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Quantum Information and

Computation for Dummies

Peter Samuelsson, Mathematical Physics

Outline

What is quantum

information?1

What is a quantum

computer?2

Why build a quantum

computer?3

How to build a

quantum computer?4

Outline

What is quantum

information?1

What is a quantum

computer?2

Why build a quantum

computer?3

How to build a

quantum computer?4

Basic unit of information

Classical bit

Quantum bit - qubit

“Quantum information is physical information that is held in the

state of a quantum system” - wikipedia

Definition

Two-state quantum system

V 1

0

Two-state classical system

More qubit ”hardware” examplesE

lectr

ons

spin

Photo

ns

polarization position

atomic level

n

n+1

Bloch sphere representation

Vector on the unit sphere

State parametrization

More qubits

- classical 00,01,10,11

2

N

- ”analog information”

Quantum vs classical

Teleportation

No cloning theoremCopy information

Quantum teleportation

Entanglement

Qubit A and B

Resource for quantum information processing

Hotly debated

non-local properties

Bob

Alic

e

Alice measures Bob must measureNo information

is transferred

Teleportation

Photon experiment

source of

entangled pairs

Alice

initial state

Bob

final state

classical communication

two-qubit

measurement

single qubit

rotation

Superdense Coding

Cryptography

- sending 2 classical bits in 1 qubit

1

Alice Bob

2 00, 01,

10, 11

300

01

10

11

Share en-

tangled pair

A rotates

- codes

B measures

- decodes

- quantum key distribution

Eve

classical

quantum

What is a quantum

computer?

Outline

What is quantum

information?1

2

Why build a quantum

computer?3

How to build a

quantum computer?4

“A quantum computer make use of quantum-mechanical phenomena, such as

superposition and entanglement, to perform operations on data.” - wikipedia

Definition

Fundamentals

Quantum computers can be programmed to carry out

the same operations as classical computers – Deutch

Any operation on a reversible Turing machine can

be simulated quantum mechanically – Benioff

A reversible Turing machine can perform the same

operations as a standard Turing machine – Bennett

1

2

3

4

Any algorithm can be

implemented on a (classical)

Turing machine – Church,Turing

One bit gates

Classical one-bit gate

= NOT aa

a c

0 1

1 0

NOT c

Single qubit gates

X

Rq

H

gateunitary transformation

operator

X

Quantum mechanics

Two qubit gates

a

bANDc a bAND

0 0 0

0 1 0

1 0 0

1 1 1

a b c

=

Classical two-bit gate

…. also AND, OR, XOR, NAND,

Two qubit gate

Controlled NOT

::

Circuits

Universal computation

Entangler

H

… and multi qubit circuits

With (for example)

Rp/4H

single qubit gates two qubit gate

any multi qubit circuit (quantum computer) can be constructed.

Quantum parallelism

f

Superposition of computations

f f f

f

Classical circuit

Quantum circuit

All function

evaluations in a

single run

Why build a quantum

computer?

Outline

What is quantum

information?1

What is a quantum

computer?2

3

How to build a

quantum computer?4

Motivation

1

Computationally hard problems

2

Quantum simulations

3Energy saving

Quantum computers can solve

certain problems much faster than

any classical computer

Grover Shor

Quantum computers can simulate

many-particle systems much faster

than any classical computer

Feynman

Quantum computers are reversible

and can consume much less energy

than (standard) classical computers

Landauer

Shor’s algorithm

Prime factorization problem: Given an integer N, find its prime factors

15 = 3 x 5 9999999942014077477 = 3162277633 x 3162277669

Time to solve:classical

quantum

Public key cryptography

email

transfers

identity

…

internet security

Product of two unknown primes

Exponential speed-up

Grovers algorithm

Unsorted data base search: Find a given element in a data base of size N

Time to solve:

classical

quantum

Quadratic speed-up

Quantum simulations

Classical computers require exponentially long time to

simulate many-particle quantum systems

Feynmans observation

Quantum computers

require polynomial timeExponential speed-up

Many-body fermion system

Lloyd

Initial state preparation

Time evolution

Charge density

Correlation functions

….

Simulate

Bosonic systems, chemical reactions, …

Outline

What is quantum

information?1

What is a quantum

computer?2

Why build a quantum

computer?3

How to build a

quantum computer?4

Di Vincenzo criteria

Requirement on physical system (hardware)

Scalable with well characterized qubits1

2

3

4

5

Initial state preparation

A universal set of quantum gates

Long (relevant) decoherence times

Reading out the result

Quantum error

correction

Linear quantum optics Trapped ions

Nuclear magnetic resonance Superconducting circuits

Electrons in solid state

Spins in quantum dots Dopants in silicon

Nitrogen vacancies in diamond Topological states in solids

Status and outlook

Achieved to date

Controlling & coupling individual qubits

Needed to beat classical computers

Full quantum computer : 104 - 105 qubits

Limited tasks: 102 - 103 qubits

Running algorithms with <10 qubits

Error correction with <10 qubits

Coherent transfer of individual qubits

…

Still plenty of work to do….