Quantum Information

1
Quantum Information

• Jan Guzowski

2
Universal Quantum Computers are Only Years Away

• From Davids 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 - 1434 BST

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 -------------------gt10-7m
• 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
Maxwells 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 -gt 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 ------gt qubit
• 0 or 1------gt
• 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 Shors
  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
Shors algorithm

• Shors 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
  ----gt 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 -----gt do nothing
• Result 01 -----gt 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)