wavefunction F, i.e. QF qF.

H
http//hyperphysics.phy-astr.gsu.edu/hbase/quantum
/qmoper.html
59
Eigenvalues and Eigenfunctions
Eigenvalue equations An operator acts on a
function to give another function Âf(x)
g(x) Where  is an operator. For example, an
operator can be defined where â represents
multiplying by a âf(x) af(x) Depending upon
the operator, the new function can be very
different from the original function. However,
in a special case, the new function is a multiple
of the original function Âf(x) lf(x) In this
case, f(x) is said to be an eigenfunction of Â
with the associated eigenvalue, l. In the case
of the Schrödinger equation with a
time-independent Hamiltonian, H is the operator,
Y is the eigenfunction and E is the eigenvalue H
Y EY
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Operators

• (1) The operators are linear, which means that
• O(Y1 Y2) OY1 OY2
• The linear character of the operator is related
  to the superposition of states and waves
  reinforcing each other in the process
• (2) The second property of the operators is that
  they are Hermitian (the 19th century French
  mathematician Charles Hermite).
• Hermitian matrix is defined as the transpose of
  the complex conjugate () of a matrix is equal to
  itself, i.e. (M)T M

In QM, the operator O is Hermitian if
C. Hermite
http//commons.wikimedia.org/wiki/ImageCharles_He
rmite_circa_1887.jpg
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Hermitian Operators

• The adjoint of an operator (A) satisfies the
  equation
• yA ly
• Hermitian operators are self-adjoint (A A)
• This has several implications
• Eigenvalues for Hermitian operators are real
• Eigenfunctions for Hermitian operators form a
  complete orthonormal set
• ?yiyjdt ?yjyidt dij Kronecker delta
• Hilbert space a complete set of N orthonormal
  functions which constitutes a basis set

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Eigenvalues of QM operator

• Postulate III
• The permissible values that a dynamical variable
  may have are those given by OF aF, where F is
  the eigenfunction of the QM operator (Hermitian)
  O that corresponds to the observable whose
  permissible real values are a.
• The is postulate can be stated in the form of an
  equation as

O F a F
operator wave function eigenvalue
wave function
Example Let F e2x and Od/dx ? dF/dx
d(e2x)/dx 2 e2x ? F is an eigenfunction of the
operator d/dx with an eigenvalue of 2.
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Quantum Mechanics
Ayi liyi The result of making a measurement of
A is one of the eigenvalues of A. That is, only a
limited set of outcomes are possible (discrete
nature of quantum mechanics). What is the value
we might expect to measure? Expectation
Value The expectation value is the average
magnitude of a property sampled over an ensemble
of identically prepared systems. The expectation
value, ltAgt, is the scalar product of Y and
AY ltAgt ?YAYdt If the wavefunction is an
eigenfunction of the operator (Y yn) ltAgt
?YAYdt ?ynAyndt ln?ynyndt ln
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Expectation Value
If a system is in state Y(t,t), the average of
any physical observable C at time t is ltCgt
?yCydt If one makes a large number of
measurements of C with identical initial state
Y(t,0), then one obtains a set of values C1, C2,
, CN. The average of C is given by the rule A
postulate of quantum mechanics is that the
integral and summation above provide the same
value, which is the expectation value.
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Quantum Mechanics
In general Y ? yn, but Hence
probability that cj is obtained in a single
measurement
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Quantum Mechanics
What does this mean? When A is measured for a
single member of an ensemble, the result is one
of the eigenvalues of A, but which one cannot be
predicted in advance. The result means that the
eigenvalue lj will be obtained in a single
measurement with the probability of cj2. So, for
a single measurement, there are specified values
of A that are possible, but over an ensemble, the
expectation value ltAgt can be a continuous value.
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Example Particle in a box

• Infinite potential well
• Particle mass m in a box length L
• U(x)0, if 0ltxltL,
• U(x)8, if xlt0 or- xgtL
• Boundary conditions on y
• Y(0)0Y(L)

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Particle in a box

• Second derivative proportional to the function
  with - sign
• Possible solutions sin(kx) and cos(kx)

k-wave vector
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Particle in a box

• Lets satisfy boundary conditions

Wave vector is quantized!
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Particle in a box

• Quantum number n

Energy is quantized!
We are not done yet, We dont know A
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Particle in a box

• We know for sure that the particles is somewhere
  in the box
• Probability to find the particle in 0ltxltL is 1
• Unitarity condition

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Particle in a box