Download Quantum information with Rydberg atoms

Quantum computing with Rydberg atoms
Klaus Mølmer
Coherence school
Pisa, September 2012
Quantum computing with Rydberg atoms
Klaus Mølmer
Coherence school
Pisa, September 2012
Quantum Computing with Rydberg
atoms
Contents:
• Introduction to Quantum Computing
• Physical implementations
• Rydberg atom quantum computing
• Reality:
• Dream:
Intel® Pentium®
Dual Core T4200-processor,
2,0 GHz,
3072 MB SDRAM.
(250 GB harddisk)
650 Euros.
Quantum computer
Processor unknown
1 kHz is fine, 100 Hz is OK
1 kbit RAM would be great !
Buy at 108 Euros
How can a small, slow, ”quantum computer”
outperform a faster PC or laptop?
Classical bit:
0 or 1
”Quantum bit”:
0 and 1
Quantum computing
Idea: ”quantum is weird”
”quantum is useful !”
A particle wave function occupies
different locations
 A computer register can deal with
several numbers at the same time.
x1 f(x1) and x2 f(x2)
in two steps
becomes:
(x1 and x2)  (f(x1) and f(x2))
in just one step.
Measurements constitute a
fundamental limitation.
We cannot measure state, a|0>+b|1>.
We can ask:
”Is the system in state |0> ?”
and the answer is
Yes with probability |a|2 ,
and the state becomes |0>
No with probability |b|2 ,
and the state becomes |1>
|a|2+|b|2=1
You get random answers, and you do not learn more
by asking twice !
Parallel processing on a single quantum computer
• a|x1> +b|x2> a|f(x1)> + b|f(x2)>
• All x, Σx cx |x>  Σx cx |f(x)>, all f(x) by a single iteration
of a single quantum register.
We only need to solve two problems:
• readout: how to get all f(x)
and not just a random f(x) ? (algorithms)
Shor and Grover !
• construction: how to build and control a
mikroscopic quantum system ? (physics)
Is a computational problem easy or hard?
How do the ressources needed scale with the
problem size?
• Addition:
1 1 1
1
1
246389135
+358901246
=605290381
needs L operations for L-digit numbers
• Multiplication:
246389135 x358901246=
needs L2 operations for L-digit numbers
Harder problems:
• Find an element out of N, who fulfils a certain condition:
max: N trials, average: N/2 attempts
√N
• Find factors of an integer N
trial and error: does 2 factor M ? does 3 ?,
… until √M (max):
~ √M attempts
Let M ~ 10L (L digit number),
then √M ~ (√10)L ~ exp(L) (exponentially hard)
 L3
Quantum computer  new scaling !!!
Shor 1994, Grover 1997, other ”single-output” problems
Shor’s algorithm
Find factor of stort large N
N=15
• Pick random a
• a=2
1=2, 22=4, 23=8,
20•=1,1,22,
• Let f(x) = ax mod N
4, 8, 1, 2 ..
1, 25=322, …
• f(x) is periodic, f(x+p)=f(x) 24•=16
p=4
• Determine numbers ap/2 ± 1 • 22 ± 1 = 3 or 5
• One of these numbers
• 3 factors15
factors N
5 factors 15
• Check ! (easy)
• Yes !
If you fail, try another a
Rydberg blockade and quantum information
Grover:
1) Σcx |x>  Σ (-1) f(x) cx |x>
(-1) if x matches the “marked“ x0
2) Inversion of cx about their mean.
x
Rydberg blockade and quantum information
Grover:
1) Σcx |x>  Σ (-1) f(x) cx |x>
(-1) if x matches the “marked“ x0
2) Inversion of cx about their mean.
~3/√N
1/√N
x
Repeat √N times
Quantum systems can explore the Hilbert space of superposition
states, and they can ”converge” on solutions to hard problems.
Must implement the calculations on a physics quantum system !
Re-cycle classical computing paradigm
• Computers represent information (data) in binary form
(bits),
• example.: 5 =1*22+0*21+1*20 =’101’
• All data manipulations are evaluations of functions based
on operations that decompose as single-bit and bit-pair
logical operations: NOT, AND, OR … .
|0>
|1>
NOT
|1>
|0>
Logical operations on quantum bits
• One-bit operation,
NOT:
01
must work ”without looking”
• Two-bit operation,
C-NOT:
(0, bit)  (0,bit)
(1, bit)  (1, NOT bit)
Qubits and gates
Orthogonal states: |0> and |1>
Orthogonal spin ½ states: |↑> and |↓>
Bloch sphere picture,
Unitary operations  rotations.
Transition: |0>  |1>
= rotation around x-axis
Phase:  a|0>+b e iφ |1>
= rotation around z-axis
Same operation can be built
In many different ways.
Basic building blocks
|0>  |0>+|1>
|1>|0> - |1>
H2 = NOT (cannot do that classically):
Hadamard gate:
|0>  |1>
|1>  |0>
Phase gate: |0>  |0>, |1>  e iφ|1>.
π/8 gate (T-gate):
|1>  e i π/4 |1>.
T and H do not commute: they span all rotations!
C-Phase/C-NOT: same operations, but only carried out of
”control-qubit” is in state |0>.
Universality
Classical computing: Gate alphabet, that allows
computation of any function (fan-out + NAND).
Surprising facts:
Hadamard, π/8 gate and C-NOT is quantum universal.
i.e. this set allows construction of any unitary operation on
full register. (operating on any qubit and qubit pair).
Hadamard, NOT and C-NOT gives no advantage for QC.
We do not know which quantum property is needed for
QC.
Quantum Error correction (Steane and Shor)
• Errors happen. Can we detect and correct them ?
• Classical approach: use copies and ”vote”:
0  000, 1  111, error: 001  000 by majority.
• In QM, copying is forbidden, and checking by measurements
collapse superposition states a|0>+b|1>  0> or |1>
• Solution: a small algorithm |0>  |000>, |1|111>
00: OK
01: NOT_3
10: NOT_2
11: NOT_1:
AARHUS
UNIVERSITET
Quantum Error correction (Steane and Shor)
• Errors happen. Can we detect and correct them ?
• Classical approach: use copies and ”vote”:
0  000, 1  111, error: 001  000 by majority.
• In QM, copying is forbidden, and checking by measurements
collapse superposition states a|0>+b|1>  0> or |1>
5-bit code can correct one flip or phase error
 multi-bit codes, topological states, gapped states
AARHUS
UNIVERSITET
How many errors can we correct?
L
N logical bits
N  N 10K physical
bits
G operations
NG = 10L
 Critical
parameter
surface
K
Error ε = 10- 3
NG=1010
AARHUS
UNIVERSITET
The five 9’s:
1-10-5=0.99999 succes
Quantum circuit and other models of QC
Quantum circuit:
Bits and sequence of gates
Cluster state computing:
Initial entangled grid of qubits.
Perform measurements only,
Program= measurement strategy: adapt observables.
AARHUS
UNIVERSITET
Dissipative quantum computing:
Gates  decay channels,
Stationary (dark) state solves problem.
QC is a great theoretical idea …
…
but it only works if we build one.
AARHUS
UNIVERSITET
Criteria and strategies for quantum computers ?
Carles Babbage
(1792-1871)
on atoms:
”… . Every atom, impressed with good and with ill,
retains at once the motions which philosophers and
sages have imparted to it, mixed and combined in ten
thousand ways with all that is worthless and base.”
Charles Babbage, Ninth Bridgewater Thesis, (1837).
Experiments by Drewsen, Aarhus.
7-bit quantum computer, 15=5*3 (i 2002)
many identical (natural) computers, majority vote
C11H5F5O2Fe
Solid state computers - gallery.
Quantum Dots (InAs/GaAs)
Atom chips
Hybrid technologies
”Self-hybridization”:
Major decoherence due to
coupling of electron spin to
nuclear spin bath.
 Identify and use the nuclear
spins (individual or collective)
as the qubits
Hybrid systems for quantum processing
Atomic systems
Atoms, ions, el. and nucl. spins
Solid, man-made, systems
Quantum dots, superconductors, cantilevers
Long coherence times
Couple weakly
Identical systems
Short coherence times
Couple strongly
Different (inhomogeneous env./prod.)
Scaling is difficult
Handling can be difficult
Scaling via microfabrication
Fixed in material
Hybrid quantum systems
Atomic systems
Atoms, ions, el. and nucl. spins
Solid, man-made, systems
Quantum dots, superconductors, cantilevers
Long coherence times
Couple weakly
Identical systems
Short coherence times
Couple strongly
Different (inhomogeneous env./prod.)
WORK
Scaling is difficult
SCALE
Scaling via microfabrication
Trapping is difficult
Fixed in material
 HYBRID 
Memory
Processing
Coupling is a problem:
•
•
•

Quite incompatible systems
Different natural frequencies
Bad spatial matching
Weak coupling strengths
Full hybrid technologies
Yale, Oxford, Zurich,
Santa Barbara,
Vienna, Saclay,
Chalmers, …
And what if we fail completely…
Niels Bohr
(about Quantum Mechanics):
” … if we should one day
wake up, and realize that
it had all been only a
dream, then I am
absolutely convinced that
we would still have
learned something !”