Quantum Information Processing

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Quantum Information Processing

• A. Hamed Majedi
• Institute for Quantum Computing (IQC)
• and
• RF/Microwave Photonics Group
• ECE Dept., University of
  Waterloo

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Outline

• Limits of Classical Computers
• Quantum Mechanics
• Classical vs. Quantum Experiments
• Postulates of quantum Mechanics
• Qubit
• Quantum Gates
• Universal Quantum Computation
• Physical realization of Quantum Computers
• Perspective of Quantum Computers

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Moores Law
The of transistors per square inch had doubled
every year since the invention of ICs.
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Limits of Classical Computation

• Reaching the SIZE Operational time limits
• 1- Quantum Physics has to be considered for
  device operation.
• 2- Technologies based on Quantum Physics
  could improve the clock-speed of microprocessors,
  decrease power dissipation miniaturize more!
  (e.g. Superconducting
• processors based on RSFQ, HTMT Technology)
• Is it possible to do much more? Is there any
  new kind of information processing based on
  Quantum Physics?

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Quantum Computation Information

• Study of information processing tasks can be
  accomplished using Quantum Mechanical systems.

Quantum Mechanics
Computer Science
Information Theory
Cryptography
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Quantum Mechanics History

• Classical Physics fail to explain
• 1- Heat Radiation Spectrum
• 2- Photoelectric Effect
• 3- Stability of Atom
• Quantum Physics solve the problems
• Golden age of Physics from 1900-1930 has been
  formed
• by Planck, Einstein, Bohr, Schrodinger,
  Heisenberg, Dirac, Born,

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Classical vs Quantum Experiments

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

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Exp. With Bullet (1)
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Exp. With Bullet (2)
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Exp. With Bullet (3)
(c)
(b)
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Exp. with Waves (1)
wave source
H1
H1
H2
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Exp. with Waves (2)
wave source
H1
H2
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Two Slit Experiment (1)
(c)
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Two Slit Experiment (2)
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Two Slit Exp. With Observer
Interference disappeared!
H1
H2
? Decoherence
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Results from Experiments

• Two distinct modes of behavior (Wave-Particle
• Duality)
• 1- Wave like 2-
  Particle-like
• Effect of Observations can not be ignored.
• Indeterminacy (Heisenberg Uncertainty
  Principle)
• Evolution and Measurement must be
  distinguished

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Stern-Gerlach Experiment
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QM Physical Concepts

• Wave Function
• Quantum Dynamics (Schrodinger Eq.)
• Statistical Interpretation (Born Postulate)

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Bit Quantum Bits (1)
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More Quantum Bits
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Qubit (1)

• A qubit has two possible states
• Unlike Bits, qubits can be in superposition state
• A qubit is a unit vector in 2D Vector Space
• (2D Hilbert Space)
• are orthonormal
  computational basis
• We can assume that

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Qubit (2)

• A measurement yields 0 with probability
  1 with
• probability
• Quantum state can not be recovered from qubit
  measurement.
• A qubit can be entangled with other qubits.
• There is an exponentially growing hidden
  quantum information.

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Math of Qubits

• Qubits can be represented in Bloch Sphere.

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Quantum Gates

• A Quantum Gate is any transformation in Bloch
  sphere allowed by laws of QM, that is a Unitary
  transformation.
• The time evolution of the state of a closed
  system is described by Schrodinger Eq.

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Example of Quantum Gates

• NOT gate

Hadamard gate
Phase gate
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Universal Computation

• Classical Computing Theorem
• Any functions on bits can be computed from
  the composition of NAND gates alone, known as
  Universal
• gate.
• Quantum Computing Theorem
• Any transformation on qubits can be done from
  composition of any two quantum gates.
• e.g. 3 phase gates 2 Hadamard gates, the
  universal computation is achieved.
• No cloning Theorem
• Impossible to make a copy from unknown qubit.
  

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Measurement

• A measurement can be done by a projection of each
  
• in the basis states, namely
  and .
• Measurement can be done in any orthonormal
  and linear combination of states .
• Measurement changes the state of the system
  can not
• provide a snapshot of the entire system.

Probabilistic Classical Bit
M
Probabilistic Classical Bit
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Multiple Qubits

• The state space of n qubits can be represented
  by Tensor
• Product in Hilbert space with
  orthonormal base vectors. E.g.
  
• states produced by Tensor Product is
  separable measurement of one will not affect
  the other.
• Entangled state can not be represented by
  Tensor Product
• E.g.

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Multiple Qubit Gates
C-NOT Gate
Any Multiple qubit logic gate may be composed
from C-NOT and single qubit gate. C-NOT Gate is
Invertible gates. There is not an irretrievable
loss of information under the action of C-NOT.
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Physics Math Connections in QIP
Postulate 1
Postulate 2
Postulate 3
Postulate 4
Isolated physical system
Evolution of a physical system
Measurements of a physical system
Composite physical system
Hilbert Space
Unitary transformation
Measurement operators
Tensor product of components
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Physical Realization of QC

• Storage Store qubits for long time
• Isolation Qubits must be isolated from
  environment to
• decrease Decoherence
• Readout Measuring qubits efficiently reliably.
• Gates Manipulate individual qubits induce
  controlled interactions among them, to do quantum
  networking.
• Precision Quantum networking measurement
  should be implemented with high precision.

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DiVinZenco Checklist

• A scalable physical system with well
  characterized qubits.
• The ability to initialize the state of the
  qubits.
• Long decoherence time with respect to gate
  operation time
• Universal set of quantum gates.
• A qubit-specific measurement capability.

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Quantum Computers

• Ion Trap
• Cavity QED (Quantum ElectroDynamics)
• NMR (Nuclear Magnetic Resonance)
• Spintronics
• Quantum Dots
• Superconducting Circuits (RF-SQUID, Cooper-Pair
  Box)
• Quantum Photonic
• Molecular Quantum Computer

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Perspective of Quantum Computation Information

• Quantum Parallelism
• Quantum Algorithms solve some of the complex
  problems efficiently (Schors algorithm, Grover
  search algorithm)
• QC can simulate quantum systems efficiently!
• Quantum Cryptography A secure way of
  exchanging keys such that eavesdropping can
  always be detected.
• Quantum Teleportation Transfer of
  information using quantum entanglement.