Quantum Information

1. Quantum Information Jan Guzowski

2. From 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 Universal Quantum Computers are Only Years Away

3. 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

4. Shrinking computer • 10-1m ------------------->10-7m • Microtechnology reaches quantum limit

5. Nanocomputer • Transistion from micro to nanotechnology with use of quantum effects • Single-electron transistor (SET)

6. Nanocomputer • Alternative to transistors: new architecture made up of ‘cells’ (this may be quantum dots) • Nano-scale classical computer

7. Nanocomputer • A true quantum computer uses quantum algorithms • A molecule as a physical implementation of qubit

8. 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

9. 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.

10. 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.

11. Quantum Information • Bit ------> qubit • 0 or 1------> • Quantum algorithm can incorporate instructions such as „... and now take a superposition of all numbers from the previous operations...”

12. 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

13. Cryptography • Breaking codes becomes possible with Shor’s quantum algorithm • Safety encoding using entanglement (cloning theorem)

14. 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)

15. 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

16. 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

17. No-cloning theorem • Assume there exists a machine M such that: • We cannot have for arbitrary because due to linearity we have:

18. 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

19. 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

20. Summary • New technologies • Quantum algorithms • Computational power (technical improvement) • Entanglement (effects impossible without quantum mechanics)