NMR Quantum Comppguting - qudev.phys.ethz.ch · Goals of this lecture Survey of NMR quantum...
Transcript of NMR Quantum Comppguting - qudev.phys.ethz.ch · Goals of this lecture Survey of NMR quantum...
NMR Quantum Computingp g
Slides courtesy of Lieven VandersypenThen: IBM Almaden, Stanford University
Now: Kavli Institute of NanoScience, TU Delft
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,
with some annotations by Andreas Wallraff.
How to factor 15 with NMR?
F FF
13
26perfluorobutadienyl
F
13C12C12C
F
F
13CFe
3
54
7p f y
iron complex
C5H5 COCO red nuclei are
qubits: F, 13C
2
Goals of this lecture
Survey of NMR quantum computing
Principles of NMR QCp QTechniques for qubit controlState of the artFuture of spins for QIPCFuture of spins for QIPCExample: factoring 15
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NMR largely satisfies the DiVincenzo criteria
Qubits: nuclear spins ½ in B0 field ( and as 0 and 1)
Quantum gates: RF pulses and delay timesQuantum gates: RF pulses and delay times
() Input: Boltzman distribution (room temperature)
Readout: detect spin states with RF coil1H
Readout: detect spin states with RF coil
Coherence times: easily several seconds 13CClCl
ClCl
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(Gershenfeld & Chuang 1997, Cory, Havel & Fahmi 1997)
Nuclear spin HamiltonianSingle spinSingle spin
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Nuclear spin HamiltonianMultiple spins
without qubit/qubitc pli gMultiple spins coupling
1H 500MHz
~ 25 mKchemical shiftsof the five F qubits
H13C15N
500126-51
25 mK
N19F31P
51470202
6qubit level separation
(at 11.7 Tesla)
Hamiltonian with RF fieldsingle-qubit rotationssingle qubit rotations
z x y
typical strength Ix, Iy : up to 100 kHzrotating wave approximation
7Rotating frame Lab frame
Nuclear spin HamiltonianCoupled spins J>0: antiferro mag.Coupled spins
coupling term
J f g
J<0: ferro-mag.
Typical values: J up to few 100 Hzyp p
16 configurations
8solid (dashed) lines are (un)coupled levels
Controlled-NOT in NMR A target bitB control bit
Before AfterA B A B
+ i
B control bit
+ i 2
+ 2
” flip A if B ”
YA Delay
2
wait
if spin B is
YA90 XA
90Delay1/2JAB
xy
z
xy
z
xy
z
xy
z
x x x x
z z z
if spin B is
z
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if spin B is x
yx
yx
yx
y
timedifferent rotation direction depending on control bit
Making room temperature spins look cold
(Cool to mK) Optical pumping
Effective pure state(Gershenfeld&Chuang Science ‘97
warm
p p p g DNP, …
(Gershenfeld&Chuang, Science 97, Cory, Havel & Fahmi, PNAS ‘97)
ner
gy
ner
gy
equalize population
low
en
hig
h en
cold
quasi cold
Look exactly like cold spins !
cold
warmquasi cold
10[Hz] [Hz]
Effective pure state preparation(1) Add up 2N-1 experiments (Knill,Chuang,Laflamme, PRA 1998)
+ + =Later (2N - 1) / N experiments (Vandersypen et al., PRL 2000)
prepare equal population (on average) and look at
(2) Work in subspace (Gershenfeld&Chuang, Science 1997)
deviations from equilibrium.
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compute with qubit states that have the same energy and thus the same population.
Read-out in NMR |0 |0
|0 |1|0 |1
Phase sensitive detection
|1 |0
Phase sensitive detectionY90z z
|0x
yx
|
positive signal for |0> (in phase)
|1 |1yY90
y
z
y
z
|1
12
xy
xy
negative signal for |1> (out of phase)
Measurements of single systems versus ensemble measurements
|00quantum state |00 + |11
single-shot bitwise |0 and |0 each bit |0 or |1
single-shot “word”wise |00 |00 or |11 QC
|0 and |0bitwise average each bit average of |0 and |1 NMR
“word”wise average |00 average of |00 and |11
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adapt algorithms if use ensemble
Quantum state tomography
after X
Look at qubits from different angles
after Yno pulse after X90
z z
after Y90
z
no pulse
xy
xy
xy
zz
xy
z
xy
z
xy
z
z zz
14
xy
xy
xy
Outline
Survey of NMR quantum computing
Principles of NMR QCTechniques for qubit controlE l f t i 15Example: factoring 15State of the artOutlook
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Off-resonance pulses and spin-selectivity
spectral content of a square pulse
off-resonant pulses induce eff. rotation in addition to z rotation in addition to x,y
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may induce transitions in other qubits
Pulse shaping for improved i l ti itspin-selectivity
gaussian pulse profile gaussian spectrumgaussian pulse profile gaussian spectrum
less cross-talk
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Missing coupling terms: Swap
How to couple distant qubits with only nearest neighbor physical couplings?
Missing couplings: swap states along qubit network
p y p g
di dSWAP12 = CNOT12 CNOT21 CNOT12 as discussed in exercise class
1 2 3 4
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“only” a linear overhead ...
Undesired couplings: refocusremove effect of coupling during delay times
bi
XB180 XB
180
opt. 1: act on qubit B
XA180XA
180 180180
opt. 2: act on qubit A
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• There exist efficient extensions for arbitrary coupling networks• Refocusing can also be used to remove unwanted Zeeman terms
opt. 2: act on qubit A
Composite pulsesExample: Y90X180Y90
corrects for corrects forcorrects for under/over-rotation
corrects for off-resonance
z z
yx y
x y
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However: doesn’t work for arbitrary input stateBut: there exist composite pulses that work for all input states
Molecule selection
A quantum computer is a known molecule.
spins 1/2 (1H, 13C, 19F, 15N, ...)
q pIts desired properties are:
spins 1/2 ( H, C, F, N, ...) long T1’s and T2’s heteronuclear, or large chemical shifts
d J li t k ( l k d) good J-coupling network (clock-speed)
stable, available, soluble, ... required to make spi s f sa e spins of same type addressable
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Quantum computer molecules (1)Grover / Deutsch-Jozsa
Cl NH H3
Q. Error correction
red nuclei are used
CCl
Cl
HO
CC
3
CH
red nuclei are used as qubits:
CCl
Cl
H
H
C
O
CH3
HOTeleportationLogical labeling / Grover Q. Error Detection
C CF F
C CH Cl
CHO
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BrF ClCl O-Na+
Quantum computer molecules (2)Deutsch-Jozsa 7-spin coherence
DD H
NC
D
CH F
D
CC
CC
HH3
O
ON
O
C
OH OH
C CF F
Order-findingC C
C CFF
F26
FFe(CH)5 (CO)2
Outline
Survey of NMR quantum computing
Principles of NMR QCTechniques for qubit controlE l f t i 15Example: factoring 15State of the artOutlook
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The good news
Quantum computations have been demonstrated in the lab
A high degree of control was reached, permitting hundreds of operations in sequence
A variety of tools were developed for accurate unitary control over multiple coupled qubits useful in other quantum computer realizations
Spins are natural, attractive qubits
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ScalingScaling
We do not know how to scale liquid NMR QCWe do not know how to scale liquid NMR QC
M i b t lMain obstacles:• Signal after initialization ~ 1 / 2n [at least in practice]
C h ti t i ll d ith l l i• Coherence time typically goes down with molecule size• We have not yet reached the accuracy threshold ...
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Main sources of errors in NMR QCQ
Early on (heteronuclear molecules)Early on (heteronuclear molecules) inhomogeneity RF field
Later (homonuclear molecules) J coupling during RF pulses
Finallydecoherencedecoherence
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Solid-state NMR ?
molecules insolid matrix
Yamaguchi & Yamamoto, 2000
Cory et al
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Electron spin qubitsp q
SL SR
Loss & DiVincenzo, PRA 1998Kane, Nature 1998
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Outline
Survey of NMR quantum computing
Principles of NMR QCTechniques for qubit controlE l f t i 15Example: factoring 15State of the artOutlook
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Quantum Factoring Find the prime factors of N: chose a and find order r
f (x) = a x mod N Results from number theory:f is periodic in x (period r)
Find the prime factors of N: chose a and find order r.
composite numbercoprime with N
f is periodic in x (period r)
gcd(a r/2 ± 1, N ) is a factor of N
Quantum factoring: find r
Quantum: ~ L 3 P Shor (1994)C l it f f t i
g
Quantum: ~ L 3 P. Shor (1994)Complexity of factoring numbers of length L: Classically: ~ e L/3
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Widely used crypto systems (RSA) would become insecure.
Factoring 15 - schematic| |0 |1 |22L 1
QFT
|x = |0 + |1 + ... + |22L-1
|x2 L bits |0
|k2L/rk
Interference
H
x a x mod N
QFT|
L=log2(15) |x | a x mod 15
|
|13 qubits
Quantum parallelism
4 qubits| | x
x0xx = ... + x2 22 + x1 21 + x0 20
ax = ... a a2 ax2 x1 x0 ...
...x a 2x a 1 x a 4
x1x2
a = 4, 11 amod 15 = 1 “easy” case
x ax a x a
period 2
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a = 2, 7, 8, 13 amod 15 = 1 “hard” casea = 14 fails
period 4
Quantum Fourier transform and the FFT
[ 1 1 1 1 1 1 1 1 ] [ 1 . . . . . . . ]
FFT
[ 1 . 1 . 1 . 1 . ]
[ 1 . . . 1 . . . ]
[ 1 ]
[ 1 . . . 1 . . . ]
[ 1 . 1 . 1 . 1 . ]
[ 1 1 1 1 1 1 1 1 ]
The FFT (and QFT) Inverts the period[ 1 . . . . . . . ]
[ 1 . . . 1 . . . ][ 1 1 ]
[ 1 1 1 1 1 1 1 1 ]
[ 1 . 1 . 1 . 1 . ][ 1 -i -1 i ]
p Removes the off-set
[ . 1 . . . 1 . . ][ . . 1 . . . 1 . ][ . . . 1 . . . 1 ]
[ 1 . -i . -1 . i . ][ 1 . -1 . 1 . -1 . ][ 1 . i . -1 . -i . ]
a = 11
|3 = |0 |0 + |1 |2 + |2 |0 + |3 |2 + |4 |0 + |5 |2 + |6 |0 + |7 |2= ( |0 + |2 + |4 + |6 ) |0 + ( |1 + |3 + |5 + |7 ) |2 after mod exp
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|4 = ( |0 + |4 ) |0 + ( |0 - |4 ) |2
p
after QFT
11 7 T l O f d d ti t t t b
Experimental approach 11.7 Tesla Oxford superconducting magnet; room temperature bore 4-channel Varian spectrometer; need to address and keep track of 7 spins
phase ramped pulsesp p p software reference frame
Shaped pulsesC t f t lk
12 34
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Compensate for cross-talk Unwind coupling during pulses
469.98 470 00 470 02 [MHz]
F13C12C
FF
Larmor frequencies 470 MHz for 19F
469.98 470.00 470.02 [MHz]1
3
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~25 mK
F13C12C
12CF
F13C
F
470 MHz for F 125 MHz for 13C
J - couplings: 2 - 220 Hz54
7 25 mK
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FC5H5 (CO)2
Fe coherence times: 1.3 - 2 s
Picture of the labPicture of the lab
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Thermal Equilibrium Spectranits
]Qubit 1
p• line splitting due to J-coupling• all lines present
art [
arb.
un
1 2 34 5
Rea
l pa
[Hz]
Qubit 2 Qubit 3uncoupled qubit states i di at d b a
rb. u
nits
]
rb. u
nits
]Qubit 2 indicated by arrows
eal p
art [
ar
eal p
art [
ar
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Re
Re
[Hz] [Hz]
Spectra after state initializationni
ts]
Qubit 1 state initializationar
t [ar
b. u
n
• only the |00 … 0 line remains• the other lines are averaged away
Qubit 1
RT spins appear cold!
Rea
l pa • the other lines are averaged awayby adding up multiple experiments
p pp[Hz]
rb. u
nits
]
rb. u
nits
]Qubit 2 Qubit 3
eal p
art [
ar
eal p
art [
ar
41[Hz] [Hz]
Re
Re
Pulse sequence (a=7)
/2 X- or Y-rotations (H and gates) i ( f i )
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> 300 pulses, 720 msX-rotations (refocusing)Z - rotations
“Easy” case (a=11)ni
ts]
321Ideal prediction
period 2
|000 0Qubit 1: |0
8 / r = 4r = 2
art [
arb.
un 321
|100 4
Ideal prediction
periodup |0 gcd(112/2 - 1, 15) = 5
gcd(112/2 + 1, 15) = 3Rea
l pa
Experiment
15 = 3 x 5[Hz]
recall
Qubit 3:rb. u
nits
]
rb. u
nits
]
eread-out scheme
Qubit 2:up |0
up/down |0,|1
eal p
art [
ar
mag
par
t [a
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Re
Im
[Hz] [Hz]
“Hard” case (a=7)ni
ts]
|000 0
period 4
8 / r = 2r = 4
art [
arb.
un |000 0
|010 2|100 4|110 6Qubit 1:
up |0
period
gcd(74/2 - 1, 15) = 3gcd(74/2 + 1, 15) = 5R
eal p
a up |0
15 3 x 5Qubit 3:
[Hz]
rb. u
nits
]
rb. u
nits
] up/down |0,|1
Qubit 2:/d |0 |1
mag
par
t [a
mag
par
t [aup/down |0,|1
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Im Im
[Hz] [Hz]
decoherence
Model quantum noise (decoherence)Spins interact with the environment
Decoherence
The decoherence model for 1 nuclear spin is well-described.
phaserandomization
energyexchange
We created a workable decoherence model for 7 coupled spins.The model is parameter free
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The model is parameter free.
Simulation ofnits
]
fundamental limit
Simulation of decoherence (1)
art [
arb.
un
fundamental limit
Rea
l pa
hard casedecoherence can be
[Hz]decoherence can be
understood and modeled
rb. u
nits
]
rb. u
nits
]
mag
par
t [a
mag
par
t [a
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Im Im
[Hz] [Hz]
Simulation ofnits
]
Simulation of decoherence (2)
fundamental limitart [
arb.
un
fundamental limit
Rea
l pa
easy casedecoherence can be
[Hz]decoherence can be
understood and modeled
rb. u
nits
]
rb. u
nits
]
eal p
art [
ar
mag
par
t [a
47
Re
Im
[Hz] [Hz]