Quantum InformationJan Guzowski

Universal Quantum Computers are Only

Years AwayFrom David’s Deutsch weblog:

„For a long time my standard answer to the question ‘how long will it be before the first universal quantum computer is built?’ was 'several decades at least’. In fact, I have been saying this for almost exactly two decades … and now I am pleased to report that recent theoretical advances have caused me to conclude that we are within sight of that goal. It may well be achieved within the next decade.

The main discovery that has made the difference is cluster quantum computation, which is a marvellous new way of structuring quantum computations which makes them far easier to implement physically.”

Tuesday, 2005/08/30 - 14:34 BST

Technology of The Future

Nanotechnology: understanding quatnum effects enables further

miniaturization Quantum algorithms:

exponential growth of computational power effective code-breaking complete security of communnication error correction quantum teleportation

Shrinking computer

10-1m ------------------->10-7m Microtechnology reaches quantum limit

Nanocomputer

Transistion from micro to nanotechnology with use of quantum effects

Single-electron transistor (SET)

Nanocomputer

Alternative to transistors: new architecture made up of ‘cells’ (this may be quantum dots)

Nano-scale classical computer

A true quantum computer uses quantum algorithms

A molecule as a physical implementation of qubit

Nanocomputer

Information Theory Information is physical Information is insensitive to exactly

how it is expressed Can have a similar role in physics to

energy and momentum Fundamental question: how the

nature allows or prevents the information to be expressed and manipulated

Maxwell’s Demon (1871)

The demon sets up a pressure difference by only raising the partition when a gas molecule approaches it from the left. This can be done in a completely reversible manner, as long as the demon's memory stores the random results of its observations of the molecules. The demon's memory thus gets hotter. The irreversible step is not the acquisition of information, but the loss of information if the demon later clears its memory.

Turing Machine (1936)

The machine's action on reading a given symbol s depends only on that symbol and the internal state G

The internal construction of the machine specified by a finite fixed list of rules of the form (s,G -> s’, G’,d).

An input `programme' on the tape is transformed by the machine into an output result printed on the tape.

Capable of efficiently simulating all classical computational methods.

Bit ------> qubit 0 or 1------> Quantum algorithm can

incorporate instructions such as „... and now take a superposition of all numbers from the previous operations...”

Quantum Information

10

Computational power 3 qubits describe 8

numbers N qubits describe 2N

numbers We can perform an

operation F simultaneously on 2N N-digit numbers

Computational power grows exponentially

Cryptography Breaking codes becomes possible

with Shor’s quantum algorithm Safety encoding using entanglement

(cloning theorem)

Classical cryptography

The encrypting and decrypting algorithms are publicly announced

The sender and the receiver share a key

Key distribution (classical) allows eavesdropping

Method of public and private key invented (based on difficulty of factorizing large integers)

Shor’s algorithm Shor’s quantum algorithm enables

factorizing large integers in „finite” time (Shor 1994)

Based on quantum Fourier transform (Coppersmith 1994, Deutsch)

Execution time grows as a quadratic function of N

Safety key distribution

Cryptosystem uses quantum entanglement: a pair of correlated particles is generated

An eavesdropper has to detect a particle to read the signal, and retransmit it in order for his presence to remain unknown.

The act of detection destroys quantum correlation ----> no-cloning theorem

Information protected by the laws of physics Complete security of communication

for arbitrary because due tolinearity we have:

No-cloning theorem Assume there exists a machine M such that:

We cannot have

1111

0000

MMR

MMR

10

MMR

MMR 110010

Quantum rerror correction

Based on classical error correction An example: information stored in a qubit

is subjected to random flips (errors) We express by means of a three-qubit

state: After a flip we make two measurments - each one

being a projection onto two state basis:

Result = 00 -----> do nothing Result = 01 -----> flip the rightmost spin etc... State reconstructed

10

10 10

111000

011100110001,101010111000

011100101010,110001111000

111000

Physical Implementation

„Repeat-untill-succes quantum computing using stationary and flying qubits” (Lim, Barret, Beige, Kok, Kwek, 2 Nov 2005)

Based on the idea of one way quantum computer : the entanglement is distributed once for all by preparing an entangled state of all the qubits (cluster state); the logic gates are then applied as sequences of only single-qubit measurements

Stationary qubits: trapped atoms, molecules, ions; quantum dots or defect centers in solids

Flying qubits: photons

Summary New technologies Quantum algorithms Computational power (technical

improvement) Entanglement (effects impossible

without quantum mechanics)