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Quantum StateProtection and Transfer
using Superconducting Qubits
Dissertation Defense of
Kyle Michael Keane
Department of Physics & Astronomy Committee:
Alexander Korotkov
Leonid Pryadko
Vivek Aji
June 29, 2012
Journal Articles1. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement
reversal,” Phys. Rev. A, 81, 040103(R), April 2010.
2. K. Keane and A. N. Korotkov, “Simple quantum error detection and correction for superconducting qubits,” arxiv:1205.1836, May 2012 (submitted to Phys. Rev. A).
1. “Decoherence suppression of a solid by uncollapsing,” Portland, OR, March 15-19, 2010), Z33.00011.
2. “Currently realizable quantum error correction/detection algorithms for superconducting qubits” Dallas, TX, March 21-25, 2011), Z33.00011.
3. “Modeling of a flying microwave qubit”Boston, MA, Feb. 27-March 2, 2012, Y29.00010
APS March Meeting Presentations
Posters1. “Theoretical analysis of phase qubits,” Quantum Computing Program Review (Minneapolis,
MN, 2009)
2. “Suppression of -type decoherence of phase qubits using uncollapsing and quantum error detection/correction,” Coherence in Superconducting qubits (San Diego, CA, 2010)
Outline
1. Introduction2. Decoherence by uncollapsing
Korotkov and Keane, PRA 2010
3. Repetitive N-qubit codes and energy relaxationKeane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012
4. Two-qubit quantum error “correction” and detectionKeane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012
5. Qubit state transferKeane and Korotkov, APS March Meeting, 2012
6. Summary
INTRODUCTIONLet’s begin with a basic
U
22
0 0 0cos2 2 2
IU
L
δ
δ
quantum variable
Superconducting Phase Qubitsstate
control
Iμw
flux bias
Ib
meas. pulse
Imeas
SQUID readout
Isq Vsq
25 mK
SQUID
flux bias
qubit
C I0 L
microwavesX-, Y-rotations
flux biasZ-rotations
operation
UΔU
|0
|1
State MeasurementSQUID-based Measurement:
lower barrierfor time t
U
relaxes
|1
|0
readout w/ SQUID
Tunneling Detected = state has been projected onto |1 and destroyedTunneling Not Detected = state has been projected onto |0
tunnels with rate
Γ 𝑡≫1
Weak Measurement
lower barrier for short time t
U
relax
|1
|0
readout w/ SQUID
Tunneling Not Detected = state projected onto|0 ORstate was |1 and didn’t have enough time to tunnel
tunnels with rate
There is a small change to the energy spacing during the lowering of the barrier
2 2
0 1 10 1
1
ie p
p
Γ 𝑡≈1
1 ( )te p t
Tunneling Detected = state has been projected onto |1 and destroyed
Uncollapsing
If tunneling does not occur, the qubit state is recoveredIn experiment, only data for cases where tunneling does not occur is kept
State Prepared
Doesn’t Tunnel Doesn’t Tunnel
Partial Measurement
Projects state toward 0 (was 1)
Partial Measurement
Projects state toward 0
π-pulse π-pulse
Zero-Temperature Energy Relaxation
11
1
1
/ 22 2 /
2 2 /
2 /
0 1 with probability
0 1
0 with probability 1
t Tt T
t T
t T
ee
e
e
This can be “unravelled” into discrete outcomes with probabilities
|0
|1
The population of the excited state moves into the ground state
DECOHERENCE SUPPRESSION BY UNCOLLAPSING
Project One
Korotkov and Keane, PRA 2010
Protection from Energy Relaxation
• Quantum Error Correction (Shor/Steane/Calderbank circa 1995)• Requires larger Hilbert space and controllable entanglement)
• Decoherence-Free Subspaces (Lidar 1998)• Requires larger Hilbert space and specfic subspaces
• Dynamical Decoupling (Lloyd and Viola 1998)• Does Not Protect Against Markovian Processes (Pryadko 2008)
Standard methods to protect against decoherence:
Our proposed method
• Simple modification of uncollapsing procedure• Our proposal was demonstrated in another system
• Requires selection of only certain cases• Similar to probabilistic QEC and linear optics QC
Ideal Procedure
storage period
11
Prepared
π-rotationPartial m
eas. (pu )
Partial meas. (p)
π-rotation
𝑒−𝑡 /𝑇1
time
axis of π-rotation
Initial value
Returned toinitial value
Similar protection for all density matrix elements
Korotkov and Keane, PRA 2010
Results
Yields a state arbitrarily close to initial
Some improvement even with naive uncollapsing strength
Korotkov and Keane, PRA 2010
Fide
lity
Measurement Strength (p)
Process with Decoherence
storage period t
11
Prepared
π-rotationPartial m
eas. (pu )
Partial meas. (p)
π-rotation
κ1
κ 𝑖≡𝑒−𝑡 𝑖 /𝑇 1
time
κ2κ3κ4
axis of π-rotation
Initial value
Pure dephasing and energy relaxation during entire process
Returned toinitial value
Korotkov and Keane, PRA 2010
Results
Pure dephasing uniformly decreases fidelity
Explains phase qubit uncollapsing experiment (Katz, 2008)
Still works with relaxation during operations
Perfect suppression requires small prob. of success
Korotkov and Keane, PRA 2010
Fide
lity
and
Prob
abili
ty
Measurement Strength (p)
Experimental Demonstration
Weak Measurementpolarization beam splitter, half wave
plate, and dark port
Optical Circuit Results
Nearly exact match to theory
Jong-Chan Lee, et. al., Opt. Express 19, 16309-16316 (2011)
Relaxationsimilar components, (except no dark port)
Protecting EntanglementInitially
entangledstate
Q1
Q2
WM π𝑻 𝟏 WM π
Entanglement is recovered
Q1
Q2
WM π𝑻 𝟏 WM π
Circumvents Entanglement Sudden Death
Same optics group did this extension experiment
Yong-Su Kim, et. al., Nature Physics, 8, 117-120 (2012)
Summary
• Does not require a larger Hilbert space• Modification of existing experiments in superconducting phase qubits• Demonstrated using photonic polarization qubit• Extended to protect entanglement
REPETITIVE CODING AND ENERGY RELAXATION
Project Two
Keane and Korotov, arxiv 2012
Motivation
|0 ⟩ |1 ⟩Bit Flip
|1 ⟩ |0 ⟩
A bit flip looks like a more difficult error
process than T1|0 ⟩ |0 ⟩
T1
|1 ⟩ |0 ⟩AND
Repetitive coding protects against bit
flips
PRO
TECT
S
????
????
?
THEREFORE…
Repetitive Quantum Codes and Energy Relaxation
|
|0N-1
tomography
T1(i)
X XAll “N-1” are 0: good Any in 1: either discard (detection) or try to correct (correction)
Encoding by N c-X gates
|
|0N-1 X
||0|0|0
||0
c-X gate (cNOT)
cNOT|
cNOT|
α | 0 |0 N −1+ β | 1 |0 N−1 →α | 0 N + β | 1 N
Syndrome Result
FAILS
Two-Qubit Encoding
syndrome
𝒔𝒊𝒏𝒈𝒍𝒆𝒒𝒖𝒃𝒊𝒕
|ψ ⟩|0 ⟩
Two qubitsEqual decoherence strength
(𝑝=1−𝑒−𝑡 /𝑇 1 )
𝐹 (ψ )=⟨ψ|ρ 𝑓𝑖𝑛𝑎𝑙|ψ ⟩
T1(i)
Keane and Korotov, arxiv 2012
Fide
lity
Decoherence Strength (p)
N-Qubit Error Detection|
|0N-1
tomography
T1(i)
X X All “N-1” are 0: keepAny in 1: discard
p
optimal, but 2 qubits are sufficient
Since the procedure works
𝐹 𝑎𝑣=∫ ⟨|ρ| ⟩𝑃𝑆
d |
~𝐹 𝑎𝑣=∫ ⟨|ρ|⟩ d |
∫ ⟨|𝑃𝑆| ⟩d |
Keane and Korotov, arxiv 2012
Fide
lity
Decoherence Strength (p)
ignore
detect
single
N-Qubit Error Correction|
|0N-1
tomography
T1(i)
X X All “N-1” are 0: keepAny in 1: cannot correct!
p
QEC is impossibleIn our paper we show
that no unitary operation can improve the fidelity
for p<0.5
Keane and Korotov, arxiv 2012
Fide
lity
Decoherence Strength (p)
ignore
correct
single
Summary
• Can be used for QED, but not for QEC of energy relaxation
• 3 qubits are optimal, but 2 qubits are sufficient
TWO-QUBIT QUANTUM ERROR DETECTION/CORRECTION
Project Three
Keane and Korotov, arxiv 2012
Two-Qubit Error “Correction”/Detection
0: good 1: either discard (only detection) or correct (if know which error)
Y/2 -Y/2
||0
tomography
X-correction needed
Y-correction needed
Z-correction needed
no correction needed (insensitive)
( )2 2YY
R
Notations:
= c-Z E1
E2
( | 0 | 1 ) | 0 cos ( | 0 | 1 ) sin ( || 1 | 0 ) |0 1i E1 = X-rotation of main qubit by arbitrary angle 2:
E1 = Y-rotation of main qubit:
E2 = Z-rotation of ancilla qubit:
( | 0 | 1 ) | 0 cos ( | 0 | | 01 ) sin ( | 1 | 0 1) |
( | 0 | 1 ) | 0 cos ( | 0 | 1 ) sin ( | 0 | 1| 0 ) | 1 E2 = Y-rotation of ancilla qubit:
( | 0 | 1 ) | 0 |( | 0 | 1 ) ( )|0 1ie
(| 00 | 01 ) (| 10 | 11 )
good
good
good
Keane and Korotov, arxiv 2012
Two-Qubit Error “Correction”/Detection
0: good 1: either discard (only detection) or correct (if know which error)
Y/2 -Y/2
||0
tomography( )
2 2YY
R
Notations:
= c-Z E1
E2
(| 00 | 01 ) (| 10 | 11 )
Various Decoherence Strengths
Fide
lity
Rotation Strength (2θ/π)
corr
det
ign
All Four Errors
Fide
lity
Rotation Strength (2θ/π)
corr
det
ign
Keane and Korotov, arxiv 2012
QED for Energy Relaxationstore in resonators
0: good 1: discard
Y/2 -Y/2
||0
tomography ( )2 2YY
R
Notations:
= c-Z T1
T1Y/2-Y/2
1100
QED of real decoherenceThe fidelity is improved
by selection of measurement result 0Fide
lity
Relaxation Strength
detect
ignore
Keane and Korotov, arxiv 2012
Almost “repetitive”
Summary
• QEC is possible for intentional errors• QED is possible for energy relaxation• Experiments can be done with superconducting
phase qubits
QUANTUM STATE TRANSFERProject Four
Keane and Korotov, APS 2012
System
Resonatoror
Phase Qubit
Transmission Line
Tunable CouplersHigh-Q Storage
Initiallyhere
Senthere
SuperconductingWaveguide
Tunable Inductance
Example from UCSB
(𝑪𝑫)=(𝑟 𝐿 𝑡𝑡 𝑟 𝑅
)(𝑨𝑩)
𝑨 𝑩
𝑪 𝑫
Tunable Parameter
1
𝑡×𝐴𝐴
𝑟 𝐿×𝐴 𝑀𝐿 𝐽
and are tuned byvarying
𝑟 𝐿
𝑡0
Korotkov, PRB 2011
Ideal ProcedureTr
ansm
issi
onCo
effici
ents
Qubitinitially is here
Qubittransferredto here
𝒕𝟏(𝒕<𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙
√𝟐𝒆− Δt / τ 𝐛𝐮−𝟏𝒕𝟐(𝒕<𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙
𝒕𝟏(𝒕>𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙
𝒕𝟐(𝒕>𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙
√𝟐𝒆+Δ t / τ 𝐛𝐮−𝟏
𝒕𝟏 𝒕𝟐Time (t)𝐭𝐦
𝒕𝒎𝒂𝒙
𝑡𝑚𝑎𝑥 ≈0.05
resonators,
𝜼=𝟎 .𝟗𝟗𝟗
Typical parameters
(UCSB)
Duration ns
ON/OFF
Desired Efficiency
Korotkov, PRB 2011
Main idea
AB
Transmission line Receiving resonator
PERFECT TRANSFER
( 𝑟1 𝑨+𝑡𝑩𝑡 𝑨+𝒓𝟐𝑩)=(
𝑟1 𝑡𝑡 𝑟 2
)(𝑨¿¿ 𝑩)“into line”
“into resonator”A B
𝒓𝟐𝑩
𝑡 𝑨
𝑡𝑩𝑟1 𝑨
τ 𝒃𝒖=τ𝒓𝒕 /|𝒕|𝟐= buildup time
resonator round trip time transmission coefficient
Korotkov, PRB 2011
Procedural RobustnessTr
ansm
issi
onCo
effici
ents
𝐭𝐦Time (t)
𝒕𝟐𝒕𝟏
�̇�=−|𝑡|2
2 τ𝑟𝑡𝐵+ 𝑡
τ𝑟𝑡𝐴
𝒕𝑩𝑨
𝒕𝟏(𝒕<𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙
√𝟐𝒆− Δt / τ 𝐛𝐮−𝟏𝒕𝟐(𝒕<𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙
𝒕𝟏(𝒕>𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙
𝒕𝟐(𝒕>𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙
√𝟐𝒆+Δ t / τ 𝐛𝐮−𝟏
𝒎𝒂𝒙
Keane and Korotkov, APS 2012
We vary the parameters in the above equations: , , and
η=𝐸 𝑓𝑖𝑛
𝐸𝑖𝑛
Shaping of ControlTr
ansm
issi
onCo
effici
ents
𝐭𝐦Time (t)
𝑡 2
𝑡1
shaping error efficiency loss
Robustness
No Problem! of 33.3 ns is 2.5 ns)
𝒎𝒂𝒙
Keane and Korotkov, APS 2012
Switching TimeTr
ansm
issi
onCo
effici
ents
𝐭𝐦𝟐 Time (t)
𝑡 2
𝑡1
𝐭𝐦
Vary and together Vary only
timing error efficiency loss
Robustness
No Problem!
ns
of 230 ns is 11.5 ns)
𝒎𝒂𝒙
Keane and Korotkov, APS 2012
Maximum Transmission CoefficientTr
ansm
issi
onCo
effici
ents
𝐭𝐦 Time (t)
𝒎𝒂𝒙
𝑡 2𝑡1
Vary only
Vary and together
amplitude error efficiency loss
Robustness
No Problem!(experiments have good control of tunable coupler)
Keane and Korotkov, APS 2012
Frequency Mismatch
ω1 ω2
ω1≠ω2
�̇�=−|𝑡|2
2 τ𝑟𝑡𝐵+ 𝑡
τ𝑟𝑡𝐴𝑒𝑖(𝜔1−ω2 )𝑡
resonator frequency
Frequency Mismatch
frequency error efficiency loss
Robustness
Requires Attention(resonator frequencies should be kept nearly equal throughout procedure)
Keane and Korotkov, APS 2012
Summary
• Robust to procedural errors (timing, shaping, maximum transmission coefficient)
• Requires active maintenance of nearly equal resonator frequencies
The second conclusion is very important for experiments — For the current solid-state tunable couplers
there is an effective frequency shift during modulation of the transmission coefficient
CLOSING REMARKSrecapitulation
Summary• Decoherence suppression by uncollapsing
– Probabilistically suppresses Markovian energy relaxation– After our proposal, it was demonstrated by another group– Extended in another experiment to entangled qubits
• N-qubit repetitive codes and relaxation– Can be used for QED, but not for QEC (2 qubits are sufficient)
• Two-qubit “QEC”/QED experiments– Can be performed with current technology
• Quantum state transfer– Robust against procedural errors– Requires resonator frequencies to be kept nearly equal
THANK YOU!
APPENDICESJust in case
Representations of Errors-Example: Energy Relaxation
11 01
10 11
1 1 1-
1- 1
p pD
p p
11 111
1t t
t T
00 11t tt t
01 011
1
2t t
t T
10 101
1
2t t
t T
0
0 0R
pK
1 0
0 1DRK
p
2 2
2 2
2
0 1 1, with probability 1
1
0 , with probability
pp
p
p
1
0
1
1
T
0 1
From the normalization requirement
Need to derive this from commutator!!!!!
Need to derive this from somewhere!!!!!
Solving these equations and combining into an operation
Choosing a specific operator sum decomposition
If you initially have a pure state, the classical mixture created by this process becomes explicit
† †R R DR DR R R R DR DR DRt K K K K P P LINK
1/1 t Tp e
This can be done for any operation however only some give physically meaningful interpretations
† †R R DR DRD K K K K
Master Equation RETURN
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