Quantum and classical computing

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Quantum and classical computing
THEORETICAL COMPUTER SCIENCE
FER 16.9.2003.
Dalibor HRG
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How to think?
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Review / Classical computing

• Classical computing
• Turing machine (A.Turing,1937.), computability
  (functions and predicates), Computational
  Complexity theory of classical computation.
• Bools algebra and circuits, today computers,
  (logic).
• Algorithms and complexity classes (P, P/poly,
  PSPACE, NP, NP-complete, BPP,) measuring how
  efficient is algorithm, can it be useful?

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Review / Classical computing

• Famous mathematical questions today
• P predicates which are decidable in polynomial
  time (head moves of Turing machine)
• PSPACE predicates decidable in polynomial space
  (cells on Turing machines track)

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Review / Classical computing

• NP we can check some solution in polynomial
  time, but finding it, is a difficult problem.
• Predicate
• SAT , HC (hamiltonian cycle),TSP (travelling
  salesman problem), 3-SAT,
• Karps reducebility
• NP complete each predicate from NP is
  reducible to 3 SAT predicate.

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Review / Classical computing
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Review / Classical computing
NANOTECHNOLOGY
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Review / Quantum computing

• (R. Feynman,Caltech,1982.) impossibility to
  simulate quantum system!
• (D. Deutsch, Oxford, CQC, 1985.) definition of
  Quantum Turing machine, quantum class (BQP) and
  first quantum algorithm (Deutsch-Jozsa).
• Postulates of quantum mechanics, superposition of
  states, interference, unitary operators on
  Hilbert space, tensorial calculation,

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Quantum mechanics

• Fundamentals dual picture of wave and particle.
• Electron wave or particle?

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Quantum mechanics
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Waves!
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Secret of the electron
Does electron interfere with itself?
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Quantum mechanics

• Discrete values of energy and momentum.
• State represent object (electrons spin, fotons
  polarization, electrons path,) and its square
  amplitude is probability for outcome when
  measured.
• Superposition of states, nothing similar in our
  life.
• Interference of states.

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Qubit and classical bit

• Bit in a discrete moment is either 0 (0V) or
  1 (5V).
• Qubit vector in two dimensional complex space,
  infinite possibilities and values.
• Physically, what is the qubit?

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Qubit
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System of N qubits

• Unitary operators legal operations on qubit.
• Unitary operators holding the lengths of the
  states. Important!!

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Tensors

• For representing the state in a quantum register.
• Example, system with two qubits
• State in this systems is

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Quantum gates

• Quantum circuits (one qubit) Pauli-X (UNOT),
  Hadamard (USRN).
• (two qubits) CNOT (UCN).

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Quantum parallelism

• All possible values of the n bits argument is
  encoded in the same time in the n qubits! This is
  a reason why the quantum algorithms have
  efficiency!

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Quantum algorithms (1)
Initial state
Quantum operators
Measurement
Time
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Quantum algorithms (2)

• Idea
• 1. Make superposition of initial state, all
  values of argument are in n qubits.
• 2. Calculate the function in these arguments so
  we have all results in n qubits.
• 3. Interference ( Walsh-Hadamard operator on the
  state of n qubits or register) of all values in
  the register. We obtain a result.

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(No-cloning theorem) Wooters Zurek 1982

• Unknown quantum state can not be cloned.
• Basis for quantum cryptology (or quantum key
  distribution).

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Quantum cryptology (1)
Alice
Bob
Quantum bits
Eve
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Quantum cryptology (2)
Public channel for authentication
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Quantum teleportation Bennett 1982

• It is possible to send qubit without sending it,
  with two classical bits as a help.

Classical bits.
EPR
Alice Bob share EPR (Einstein,Podolsky,Rosen)
pair.
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Present algorithms?

• Deutsch-Josza
• Shor - Factoring 1994.,
• Kitaev - Factoring
• Grover - Database searching 1996.,
• Grover - Estimating median

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Who is trying?

• Aarhus
• Berkeley
• Caltech
• Cambridge
• College Park
• Delft
• DERA (U.K.)
• École normale supérieure
• Geneva
• HP Labs (Palo Alto and Bristol)
• Hitachi
• IBM Research (Yorktown Heights and Palo Alto)
• Innsbruck
• Los Alamos National Labs
• McMaster
• Max Planck Institute-Munich

• Melbourne
• MIT
• NEC
• New South Wales
• NIST
• NRC
• Orsay
• Oxford
• Paris
• Queensland
• Santa Barbara
• Stanford
• Toronto
• Vienna
• Waterloo
• Yale
• many others

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Corporations?
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Corporations?