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

3. 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

5. Moore’s Law
The # of transistors per square inch had doubled every year since the invention of ICs.

7. 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?

8. Quantum Computation & Information
Study of information processing tasks can be accomplished using Quantum Mechanical systems.
Quantum
Mechanics
Computer
Science
Information
Theory
Cryptography

9. 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 has been formed
by Planck, Einstein, Bohr, Schrodinger, Heisenberg, Dirac, Born, …

10. Classical vs Quantum Experiments
Classical Experiments
Experiment with bullets
Experiment with waves
Quantum Experiments
Two slits Experiment with electrons
Stern-Gerlach Experiment

11. Exp. With Bullet (1)
detector
wall
P1(x)
wall H1 H2
(a)
Gun

12. Exp. With Bullet (2)
detector
wall
P2(x)
wall H1 H2
(a)
Gun

13. Exp. With Bullet (3)
P2(x)
P1(x)
(c)
wall H1 H2
(a)
Gun
(b)
(c)

14. Exp. with Waves (1)
detector
wall
wave source H1
I1(x)
I2(x)
(b) H1 H2

15. Exp. with Waves (2) detector wall wave source I1(x) I2(x) (b) (c) H1

16. Two Slit Experiment (1) detector wall P2(x) P1(x) (b) (c)
Results intuitively expected
wall H1 H2
(a)
source of electrons
(c)

17. Two Slit Experiment (2) detector wall P2(x) P1(x) (b) (c)
Results observed
wall H1 H2
(a)
source of electrons

18. Two Slit Exp. With Observer
light source
detector
wall
P2(x)
P1(x)
(b)
(c)
Interference disappeared! H1
source of electrons H2
⇨ “Decoherence”

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

20. Stern-Gerlach Experiment

21. QM Physical Concepts Wave Function Quantum Dynamics (Schrodinger Eq.)
Statistical Interpretation (Born Postulate)

22. Bit & Quantum Bits (1)
V(t) t 1

23. More Quantum Bits

24. 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 &
&
&

25. 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.

26. Math of Qubits
Qubits can be represented in Bloch Sphere.

27. 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.

28. Example of Quantum Gates
NOT gate: X
• Z gate: Z
• Hadamard gate: H
• Phase gate: P

29. 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.

30. Measurement M 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

31. 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.

32. 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.

33. 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

34. 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.

35. 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.

36. 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 …

37. Spintronics
Cavity QED
Atom Chip
RF-SQUID
Cooper
Pair Box

38. Perspective of Quantum Computation & Information
Quantum Parallelism
Quantum Algorithms solve some of the complex problems efficiently (Schor’s 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.