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Transcript of David G. Cory Department of Nuclear Engineering Massachusetts Institute of Technology Using Nuclear...
David G. CoryDepartment of Nuclear
EngineeringMassachusetts Institute of
Technology
Using Nuclear Spins forQuantum
Information Processing
and Quantum
Computing
Quantum Information Processing
• The precise control of a set of coupled
2-level systems.
HHintint==IIIIzz
interaction with B fieldinteraction with B field
E
|0 >
|1 >
E
| >
| >
• Qubit can be in a continuum of states
• Most states are superpositions of 0 and 1
0
1
“0 and 1”qubit spin
Addressable Qubits• Chemically distinct spins
HHintint==IIIIzz++SSSSzz+2πJIzSz
interaction with B fieldinteraction with B field
I S
2-3 Dibromothiophene
coupling between spinscoupling between spins
JIS
External Hamiltonian
– Experimentally Controlled Hamiltonian:
– Total Hamiltonian:
HHextext(t)(t) ==RFxRFx(t)(t)··(I(Ixx+S+Sxx)+)+RFyRFy(t)(t)··(I(Iyy+S+Syy))
HHtotaltotal(t)(t)
controlled viacontrolled via
HHextext(t)(t)
I SJIS
9.6 T
RF wave
spins couple to RF fieldspins couple to RF field
HHtotal total (t)(t) = H= Hintint + H + Hextext(t)(t)
Single Qubit Gates
I S
qubit selectiveinversion pulse
Conditional Qubit Gates
I S
selectiveπ/2
coupling
selectiveπ/2
Quantum & Classical ChannelQuantum & Classical Channel
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
Partial Trace
System
Environment
0.95 0.99
0.98 1.000.96
0.99
Decoherence Free Subspacephase noise:
φjeφje−
Collective phase noise:φje
φje−
φjeφjeφje−
φje−
φjeφje−
φ2je
φ2je−
EncodeEncode
Logical Logical QubitQubit
DFS for Memory
30 60 900.4
0.6
0.8
1
Info
rma
tio
n
Noise strength (Hz)
Encoded
Un-Encoded
EngineeredEngineered
NoiseNoise
Encode Decode
Samples and Hamiltonians
Alanine (3 Qubits)Alanine (3 Qubits) Crotonic Acid (4 Qubits)Crotonic Acid (4 Qubits)
C4C1
C2
C3
J12
J23
J34
∑∑∑= <=
+=n
k kl
lz
kzkl
kz
n
kk IIJIH
11int 2πω
C3C1 C2
J12 J23
J13
CN
N
C
C
CC
C
O–
ND3+
D H
H
H H O
H
Control of EntanglementControl of Entanglement
( )1110002
1+=GHZ
X Measurement C=0.92
Z Measurement C=0.89
GHZ StateC=0.88
Traced state C=0.71 Traced state
( )1101010113
1++=W
Z Measurement C=0.80
X Measurement C=0.77
W StateC=0.73
Pseudopure state - Product of two Singlet states
(real part of the density matrix)
Strong Measurement
Entangle bits 1 & 2, and bits 3 & 4
Map bits 2 & 3 onto the Bell basis
H
Measurement of bits 2 & 3
|01> + |10>
|01> + |10>
Final Results – After Selective Strong Measurement in the Bell basis
n (number of H-CNot pairs)
Fin
al C
orre
lati
on
C
C
H
0
0
0 H
H( )n
H
|000 GHZ|000 for n = 0, 8, 16, …|100 for n = 4, 12, 20, ...
Output for n = 128
Correlation: 96.65%
Correlation: 90.89%
Output for n = 64
16
Output state
Quantum Fourier Transform
• Shor’s algorithm
• Quantum simulations
• Quantum chaos
Input state
QFT SuperoperatorFidelity = .99 Fidelity = .80
QFT Superoperator
Theoretical QFT Superoperator
Experimental QFT Superoperator
Statistical Verification of Control
Q =2−2nq
Tr ρi2
[ ]i=1
nq
∑ = number of qubits
reduced density matrix for qubit i
qn
iρ
m = 16 (+) m = 24 (.) m = 32 (x) m = 40 (o)
Inset: average Q approaches CUE average exponentially.
Random Circuit on nq = 8 qubits
= 4, 6, 8, 10.
P(Q)
0.999
0.99
0.9
Gat
e F
idel
ity
0.9999
RF Power (Hz)
Strongly Modulated Pulses
modulation frequency
add
ress
abil
ity
Why is quantum noise so bad?Why is quantum noise so bad?Consider an entangled state:
If any are disturbed, they all collapse
How do we protect our information?How do we protect our information?
To make matters worse:•Can’t Copy•Can’t Look No Majority Coding
Quantum Error CorrectionIf the noise is sufficiently weak compared with your control rates then you correct a subset of errors with only a finite accuracy requirements.
Encoder
Memory
Extra Bits
RegularlyCorrect Error
Decoder
Discarded Bits
Arbitrary Collective NoiseExpand your state space in the joint eigenstates of J2 and Jz
€
J =J32( ) ⊕ J
12( ) ⊗ J
12( )⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
J can’t distinguish between the 2 paths to the (1/2) state.
(1/2)1 (1/2)2
(0)12 (1/2)3
(1)12 (1/2)3
(1/2)123
(1/2)123
(3/2)123
Encode 22
22
==expexp((--i2i2ππ/3)/3)
Encode
λλ
λλ
zz
zz
0 10 20 300.4
0.6
0.8
1
Noise Strength (Hz)Noise Strength (Hz)
Encoded, Y, Z Noise
No Encoding, No Encoding, YY Noise Noise
Info
rma
tion
Info
rma
tion
Weak NoiseWeak Noise
Experimental Results
Strong Noise Limit
Z-X Noise 0.24Un-Encoded
0.70NS-Encoded
No Noise0.700.70
Z-X NoiseZ-Y Noise
Info
Nuclear Spins in the Solid StateM = -N/2
N/2
N/2 -1
N/2 -2
-N/2 +1
-N/2 +2
0Liquid state is a good test-bed for QIP,
not a scalable approach to QC.
Solid State appears to be scalable.
Spin Hamiltonian
Zeeman Hamiltonian + +
= B0
HZ = hωIz
i
i∑
HD = hωD
ij 3Izi Iz
j −I i ⋅ I j( )
i<j∑
Secular Dipolar Hamiltonian
Htot = HZ + HD
>> Dij)
• use chemistry locally for error correction
• use spatial addressing to define qubits (magnetic field gradients)
++
• dipolar to nearest neighbor coupling
• single spin detection
++
The selective decoupling problem• Consider a system consisting of pairs of spins.
• nearest neighbor coupling is well defined (d), • there is a quasi-continuous broadening that arises from coupling
to distant spins (D ) • Typically d >> D.
increasing D decreasing Dd is fixed
dij =
γ2h
rij3 3cos2θij −1( )
Dipolar -> nn
€
˜ H D(t)=h2
dij σziσz
j +σyiσy
j −2σxi σx
j( )
i<j∑ +
3h2
dij
σziσz
j −σyiσy
j( )cos
2ω1
ωmsinωmt
⎛
⎝ ⎜
⎞
⎠ ⎟
−σziσy
j +σyiσz
j( )sin
2ω1
ωmsinωmt
⎛
⎝ ⎜
⎞
⎠ ⎟
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
⎫
⎬
⎪ ⎪
⎭
⎪ ⎪
i<j∑
The RF Hamiltonian is
€
HRF
( t ) = h ω1
cos ωm
t σx
and the corresponding RF propagator is
€
URF
( t ) = exp − i
ω1
ωm
sin ωm
t σx
⎛
⎝
⎜
⎞
⎠
⎟
The interaction frame dipolar Hamiltonian is time-dependent
Status of NMR QIP
The functions J0(x) (blue) and H0(x) (red).
€
˜ H D(t)=h2
dij σziσz
j +σyiσy
j −2σxi σx
j( )
i<j∑ +
3h2
dij
J 02ω1
ωm
⎛
⎝ ⎜
⎞
⎠ ⎟ σz
iσzj −σy
iσyj
( )
−H02ω1
ωm
⎛
⎝ ⎜
⎞
⎠ ⎟ σz
iσyj +σy
iσzj
( )
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
⎫
⎬
⎪ ⎪
⎭
⎪ ⎪
i<j∑
The zero-order average Hamiltonian is,
where J0(x) and H0(x) are the zeroth order Bessel and Struve functions
Starting from IxA
without control (left) and with control (right)
BLUE: IxA; BROWN: Ix
B; BLACK: IxC
0 0.2 0.4 0.6 0.8 1.02
1.5
1
0.5
0
0.5
1
1.5
2
fraction of cycle time for weaker coupling0 0.2 0.4 0.6 0.8 1.0
0.5
0
0.5
1
1.5
2
fraction of cycle time for weaker coupling
BC coupling = 1/8 AB coupling
Starting from IxA
without control (left) and with control (right)
Quantum Simulation, Spin Diffusion
€
Hdiff = bi,j(σ i+σ j
−+σ i−σ j
+)i,j∑
spin-spin correlation time ~ 6 µs
diffusion time 10 -> 100 s
steps ~ 10^8
mean displacement ~ 1 µm
# spins involved 10^11
To connect with theorists combine this
with nn coupling scheme.
Diffusion measurements
k2s/cm2)
k2scm2)
(~10-12 cm2/s)D=D|| [001] [111]
Zhang et al. (T1 = 114-157 s) 7.14 ±0.52 5.31±0.34
Boutisetal.(T1=256-288s) 6.4±0.9 4.4±0.5
29±3 33±4
k2s/cm2)
k2s/cm2)
1 single spin:Result of a quantum computation
Set of N spins:Collective measurement
Transfer of polarization Transfer of polarization single spin single spin transducer spins. transducer spins.
The final state of transducer spins is determined by the The final state of transducer spins is determined by the
state of the controlling (single) spinstate of the controlling (single) spin
Spin TransducerSpin Transducer
1H19F
Global CNOT
• Ideal behavior:
|0>0|00000…0> |0>0 |00000…0>1-N
|1>0|00000…0>1-N |1>0 |11111…1>1-N
(initial state) (final state)
|0>
|2>|1>
|N>
…… Tra
ce
Mea
sure
men
t
Series of Cnots
Requisites on Control
Addressable spins.
Interactions w/ single spin
Control operator
Entanglement none
Gates or basic steps # for max
Contrast
Gate: CNOT
n
Final Contrast 2
−
=
+ +∏ 01
0 EE xi
n
i
σ −
=
+ +∏ 01
0 EE xi
n
i
σ( )
−−
=
+
=
−
=
+
=
+
+⎭⎬⎫
⎥⎦
⎤⎢⎣
⎡+−
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+−
∏∏
∏∏
022
1
2210
2/2
12/2
Ei
EEE
i
n
ii
n
i
z
i
n
ii
n
i
x
σσσ
σ−
=
+ +∑∏ 0}{
2/
10 EE
i
xi
n
i
σ
Maximum Entanglement Scheme
ρ0|0 00..0> |>(|1 00..>+|0 00..>)/ 2 |>(|1 10..>+|0 00..>)/ 2
|> (|1 11..>+|0 00..>)/ 2 |1>(|0 11..>+|1 00..>)/ 2 |1>(|0 11..>+|1 01..>)/ 2
|1> (|0 11..>+|1 11..>)/ 2 |1>(|0 11..>+ |1 11..>- |1 11..>+|0 11..>)/2 =
ρfin |1> |0 111111...1> vs. |0> |0 00000..0>
U1=UHAD U2=UCNOT12*… UCNOT1N U3= UCNOT01 U4=U2-1 U5=U1
-1=UHAD
|0>
|2>|1>
|N>
H ………… H
Mea
sure
men
t
UU11
UU22
UU33
UU44
UU55
Series of CnotsEntanglement
w/ CnotsEntanglement w/
MQCPerturbative
approach
Requisites on Control
Addressable spins.
Interactions w/ single spin
Addressable spins.
Only one interaction w/ single spin
• Collective control
• Refocusing of the control operator
• Collective control
Control operator
Entanglement none Cat State Cat StateGround State of
Heisenberg Hamiltonian
Gates or basic steps # for max
Contrast
Gate: CNOT
n
Gate:CNOT
2n-1Sequence: DQ &
Dipolar Ham:
2n +1CNOT
Sequence: DQ1 & Dipolar
Hamiltonian: ~n
Final Contrast 2 2 ~1 ~1
−
=
+ +∏ 01
0 EE xi
n
i
σ −
=
+ +∏ 01
0 EE xi
n
i
σ( )
−−
=
+
=
−
=
+
=
+
+⎭⎬⎫
⎥⎦
⎤⎢⎣
⎡+−
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+−
∏∏
∏∏
022
1
2210
2/2
12/2
Ei
EEE
i
n
ii
n
i
z
i
n
ii
n
i
x
σσσ
σ−
=
+ +∑∏ 0}{
2/
10 EE
i
xi
n
i
σ
Creation of Cat State
With the N-quantum Grade Raising operator:
the pure state is transformed into the cat state:
( )∏∏ −+ +=k
k
k
kNGR bH σσ)(
2
111000000
)(4 KK
K+
=− N
GRHbie
π
Series of CnotsEntanglement
w/ CnotsEntanglement w/
MQCPerturbative
approach
Requisites on Control
Addressable spins.
Interactions w/ single spin
Addressable spins.
Only one interaction w/ single spin
Collective control
Refocusing of the control operator
• Collective control
Control operator
Entanglement none Cat State Cat StateGround State of
Heisenberg Hamiltonian
Gates or basic steps # for max
Contrast
Gate: CNOT
n
Gate:CNOT
2n-1Sequence: DQ &
Dipolar Ham:
2n +1CNOT
Sequence: DQ1 & Dipolar
Hamiltonian: ~n
Final Contrast 2 2 ~1 ~1
−
=
+ +∏ 01
0 EE xi
n
i
σ −
=
+ +∏ 01
0 EE xi
n
i
σ( )
−−
=
+
=
−
=
+
=
+
+⎭⎬⎫
⎥⎦
⎤⎢⎣
⎡+−
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+−
∏∏
∏∏
022
1
2210
2/2
12/2
Ei
EEE
i
n
ii
n
i
z
i
n
ii
n
i
x
σσσ
σ−
=
+ +∑∏ 0}{
2/
10 EE
i
xi
n
i
σ
• Limited number of spins.
• Simulated all the pulse sequences needed for
the wanted propagators,
• varying the # of repetitions of the cycle.
Quantum Transducer• Model:
– linear chain of spins.
– all the couplings are taken into account.
Results: Entanglement Scheme
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
# of repetitions
Entanglement of the 1st SpinGlobal Entanglement
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
# of repetitions
Contrast
• Simulations (8 spins)
Global EntanglementEntanglement of 1st
spin
1
Conclusions• Liquid state NMR is a useful QIP test-bed ~ 8 qubits.• Systems of dipole coupled spins are universal for QIP.• Solid state, nuclear spin approaches appear to be scalable.• For selected problems we are already beyond the
capabilities of classical computing.• Introduced a quantum transducer that uses entanglement to
make a classical measurement that could not otherwise be realized.
Dr. Timothy HavelProfessor Seth LloydProfessor Raymond LaflammeDr. Chandrasekhar RamanathanDr. Chandrasekhar RamanathanDr. Joseph EmersonDr. Joseph EmersonDr. Grum TeklemariamDr. Marco Pravia Dr. Evan Fortunato Dr. Greg Boutis Dr. Yaakov WeinsteinNicolas BoulantPaola Cappellaro Zhiying (Debra) ChenHyung Joon ChoDaniel GreenbaumJonathan HodgesSuddhasattwa Sinha