Transcript No Slide Title
Qu
antum
What is
In
formation
and
T
echnology
?
Prof. Ivan H. Deutsch Dept. of Physics and Astronomy University of New Mexico
Second Biannual Student Summer Retreat of the SQuInT Network
St. John’s College, Santa Fe, New Mexico June 25, 2001 – June 29, 2001
Outline
• What is
classical information
?
• What is
quantum information
?
• What is
quantum computing
?
• How would we
implement
these ideas?
References: • M. A. Nielsen and I. Chuang,
Quantum computation and quantum information”
(Cambridge Press, 2000).
• Prof. Preskill’s notes: http://www.theory.caltech.edu/people/preskill/ph229/ •
Introduction to Quantum Computation and Information
, H-K Lo, S. Popescu, T. Spliller eds., (World Scientific, 1998).
• Special Issue of Physical Implementations: Fortschritte der Physik
48
2000.
U I O
Q Z W
Quantifying Information
3 bit message:
{000, 001, 010, 011, 100, 101, 110, 111}
With no
prior knowledge
:
Information
log 2 (# of possibities) = log 2 (8) = 3 bits
With
a priori knowledge (probabilistic)
:
Probability 0 =
p
Probability 1 = 1-
p
• “Typical word” of length
N
has:
Np
0’s and
N
(1-
p
) 1’s • #
typ
N
Np N
!
(
Np
)!
N
(1
p
) !
• Information in a typical word = log 2 (#
typ
)
Shanon Information
Shanon Information =
Average information/ letter
H
(
p
) log 2 (#
typ
)
p
log 2 (
p
) (1
N p
)log 2 1
p
Generally, given alphabet:
{
a j
|
j
1,..,
n
}with probability
p j H
(
A
)
n
j
1
p j
log 2 (
p j
)
=
Entropy
log 2 (
p j
) Information in
a j
0
H
(
A
) log 2 (
n
)
Mutual Information
Suppose:
Alice send message
A
Bob receives message with probability
B p(A)
with probability
p(B)
How much information can Alice communicate to Bob?
Degree of correlation:
p
(
A
;
B
)
p
(
A
)
p
(
B
)
p
(
A
,
B
) Joint probability
I
(
A
;
B
) log(
p
(
A
;
B
))
I
(
B
;
A
)
• Information Bob gets from Alice • Information common to
A
and
B
Quantum Information
Information encoded in a
quantum state
Qubit:
Two-level quantum system
0 cos q / 2) 0 q 0 , 1 “Logical Basis” f
e i
f sin q / 2 1
Bloch sphere
1 Tremendous amount of information
encoded
in
e
n
( q , f ) N S Given finite precision to specify q,f Alice can specify
many
different
Inaccessible Information
• Alice encodes 2 “ classical bit” in a qubit Message 0 1 1 2 2 3 3 0 1 2 3 • Bob decodes through Stern-Gerlach apparatus.
1 2 0 3
Quantum information
cannot be read
1
Holevo Bound
Alice cannot send more than one bit of information to Bob per qubit!
I
(
A
;
B
) 1bit
No Cloning
e.g. copy the unknown qubit onto an “ancilla” qubit • Transformation on basis states 0 0 0 0 1 0
• Linearity
0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 0
Quantum Information cannot be copied!
Information Gain Disturbance
Attempt to copy “distinguishing information” into an ancilla.
u
v
u
If
u
u
v
0 1
v v
v v
Any attempt to distinguish between two
non-orthogonal
necessarily results in a disturbance of the states states
Composite Systems
Classical
• One bit:
A
={
a| a=
0 or 1} • Two bits:
A
B
{(
a
,
b
) |
a
,
b
product • N
bits: Space has 2
N
Quantum
0 or1} configurations.
Cartesian
• One qubit:
A
,
c
0 0
c
1 1 ,
c
0 ,
c
1 finite precision complex numbers • Two qubits:
Tensor product A
B
AB
c
00 0,0
c
01 0,1
c
10 1, 0
c
11 1,1
• N N
qubits: Space has states, with
x
bits for each
c
.
Entanglement
(Pure) states
N
-qubits are generally “
entangled
”
1 2
N
Nonclassical correlations
e.g. Two qubits, Bell’s Inequalities Alice receives a random bit 0 1 ( ) 0 0
A
1
B
1
A
0 1
B
/ 2 Bob receives a random bit
•
Quantum-Information cannot be read.
•
Quantum-Information cannot be copied.
•
Nonorthogonal states cannot be distinguished.
•
Exponential growth in
inaccessible
information.
•
Quantum correlations cannot be used for communicating classical information
•
Measurement is
irreversible - collapse of the wave function
.
Quantum Information as a
Information-Gain/Disturbance:
• Quantum - Secret key distribution
Tensor Structure of Composites (Entanglement)
• Quantum dense coding - Sending two classical bits with one classical bit plus EPR.
• Quantum teleportation - Communicating a qubit •
Elements of Quantum Computation
in
out
Take advantage of exponentially large state space
• Quantum Register: quantum state of the system • Quantum Logic Gates: unitary transformation on subsystems • Error-tempering: • Measurement: correct/suppress errors read out classical information
Quantum Parallelism
(Deutsch)
E.g. 3-qubit “quantum register”
0 bits of classical input information.
1 0 2 3 4 5 6 7 1 0 2 3 4 5 6 7 3 bits of classical output information.
Quantum Computing: Quantum Control in
in
out
Many - body State
Complexity:
Asymptotic Behavior Given n bits to specify the state, how do the resources scale as n
Resources:
• Time • Energy • Space •
“
Easy
”
Polynomial in
n • “
Hard
”
Exponential in
n
Quantum Logic Gates
• Single qubit:
U
0 3
i
0 1,
c i
i
,
i
1,2,3
c i
2 1 Pauli
Rotation on Bloch sphere
NOT: Hadamard
:
1
H
0 1 1 0 1 3 2 •
Two qubit:
“Entangling unitary”
0 1 1 0 0 1 0 0 1 1 / 2 / 2 Controlled NOT: Control 0 0 0 0 0 1 1 0 0 1 1 1
x y
x x
Target
y
Flip the state of the target bit conditonal on the state of the control bit 3 1 1 1 1 0 0 2 1 0 0 0 2 1 1
Universality
• Logic Gates: Basic building blocks
U
:H
a i
1
i
2
i N
i
1
i
2
i N
Unitary acts on combinations of qubits
R
e
n
( q );
cNOT
Single qubit rotation Two-qubit entangling gate
Entangling Gate:
ˆ 12 ˆ 1 ˆ 2
Efficient algorithms: # of gates not exponential in N
Quantum Circuits
0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1
CNOT from CPHASE
1 1
(I)
1 0 1 2
(II)
1 1 0
(III)
0 1 2
(IV)
Deutsch’s Problem
Coupling Between Qubits
• Entanglement
Coupling To External Drive
• Unitary evolution
The Tao of Quantum Computation
Coherence
Coupling to the Environment
Errors
Physical Implementations
Atomic-Molecular Optical Systems (Gas Phase)
• Ion Traps • Cavity
QED
• Neutral Atom Traps • Linear/Nonlinear Optics
Condensed Matter (Liquid or Solid Phase)
• Semiconductors (electronics) • Nuclear/Electron Magnetic Resonance (liquid, solid, spintronics) • Superconductors (flux or charge qubits) • Electrons floating on liquid helium
• Ion Trap + + + • Strong Qubit Coupling: Coulomb Repulsion • Strong Coupling to Environment - Technical Noise
• Coherent photon exchange • Spontaneous emission + + -
Cavity QED:
• Interactions turned “on” and “off” • Real photon exchange • Strong-coupling regime (enhance coherence) Optical Lattice: + + + + + + • Virtual photon exchange • Near-field interaction dominates + + + + + + -
Ideal:
in
Reality:
in
Measurement
$
out
out
j
ˆ
j
†
j
ˆ
j
ˆ
j
1 ˆ †
j
Summary
• Information Processing constrained by
physical laws
.
• Quantum Information:
Information-gain/disturbance.
Exponential growth of state space. Entanglement.
• Quantum Computation - asymptotic savings of
physical resources
.
• Physical Implementation -
Quantum Control of Many-body System
!
Classical Physics Digital: Particles Analog: Waves Quantum Physics