Quantum Information and Computation for Dummies Peter Samuelsson, Mathematical Physics
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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
0 1 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” examples
Ele
ctro
ns
spin P
hoto
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 theorem Copy 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 measure No 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
3 00 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 a a a c 0 1 1 0
NOT c
Single qubit gates
X
Rθ
H
gate unitary transformation operator
X
Quantum mechanics
Two qubit gates
a b
AND c a b AND 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)
Rπ/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
3 Energy 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 time Exponential speed-up
Many-body fermion system
Lloyd
v Initial state preparation v Time evolution v Charge density v Correlation functions v ….
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 qubits 1
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 Charge states in quantum dots
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….
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