Quantum and classical computing

Dalibor HRG

EECS FER 16.9.2003.

How to think?

Review / Classical computing

Classical computing:

Turing machine (A.Turing,1937.), computability (functions and predicates), Computational Complexity – theory of classical computation.

Bool’s 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?

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 machine’s track)

Review / Classical computing

P PSPACE ?

P NP ?

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,…

Karp’s reducebility: NP – complete: each predicate from NP is reducible to 3– SAT predicate.

*: , {0,1}F B B B

3 , SAT SAT HC TSP

Review / Classical computing

Review / Classical computing

NANOTECHNOLOGY

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,…

Quantum mechanics

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

Quantum mechanics

Waves!

Secret of the electron

Does electron interfere with itself?

Quantum mechanics

Discrete values of energy and momentum. State represent object (electron’s spin, foton’s

polarization, electron’s path,…) and its square amplitude is probability for outcome when measured.

Superposition of states, nothing similar in our life.

Interference of states.

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?

Qubit

0 1

20 1

0 1

, 0 , 1C C

System of N qubits

Unitary operators: legal operations on qubit.

Unitary operators: holding the lengths of the states. Important!!

{0,1}nx

x

x

2

{0,1}

1n

xx

† † U U UU I

Tensors

For representing the state in a quantum register. Example, system with two qubits:

State in this systems is:

0 10 ,

2

0 1 10 0 0 0 1

2 2

Quantum gates

Quantum circuits (one qubit): Pauli-X (UNOT), Hadamard (USRN).

(two qubits): CNOT (UCN).

10

01

NOTU

NOTU

0 1- 2

10

0 1 2

11

2

SRNU

SRNU

NOTU

SRNU

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!

1 . . . 111 . . . 11 n

0...00...0...111...11 22

1

01 ... 01 01 22

1

1 11

n

n

UUUn SRNSRNSRN

Quantum algorithms (1)

00

00

Time

Quantum operatorsInitial state

Measurement

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.

12

0

n

ix xα f

12

0

)(n

ix xfα

(No-cloning theorem) Wooters & Zurek 1982

Unknown quantum state can not be cloned.

Basis for quantum cryptology (or quantum key distribution).

ψψψ 0

0 1

Quantum cryptology (1)

Quantum bitsAlice Bob

Eve

Quantum cryptology (2)

Public channel for authentication

Quantum teleportation Bennett 1982

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

Alice

Bob

EPR

Alice & Bob share EPR (Einstein,Podolsky,Rosen) pair.

Classical bits.

Present algorithms?

Deutsch-JoszaShor - Factoring 1994., Kitaev - Factoring Grover - Database searching 1996.,Grover - Estimating median

O N

3O n

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…

Corporations?

Corporations?