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
3
Your Computer
Moore’s Law
The # of transistors per square inch had doubled every year since the invention of ICs.
4
How small can they be?
Here Quantum mechanics comesinto play
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.
QuantumMechanics
ComputerScience
InformationTheory
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)
Gun
wall
H1
H2
(a)
detector
wall
P1(x)
Exp. With Bullet (2)
Gun
wall
H1
H2
(a)
detector
wall
P2(x)
Exp. With Bullet (3)
Gun
wall
H1
H2
(a)
P2(x)
P1(x)
(c)(b) (c)
(x))P(x)(P(x)P 2121
12
Exp. with Waves (1)
wave source H1
H2
H1
detector
wall
I1(x)
I2(x)
(b)
Exp. with Waves (2)detector
wall
I1(x)
I2(x)
(b) (c)
2
2112 (x)(x)(x)I hh
H1
H2
wave source
Two Slit Experiment (1)
source of electrons
wall
H1
H2
(a)
detector
wall
P2(x)
P1(x)
(b) (c)(c)
(x))P(x)(P(x)P 2121
12
Results intuitively expected
Two Slit Experiment (2)
source of electrons
wall
H1
H2
(a)
detector
wall
P2(x)
P1(x)
(b) (c)
?(x)P12
Results observed
Two Slit Exp. With Observer
source of electrons
detector
wall
P2(x)
P1(x)
(b) (c)
(x)P(x)P(x)P 2112
Interference disappeared!
light source
“⇨ Decoherence”
H1
H2
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)
t 1
V(t)
t0
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 &
&
&
1
01
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
P• Phase gate:
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.
M
Probabilistic Classical Bit
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
A
B
A
B A
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.
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
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
RF-SQUID
Cooper
Pair Box
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.