Light and MatterTim Freegarde

School of Physics & Astronomy

University of Southampton

Quantum computation

2

Binary computing elements

• e.g. half-adder circuit

• any computer can be built from 2-bit logic gates

A B C

0 0 10 1 11 0 11 1 0

A

BC

NAND

A

BD

C

A B C D

0 0 0 00 1 0 11 0 0 11 1 1 0

A B C

0 0 00 1 11 0 11 1 0XOR

A

BC

carrysum

• gates are not reversible: output does not define input

HALF-ADDER

3

Reversible binary computing elements

• e.g. half-adder circuit

• any computer can be built from 2-bit logic gates

A B C

0 0 10 1 11 0 11 1 0

A

BC

NAND

A

BD

C

A B C D

0 0 0 00 1 0 11 0 0 11 1 1 0

A B C

0 0 00 1 11 0 11 1 0XOR

A

BC

A

0011

A

A

0011

carrysum

• gates are not reversible: output does not define input

HALF-ADDER

• for reversible gates, additional outputs needed

4

Reversible binary computing elements

• e.g. half-adder circuit

• any computer can be built from 2-bit logic gates

A B C

0 0 10 1 11 0 11 1 0

A

BC

NAND

A B C D

0 0 0 00 1 0 11 0 0 11 1 1 0

A B C

0 0 00 1 11 0 11 1 0XOR

A

BC

A

0011

A

A

0011

carrysum

• gates are not reversible: output does not define input

A

BD

A

C0

B

A

0

D

A

C

HALF-ADDER

• for reversible gates, additional outputs needed

CNOT

CCNOT (Toffoli)

5

Thermodynamics of computation

• e.g. entropy

• thermodynamic quantities are associated with any physical storage of information

0 1WkS log

• setting a binary bit reduces entropy by

2log

1log2log

k

kkS

• hence energy consumption

2logTkSTQ • reversible logic does not change ; no energy

consumed if change is slowW

• note that conventional logic gates consume kT610

6

Quantum computing

• electronic or nuclear spin of atom or molecule

• each data bit corresponds to a single quantum property

• electronic state of atom or molecule• polarization state of single photon• vibrational or rotational quantum number

• e.g. electron spins in magnetic field gradient• electromagnetic interactions between

trapped ions lift degeneracies in radiative transitions 11

1001

00

CNOT

00

11

1

E D C B A

B

A• evolution described by Schrödinger’s

equation

• operations carried out as Rabi -pulses

7

Quantum computing

• tiny, reversible quantum bits (qubits) for small, fast, low power computers

• complex wavefunctions may be superposed:

11

1001

00

CNOT

00

11

1

E D C B A

B

A

• parallel processing: result is

111001001010 AB

11100100 FFFF

• classical read-out: probabilistic results

• limited algorithms:• factorization (encryption security)• parallel searches (data processing)

8

Quantum computing

• extension of computing from real, binary numbers to complex, continuous values

• extension of error-correction algorithms from digital computers to analogue computers

11

1001

00

CNOT

00

11

1

E D C B A

B

A

• link between numerical and physical manipulation

• extension of quantum mechanics to increasingly complex ensembles

• is quantum mechanics part of computation, or computation part of quantum mechanics?

• statistical properties (the measurement problem)

9

observe describe understand

predict exploit

quantum optics

quantum mechanics

Quantum information processing

classical mechanics

Kepler 1571 Newton 1642Galileo 1564 H G Wells 1866 A C Clarke 1917

Planck 1858 Einstein 1879 Townes 1915

Schawlow 1921

Fraunhofer 1787 Balmer 1825

Compton 1892 Hertz 1887

De Broglie 1892

Schrödinger 1887

Heisenberg 1901

Feynman 1918

10

Further reading

• R P Feynman, Feynman Lectures on Computation, Addison-Wesley (1996)

• A Turing, Proc Lond Math Soc ser 2 442 230 (1936)

• C H Bennett, P A Benioff, T J Toffoli, C E Shannon

• D Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc Roy Soc Lond A 400 97 (1985)

• www.qubit.org

• D P DiVicenzo, “Two-bit gates are universal for quantum computation,” Phys Rev A 51 1015 (1995)