Quantum Information Processing
A. Hamed Majedi
Institute for Quantum Computing (IQC)
and
RF/Microwave & Photonics Group
ECE Dept., University of Waterloo
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
Moore’s Law
The # of transistors per square inch had doubled every year since the invention of ICs.
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?
Quantum Computation & Information
• Study of information processing tasks can be accomplished
using Quantum Mechanical systems.
Quantum
Mechanics
Computer
Science
Information
Theory
Cryptography
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, …
Classical vs Quantum
Experiments
• Classical Experiments
 Experiment with bullets
 Experiment with waves
• Quantum Experiments
 Two slits Experiment with electrons
Stern-Gerlach Experiment
Exp. With Bullet (1)
detector
H1
Gun
P1(x)
H2
wall
wall
(a)
Exp. With Bullet (2)
detector
H1
Gun
H2
P2(x)
wall
wall
(a)
Exp. With Bullet (3)
H1
Gun
P1(x)
H2
P12 (x)  12 (P1 (x)  P2 (x))
P2(x)
wall
(a)
(b)
(c)
Exp. with Waves (1)
detector
wave
source
H1
H1
I1(x)
H2
I2(x)
wall
(b)
Exp. with Waves (2)
detector
wave
source
I1(x)
H1
H2
I 12 (x)  h1 (x)  h2 (x)
I2(x)
wall
(b)
(c)
2
Two Slit Experiment (1)
detector
Results intuitively
expected
P1(x)
H1
H2
source of
electrons
P12 (x)  12 (P1 (x)  P2 (x))
P2(x)
wall
(a)
wall
(b)
(c)
Two Slit Experiment (2)
detector
Results
observed
P1(x)
H1
H2
source of
electrons
P12 (x)  ?
P2(x)
wall
(a)
wall
(b)
(c)
Two Slit Exp. With Observer
light
source
detector
Interference
disappeared!
P1(x)
H1
source of
electrons
H2
P12 (x)  P1 (x)  P2 (x)
P2(x)
⇨ “Decoherence”
wall (b)
(c)
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
Stern-Gerlach Experiment
S
N
QM Physical Concepts
• Wave Function
• Quantum Dynamics (Schrodinger Eq.)
• Statistical Interpretation (Born Postulate)
Bit & Quantum Bits (1)
V(t)
1
t
V(t)
0
t
More Quantum Bits
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
&
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.
Math of Qubits
• Qubits can be represented in Bloch Sphere.



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.
Example of Quantum Gates
• NOT gate:
X
• Z gate:
Z
• Hadamard gate:
H
• Phase gate:
P
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.
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
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.
Multiple Qubit Gates
C-NOT Gate
A
A
B
B A
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.
Physics & Math Connections in QIP
Postulate 1
Isolated physical system
Hilbert Space
Postulate 2
Evolution of a physical
system
Unitary
transformation
Postulate 3
Measurements of a
physical system
Measurement
operators
Postulate 4
Composite physical
system
Tensor product
of components
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.
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.
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
…
Spintronics
Cavity QED
Atom Chip
Cooper
Pair Box
RF-SQUID
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.