Transcript Introduction to Quantum Computing Lecture 3: Qubit

Introduction to Quantum
Computing
Lecture 3: Qubit Technologies
Rod Van Meter
[email protected]
June 27-29, 2005
WIDE University School of Internet
with help from K. Itoh, E. Abe,
and slides from T. Fujisawa (NTT)
Course Outline
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Lecture 1: Introduction
Lecture 2: Quantum Algorithms
Lecture 3: Devices and Technologies
Lecture 4: Quantum Computer Architecture
Lecture 5: Quantum Networking & Wrapup
Lecture Outline
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Brief review
DiVincenzo's criteria & tech overview
Technologies:
All-Si NMR, quantum dots, superconducting,
ion trap, etc.
Comparison
Superposition and ket Notation
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Qubit state is a vector
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two complex numbers
|0> means the vector for 0;
|1> means the vector for 1;
|00> means two bits, both 0;
|010> is three bits, middle one is 1;
etc.
A qubit may be partially both!
(just like the cat, but stay tuned for
measurement...)
complex numbers are wave fn amplitude;
square is probability of 0 or 1
1-qubit state and Bloch
sphere(Phase)
1-qubit basis states
1
0
z
Bloch
sphere
0 1
0
1
0
superposition
0
1
2
2
=1
1
,
C
0
ei cos 0 ei sin 1
2
global phase has no2
y
effect
2
0 i1
x
2
1
relative
phase
cos 0 ei sin 1
2
2
Reversible Gates
NOT
Control-NOT
x
x
X
or
Control-Control-NOT
(Toffoli gate)
Control-SWAP
(Fredkin gate)
Bennett, IBM JR&D
Nov. 1973
This set can be used for classical reversible
computing, as well;
brings thermodynamic benefits.
Quantum Algorithms
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Deutsch-Jozsa(D-J)
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Grover's search algorithm
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Phys. Rev. Lett., 79, 325 (1997)
Shor's algorithm for factoring large
numbers
–
D. Deutsch
Proc. R. Soc. London A, 439, 553 (1992)
SIAM J. Comp., 26, 1484 (1997)
R. Jozsa
L. K. Grover
P. W. Shor
Grover's Search
N
n
2 unordered items, we're looking for item “β”
Classically, would have to
check about N/2 items
β
Hard task!!
N  2 n items
To use Grover's algorithm, create superposition of all N inputs
N
repetitions of the Grover iterator G will produce “β”
1
N 1
N
x 0
x
Shor's Factoring Algorithm
66554087 ? 6703 9929
An “efficient” classical algorithm for this
problem is not known.
Using classical number
field sieve, factoring an
L-bit number is
3
2
k L log L
Oe
Using quantum Fourier
transform (QFT), factoring
an L-bit number is
O L
3
Superpolynomial speedup! (Careful, technically
not exponential speedup, and not yet proven that
no better classical algorithm exists.)
Summary: Characteristics of QC
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Superposition brings massive parallelism
Phase and amplitude of wave function used
Entanglement
Unitary transforms are gates
Measurement both necessary and problematic
when unwanted
So, can we surpass a classical computer with a
quantum one?
Depends on discovery of quantum algorithms
and development of technologies
Lecture Outline
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Brief review
DiVincenzo's criteria & tech overview
Technologies:
All-Si NMR, quantum dots, superconducting,
ion trap, etc.
Comparison
DiVincenzo’s Criteria
1.
2.
3.
4.
5.
Well defined extensible qubit array
Preparable in the “000…” state
Long decoherence time
Universal set of gate operations
Single quantum measurements
Qubit initialization
Execution of an
algorithm
Read the result
Must be done within
decoherence time!
Qubit Representations
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Electron: number, spin, energy level
Nucleus: spin
Photon: number, polarization, time, angular
momentum, momentum (energy)
Flux (current)
Anything that can be quantized and follows
Schrodinger's equation
Problems
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Coherence time
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Gate time
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NMR-based systems slow
(100s of Hz to low kHz)
Gate quality
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nanoseconds for quantum dot, superconducting
systems
generally, 60-70% accurate
Interconnecting qubits
Scaling number of qubits
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largest to date 7 qubits, most 1 or 2
A Few Physical Experiments
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IBM, Stanford, Berkeley, MIT
(solution NMR)
NEC (Josephson junction charge)
Delft (JJ flux)
NTT (JJ, quantum dot)
Tokyo U. (quantum dot, optical lattice, ...)
Keio U. (silicon NMR, quantum dot)
Caltech, Berkeley, Stanford (quantum dot)
Australia (ion trap, linear optics)
Many others (cavity QED, Kane NMR, ...)
Physical Realization
Cavity QED
Ion trap
Magnetic resonance
Superconductor
Examples of Qubits
Neutral atoms
Atomic
Flux states in
in optical lattices
cavity QED
superconductors
Optically driven
electronic states
Entanglement (4)
in quantum dots
Trapped ions
Solution NMR
Optically driven
Solid-state
Shor (7)
systems
All-silicon quantum computer
Crystal lattice
spin states in
Rabi (1)
Charge states in
superconductors
quantum dots
Electronically
driven
electronic states
in quantum dots
Electrons
floating
on liquid
helium
Single-spin
Impurity spins in
Electronically driven
Others: Nonlinear
MRFM
semiconductors
spin states in quantum dots
optics, STM etc
Lecture Outline
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Brief review
DiVincenzo's criteria & tech overview
Technologies:
All-Si NMR, quantum dots, superconducting,
ion trap, etc.
Comparison
Technologies Reviewed
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Liquid NMR
Solid-state NMR
Quantum dots
Superconducting Josephson junctions
Ion trap
Optical lattice
All-optical
Liquid Solution NMR
Billions of molecules are used,
each one a separate quantum
computer. Most advanced
experimental demonstrations to
date, but poor scalability as
molecule design gets difficult and
SNR falls. Qubits are stored in
nuclear spin of flourine atoms and
controlled by different frequencies
of magnetic pulses.
Used to factor 15 experimentally.
Vandersypen, 2000
All-Silicon Quantum Computer
105
29Si
atomic chains in
28Si
matrix work like
molecules in solution NMR QC.
Many techniques used
for solution NMR QC are
available.
2
1
4
3
6
5
A large field gradient separates
Larmor frequencies of the nuclei
within each chain.
Copies
8
No impurity dopants or
7
9
electrical contacts are
needed.
T.D.Ladd et al., Phys. Rev. Lett. 89, 017901 (2002)
Overview
L 400Î ¼
m
W 4Î ¼
m
H 10Î m
¼
s 2.1Î ¼
m
l 300Î ¼
m
t 0.25Î m
¼
w 4Î ¼
m
dBz/dz = 1.4 T/mm
B0 = 7 T
Active region
100m x 0.2m
Keio Choice of System and
Material
Two incompatible conditions to realize quantum computers:
1. Isolation of qubits from the environment
2. Control of qubits through interactions with the environment
System: NMR
1.
2.
Weak ensemble measurement
Established rf pulse techniques for manipulation
Material: silicon
1.
2.
Longest possible decoherence time
Established crystal growth, processing and
isotope engineering technologies
Fabrication: 29Si Atomic Chain
Regular step arrays on slightly
miscut
28Si(111)7x7
surface
(~1º from normal)
Steps are straight, with about 1
kink in 2000 sites.
28Si
29Si
STM image
J.-L.Lin et al., J. Appl. Phys 84, 255 (1998)
Kane Solid-State NMR
Qubits are stored in the spin of
the nucleus of phosphorus
atoms embedded in a zero-spin
silicon substrate. Standard
VLSI gates on top control
electric field, allowing electrons
to read nuclear state and
transfer that state to another P
atom.
Kane, Nature, 393(133), 1998
Semiconductor nanostructure
High electron mobility transistor (HEMT)
Quantum dot structure
# of electrons
N < 30
http://www.fqd.fujitsu.com/
Quantum dot in the Coulomb blockade regime
Single-electron transistor (SET)
Coulomb blockade region, (the number
of electrons is an integer, N)
orbital degree of freedom
&
spin degree of freedom
(spin qubit)
double quantum dot
bonding & anti-bonding orbitals
(charge qubit)
Quantum dot artificial atom
500 nm
|(r)|2
1s
2p
3d
S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996).
Environment surrounding a QD
dissipation (T1), decoherence (T2) & manipulation
coherency of the system
Double quantum dot device
A double quantum dot
fabricated in AlGaAs/GaAs 2DEG
(schematically shown by circles)
cf. CPB charge qubit (Y. Nakamura ’99)
Approximate number of electrons:
10 ~ 30
Charging energy of each dot: ~ 1 meV
Lattice temp. ~ 20 mK (2 ueV)
Typical energy spacing in each dot: ~100 ueV
Electron temp. ~100 mK (9 ueV)
Electrostatic coupling energy:~200 ueV
Measurement system
dilution refrigerator
Tlat ~ 20 mK
Telec ~ 100 mK
B = 0.5 T
Double quantum dot
AlGaAs/GaAs 2DEG
EB litho., fine gates,
ECR etching
1 mm
“thin coax”
filtering
Josephson Junction Charge
(NEC)
One-qubit device can control the
number of Cooper pairs of
electrons in the box, create
superposition of states.
Superconducting device, operates
at low temperatures (30 mK).
Nakamura et al., Nature, 398(786), 1999
Two-qubit device
Pashkin et al., Nature, 421(823), 2003
JJ Phase (NIST, USA)
Qubit representation is phase of current oscillation.
Device is physically large enough to see!
J. Martinis, NIST
JJ Flux (Delft)
The qubit representation is
a quantum of current (flux)
moving either clockwise or
counter-clockwise around
the loop.
Ion Trap
Ions (charged atoms) are
suspended in space in an
oscillating electric field.
Each atom is controlled
by a laser.
NIST, Oxford, Australia, MIT, others
Ion Trap
Number of atoms that can
be controlled is limited,
and each requires its own
laser. Probably <100 ions
max.
NIST, Oxford, Australia, MIT, others
Optical Lattice (Atoms)
Neutral atoms are
held in place by
standing waves from
several lasers.
Atoms can be
brought together to
execute gates by
changing the waves
slightly.
Also used to make
high-precision atomic
clocks.
Deutsch, UNM
All-Optical (Photons)
All-optical CNOT gate composed from
beam splitters and wave plates.
O'Brien et al., Nature 426(264), 2003
Lecture Outline
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Brief review
DiVincenzo's criteria & tech overview
Technologies:
All-Si NMR, quantum dots, superconducting,
ion trap, etc.
Comparison
Comparison
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NMR (Keio, Kane): excellent coherence times, slow
gates
– nuclear spin well isolated from environment
– Kane complicated by matching VLSI pitch to
necessary P atom spacing, and alignment
Superconducting: fast gates, but fast decoherence
Quantum dots: ditto
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electrons in solid state easily influenced by
environment
Comparison (2)
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Ion trap: medium-fast gates, good coherence time
(one of the best candidates if scalability can be
addressed)
Optical lattice (atoms): medium-fast gates, good
coherence time; gates and addressability of
individual atoms need work
All-Optical (photons): well-understood technology
for individual photons, but hard to get photons to
interact, hard to store
By the Numbers
Technology
All-Si NMR
Kane NMR
Ion Trap
Optical Lattice
Quantum Dot
Josephson Junction
All-Optical
Decoherence Time
25 seconds
seconds
seconds
seconds
low microsecs
microseconds
N/A
Gate Time
high millisecs
high millisecs
low millisecs
low millisecs
nanosecs
nanosecs
N/A
Apples-to-apples comparison is difficult, and
coherence times are rising experimentally.
Gates
10000?
1000?
1000?
?
100?
10-10000?
N/A
Quantum Error Correction and the
Threshold Theorem
Our entire discussion so far has been on “perfect”
quantum gates, but of course they are not perfect.
Various “threshold theorems” have suggested that
we need 10^4 to 10^6 gates in less than the
decoherence time in order to apply quantum error
correction (QEC). QEC is a big enough topic to
warrant several lectures on its own.
Wrap-Up
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Qubits can be physically stored on electrons
(spin, count), nuclear spin, photons
(polarization, position, time), or phenomena
such as current (flux); anything that is
quantized and subject to Schrodinger's wave
equation.
More than fifty technologies have been
proposed; all of these described are relatively
advanced experimentally.
Wrap-Up (2)
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Many technologies depend on VLSI
Most are one or two qubits
Have not yet started our own Moore's Law
doubling schedule
Several years yet to true, controllable, multiqubit demonstrations
10-20 years to a useful system?
Upcoming Lectures
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Quantum computer architecture
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Quantum networking
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how do you scale up? how do you build a
computer out of this? what matters?
Quantum key distribution
Teleportation
Wrap-Up and Review