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Quantum Information Processing

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A. Hamed Majedi

Institute for Quantum Computing (IQC)

and

RF/Microwave & Photonics Group

ECE Dept., University of Waterloo

- 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

The # of transistors per square inch had doubled every year since the invention of ICs.

- 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?

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

Quantum

Mechanics

Computer

Science

Information

Theory

Cryptography

- 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 Experiments
- Experiment with bullets
- Experiment with waves

- Quantum Experiments
- Two slits Experiment with electrons
- Stern-Gerlach Experiment

detector

P1(x)

Gun

wall

H1

H2

wall

(a)

detector

Gun

P2(x)

wall

H1

H2

wall

(a)

P1(x)

Gun

P2(x)

(c)

H1

H2

wall

(a)

(b)

(c)

detector

I1(x)

I2(x)

(b)

wall

wave source

H1

H1

H2

detector

I1(x)

I2(x)

(b)

(c)

wall

wave source

H1

H2

Results intuitively expected

P1(x)

H1

detector

H2

P2(x)

source of electrons

wall

(c)

(a)

(b)

wall

(c)

Results observed

P1(x)

H1

detector

H2

P2(x)

source of electrons

wall

(c)

(a)

(b)

wall

light source

P1(x)

detector

P2(x)

source of electrons

(c)

(b)

wall

Interference disappeared!

H1

H2

⇨ “Decoherence”

- 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

S

N

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

V(t)

1

t

V(t)

0

t

- 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 &

&

&

- 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.

- Qubits can be represented in Bloch Sphere.

- 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.

P

X

• Z gate:

Z

H

- NOT gate:

• Hadamard gate:

• Phase gate:

- 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.

- 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

- The state space of nqubits 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.

• Entangledstate can not be represented by Tensor Product

E.g.

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.

- Storage:Store qubits for long time
- Isolation:Qubits must be isolated fromenvironment 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.

- 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.

- 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

- 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.