Quantum Parallelism and the Exact Simulation of Physical Systems

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Quantum Parallelism and the Exact Simulation of
Physical Systems

• Dan Cristian Marinescu
• School of Computer Science
• University of Central Florida
• Orlando, Florida 32816, USA

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Frontier(s)from Websters unabridged dictionary.

• The part of a settled or civilized country
  nearest to an unsettled or uncivilized region.
• Any new or incompletely investigated field of
  learning or thought.

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What is a Quantum computer?

• A device that harnesses quantum physical
  phenomena such as entanglement and superposition.
  
• The laws of quantum mechanics differ radically
  from the laws of classical physics.
• The unit of information, the qubit can exist as
  a 0, or 1, or, simultaneously, as both 0 and 1.

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Does quantum computing represent the frontiers
of computing?

• Is it for real? Can we actually build quantum
  computers? - Very likely, but it will take
  some time.
• If so, what would a quantum computer allow us to
  do that is either unfeasible or impractical with
  todays most advanced systems? Exact
  simulation of physical systems, among other
  things.
• Once we have quantum computers do we need new
  algorithms? Yes, we need quantum
  algorithms.
• Is it so different from our current thinking that
  it requires a substantial change in the way we
  educate our students? Yes, it does.

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Quantum computers now and then

• All we have at this time is a 7 (seven) qubit
  quantum computer able to compute the prime
  factors of a small integer, 15.
• To break a code with a key size of 1024 bits
  requires more than 3,000 qubits and 108 quantum
  gates.

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Approximate computer simulation of physical
systems

• Eniac and the Manhattan project. The first
  programs to run, simulation of physical
  processes.
• Computer simulation new approach to scientific
  discovery, complementing the two well established
  methods of science experiment and theory.
• Approximate simulation based upon a model that
  abstracts some properties of interest of a
  physical system.

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Exact simulation of physical systems

• How far do we want to go at the microscopic
  level? Molecular, atomic, quantum? - All of the
  above.
• What about cosmic level? - Yes, of course.
• Is it important? - - Yes (Feynman,
  1981) .
• Who will benefit?
• Natural sciences ? physics, chemistry, biology,
  astrophysics, cosmology,.
• Application ? nanotechnology, smart materials,
  drug design,

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Large problem state space

• From black hole thermodynamics a system
  enclosed by a surface with area A has a number of
  observable states
• c 3 x1010 cm/sec
• h 1.054 x 10-34 Joules/second
• G 6.672 x 10-8 cm3 g-1 sec-2
• For an object with a radius of 1 Km ? N(A)
  e80

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Acknowledgments

• Some of the material presented is from the book
• Approaching Quantum Computing
• by Dan C. Marinescu and Gabriela M. Marinescu
• to be published by Prentice Hall in June 2004
• Work supported by National Science Foundation
  grants MCB9527131, DBI0296107,ACI0296035, and
  EIA0296179.

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Contents

• Computing and the Laws of Physics
• Quantum Mechanics Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Deutschs Algorithm
• Virus Structure Determination and Drug Design
• Summary

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The limits of solid-state technologies

• For the past two decades we have enjoyed Gordon
  Moores law. 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 are subject to Heissenberg
  uncertainty

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Power dissipation and circuit density

• The computer technology vintage year 2000
  requires some 3 x 10-18 Joules per elementary
  operation.
• 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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Contents

• Computing and the Laws of Physics
• Quantum Mechanics Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Deutschs Algorithm
• Virus Structure Determination and Drug Design
• Summary

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A happy marriage

• The two greatest discoveries of the 20-th century
• quantum mechanics
• stored program computers
• led to the idea of
• quantum computing and
• quantum information theory

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Quantum Quantum mechanics

• Quantum ? Latin word meaning some quantity. In
  physics it is used with the same meaning as the
  word discrete in mathematics.
• Quantum mechanics ? a mathematical model of the
  physical world.

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Heissenbergs uncertainty principle

• ... Quantum Mechanics 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.' (Max
  Borns Nobel prize lecture on December 11, 1954)

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Milestones in quantum computing

• 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 integers.

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Deterministic versus probabilistic photon behavior
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Contents

• Computing and the Laws of Physics
• Quantum Mechanics Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Virus Structure Determination and Drug Design
• Summary

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One qubit

• Mathematical abstraction
• Vector in a two dimensional complex vector space
  (Hilbert space)
• Diracs notation
• ket ? column
  vector
• bra ? row vector
• bra ? dual vector (transpose and complex
  conjugate)

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State description
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A bit versus a qubit

• A bit
• Can be in two distinct states, 0 and 1
• A measurement does not affect the state
• A qubit
• can be in state or in state or in
  any other state that is a linear combination of
  the basis state
• When we measure the qubit we find it
• in state with probability
• in state with probability

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Qubit measurement
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Two qubits

• Represented as vectors in a 2-dimensional Hilbert
  space with four basis vectors
• When we measure a pair of qubits we decide that
  the system it is in one of four states
• with probabilities

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Two qubits
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Measuring two qubits

• Before a measurement the state of the system
  consisting of two qubits is uncertain (it is
  given by the previous equation and the
  corresponding probabilities).
• After the measurement the state is certain, it is
  
• 00, 01, 10, or 11 like in the case of a
  classical two bit system.

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Measuring two qubits (contd)

• What if we observe only the first qubit, what
  conclusions can we draw?
• We expect the system to be left in an uncertain
  sate, because we did not measure the second qubit
  that can still be in a continuum of states. The
  first qubit can be
• 0 with probability
• 1 with probability

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Measuring two qubits (contd)

• Call the post-measurement state when we
  measure the first qubit and find it to be 0.
• Call the post-measurement state when we
  measure the first qubit and find it to be 1.

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Measuring two qubits (contd)

• Call the post-measurement state when we
  measure the second qubit and find it to be 0.
• Call the post-measurement state when we
  measure the second qubit and find it to be 1.

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Bell states - a special state of a pair of qubits

• If and
• When we measure the first qubit we get the
  post measurement state
• When we measure the second qubit we get the
  post mesutrement state

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This is an amazing result!

• The two measurements are correlated, once we
  measure the first qubit we get exactly the same
  result as when we measure the second one.
• The two qubits need not be physically constrained
  to be at the same location and yet, because of
  the strong coupling between them, measurements
  performed on the second one allow us to determine
  the state of the first.

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Entanglement (Verschrankung)

• Discovered by Schrodinger.
• An entangled pair is a single quantum system in a
  superposition of equally possible states. The
  entangled state contains no information about the
  individual particles, only that they are in
  opposite states.
• Einstein called entanglement Spooky action at a
  distance".

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Classical gates

• Implement Boolean functions.
• Are not reversible (invertible). We cannot
  recover the input knowing the output.
• This means that there is an irretrievable loss of
  information.

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One qubit gates

• I ? identity gate leaves a qubit unchanged.
• X or NOT gate? transposes the components of an
  input qubit.
• Y gate.
• Z gate ? flips the sign of a qubit.
• H ? the Hadamard gate.

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Identity transformation, Pauli matrices, Hadamard
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CNOT a two qubit gate

• Two inputs
• Control
• Target
• The control qubit is transferred to the output as
  is.
• The target qubit
• Unaltered if the control qubit is 0
• Flipped if the control qubit is 1.

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The two input qubits of a two qubit gates
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State space dimension of classical and quantum
systems

• Individual state spaces of n particles combine
  quantum mechanically through the tensor product.
  If X and Y are vectors, then
• their tensor product X Y is also a vector,
  but its dimension is
• dim(X) x dim(Y)
• while the vector product X x Y has dimension
• dim(X)dim(Y).
• For example, if dim(X) dim(Y)10, then the
  tensor product of the two vectors has dimension
  100 while the vector product has dimension 20.

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Parallelism and Quantum computers

• In quantum systems the amount of parallelism
  increases exponentially with the size of the
  system, thus with the number of qubits (e.g. a 21
  qubit quantum computer is twice as powerful as a
  20 qubit quantum computer).
• A quantum computer will enable us to solve
  problems with a very large state space.

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Contents

• Computing and the Laws of Physics
• Quantum Mechanics Quantum Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Deutschs Algorithm
• Virus Structure Determination and Drug Design
• Summary

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A quantum circuit

• Given a function f(x) we can construct a
  reversible quantum circuit consisting of Fredking
  gates only, capable of transforming two qubits as
  follows
• The function f(x) is hardwired in the circuit

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A quantum circuit (contd)

• If the second input is zero then the
  transformation done by the circuit is

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A quantum circuit (contd)

• Now apply the first qubit through a Hadamad gate.
• The resulting sate of the circuit is
• The output state contains information about f(0)
  and f(1).

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Quantum parallelism

• The output of the quantum circuit contains
  information about both f(0) and f(1). This
  property of quantum circuits is called quantum
  parallelism.
• Quantum parallelism allows us to construct the
  entire truth table of a quantum gate array having
  2n entries at once. In a classical system we can
  compute the truth table in one time step with 2n
  gate arrays running in parallel, or we need 2n
  time steps with a single gate array.
• We start with n qubits, each in state 0gt and we
  apply a Walsh-Hadamard transformation.

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Contents

• Computing and the Laws of Physics
• Quantum Mechanics and Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Deutschs Algorithm
• Virus Structure Determination and Drug Design
• Summary

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Deutschs problem

• Consider a black box characterized by a transfer
  function that maps a single input bit x into an
  output, f(x). It takes the same amount of time,
  T, to carry out each of the four possible
  mappings performed by the transfer function f(x)
  of the black box
• f(0) 0
• f(0) 1
• f(1) 0
• f(1) 1
• The problem posed is to distinguish if

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A quantum circuit to solve Deutschs problem
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Evrika!!

• By measuring the first output qubit qubit we are
  able to determine performing
  a single evaluation.

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Contents

• Computing and the Laws of Physics
• Quantum Mechanics and Computers
• Qubits and Quantum Gates
• Quantum Parallelism
• Deutschs Algorithm
• Virus Structure Determination and Drug Design
• Summary

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Sindbis virus reconstruction and pseudo-atomic
modeling. The reconstruction computed at 11Å
(top half of first two panels) and 22Å (bottom
half of first two panels), viewed along a
two-fold axis and represented as a surface-shaded
solid (left panel) and as a thin, non-equatorial
section (middle panel). The 11Å reconstruction,
which shows significantly greater detail compared
to that in the 22Å map available approximately
one year ago, provides more accurate data for
fitting atomic models as illustrated in the right
panel, which is an enlarged view of the boxed
area in the middle panel.
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Final remarks

• A tremendous progress has been made in quantum
  computing and quantum information theory during
  the past decade.
• Motivation ? the incredible impact this
  discipline could have on how we store, process,
  and transmit data and knowledge in this
  information age.

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Final remarks (contd)

• Computer and communication systems using quantum
  effects have remarkable properties.
• Quantum computers enable efficient simulation of
  the most complex physical systems we can
  envision.
• Quantum algorithms allow efficient factoring of
  large integers with applications to cryptography.
  
• Quantum search algorithms speedup considerably
  the process of identifying patterns in apparently
  random data.
• We can improve the security of our quantum
  communication systems because eavesdropping on a
  quantum communication channel can be detected
  with high probability.

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Summary

• Quantum computing and quantum information theory
  is truly an exciting field.
• It is too important to be left to the physicists
  alone.