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 electronsStern-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)

22112 (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 &

&

&

10

1

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

CooperPair 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.