Light
and
Matter
Quantum computation
Tim Freegarde
School of Physics & Astronomy
University of Southampton
Binary computing elements
• any computer can be built from 2-bit logic gates
A
• e.g. half-adder circuit
• gates are not reversible: output does not define
input
A
B
C
0
0
1
1
0
1
0
1
1
1
1
0
A
B
C
0
0
1
1
0
1
0
1
0
1
1
0
A
B
C
D
0
0
1
1
0
1
0
1
0
0
0
1
0
1
1
0
C
B
NAND
A
C
B
XOR
A
B
D
C
HALF-ADDER
carry
sum
2
Reversible binary computing elements
• any computer can be built from 2-bit logic gates
A
• e.g. half-adder circuit
• gates are not reversible: output does not define
input
• for reversible gates, additional outputs needed
A
B
C
0
0
1
1
0
1
0
1
1
1
1
0
A
B
C
A
0
0
1
1
0
1
0
1
0
1
1
0
0
0
1
1
A
B
C
D
A
0
0
1
1
0
1
0
1
0
0
0
1
0
1
1
0
0
0
1
1
C
B
NAND
A
A
C
B
XOR
A
B
D
C
HALF-ADDER
carry
sum
3
Reversible binary computing elements
• any computer can be built from 2-bit logic gates
A
• e.g. half-adder circuit
• gates are not reversible: output does not define
input
• for reversible gates, additional outputs needed
A
B
C
0
0
1
1
0
1
0
1
1
1
1
0
A
B
C
A
0
0
1
1
0
1
0
1
0
1
1
0
0
0
1
1
A
B
C
D
A
0
0
1
1
0
1
0
1
0
0
0
1
0
1
1
0
0
0
1
1
C
B
NAND
A
A
C
B
XOR
A
A
B
0
D
C
A
A
CNOT
B
0
CCNOT (Toffoli)
HALF-ADDER
D
C
carry
sum
4
Thermodynamics of computation
• thermodynamic quantities are associated with any
physical storage of information
• e.g. entropy
0
1
S  k log W
• setting a binary bit reduces entropy by
S  k log 2  k log 1
 k log 2
• hence energy consumption
Q  T S  k T log 2
• reversible logic does not change
change is slow
W ; no energy consumed if
• note that conventional logic gates consume
 106 kT
5
Quantum computing
• each data bit corresponds to a single quantum
property
•
•
•
•
electronic or nuclear spin of atom or molecule
electronic state of atom or molecule
polarization state of single photon
vibrational or rotational quantum number
1
• e.g. electron spins in magnetic field gradient
• operations carried out as Rabi  -pulses
• evolution described by Schrödinger’s equation
E
1
B
A
0
0
• electromagnetic interactions between trapped ions
lift degeneracies in radiative transitions
1
D
C
11
B
A
CNOT
10
01
00
6
Quantum computing
• tiny, reversible quantum bits (qubits) for small,
fast, low power computers
• complex wavefunctions may be superposed:
0
 1 B  0  1 A  00  01  10  11
1
1
1
B
A
• parallel processing: result is
F  00   F  01   F  10   F  11 
0
0
E
D
• classical read-out: probabilistic results
• limited algorithms:
• factorization (encryption security)
• parallel searches (data processing)
C
11
B
A
CNOT
10
01
00
7
Quantum computing
• extension of computing from real, binary numbers
to complex, continuous values
• extension of error-correction algorithms from
digital computers to analogue computers
1
• link between numerical and physical manipulation
• is quantum mechanics part of computation, or
computation part of quantum mechanics?
E
• statistical properties (the measurement problem)
1
B
A
0
0
• extension of quantum mechanics to increasingly
complex ensembles
1
D
C
11
B
A
CNOT
10
01
00
8
Quantum information processing
classical
mechanics
Galileo 1564
Kepler 1571
Newton 1642
H G Wells 1866
A C Clarke 1917
quantum
optics
observe
Fraunhofer 1787
describe
Balmer 1825
Planck 1858
understand
Einstein 1879
predict
exploit
Townes 1915 Schawlow 1921
quantum
mechanics
Compton 1892
Hertz 1887 De Broglie 1892
Schrödinger 1887 Heisenberg 1901
Feynman 1918
9
Further reading
• R P Feynman, Feynman Lectures on Computation,
Addison-Wesley (1996)
• A Turing, Proc Lond Math Soc ser 2 442 230 (1936)
• D Deutsch, “Quantum theory, the Church-Turing principle and the universal
quantum computer,” Proc Roy Soc Lond A 400 97 (1985)
• D P DiVicenzo, “Two-bit gates are universal for quantum computation,” Phys Rev
A 51 1015 (1995)
• C H Bennett, P A Benioff, T J Toffoli, C E Shannon
• www.qubit.org
10