PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON …
Transcript of PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON …
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PROTECTING QUANTUM SUPERPOSITIONS
IN JOSEPHSON CIRCUITS
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Acknowledgements to Yale quantum information team members:
M. Hays
S. Mundhada
A. Narla
K. Serniak
V. Sivak
K. Sliwac
S. Touzard
R. Heeresd
Y. Liu
W. Pfaff
N. Ofek
U. Vool
C. Wangf
Z. Wang
G. de Lange
A. Grimm
A. Kou
Z. Leghtasb
M. Hatridgea
S. Shankar
L. Frunzio
S. Girvin
L. Glazman
L. Jiang
M. Mirrahimi
R. Schoelkopf
PROTECTING QUANTUM SUPERPOSITIONS
IN JOSEPHSON CIRCUITS
C. Axline
J. Blumoff
K. Chou
Y. Gao
A. Petrenkoc
P. Rheinhold
A. Royf
Now at: aU. Pittsburgh, bENS Paris, cQCI, dCEA Saclay, fU. M. Amherst, fU. Aachen
Sponsors:
Michel Devoret, Yale University
Capri Spring School 2017
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LECTURE I: INTRODUCTION AND OVERVIEW
PROTECTING QUANTUM SUPERPOSITIONS
IN JOSEPHSON CIRCUITS
LECTURE II: CAT CODES
LECTURE III: PARITY MONITORING AND CORRECTION
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PROTECTING CLASSICAL INFORMATION
1
0
0 1
ONE BIT OF INFORMATION
ENCODED IN A SWITCH
ENERGY BARRIER
PROTECTS INFORMATION
energy of switch
switchposition
BU k T
error expB
U
k T
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ELEMENT OF QUANTUM INFORMATION: QUBIT
X
Y
Z
MUST PRESERVE ALL SUPERPOSITIONS, I.E. THE ENTIRE SPHERE
Bloch sphere
cos 0 exp sin 12 2
i
0 1
Two basis quantum states:
Generic superposition:
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CARDINAL POINTS OF THE BLOCH SPHERE
X
Y
Z
1
1
1
1
1
1
Z
Z
X
X
Y
Y
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DECOHERENCE:
LOSS OF
INFORMATION
X
Y
"T1" "Tf"
Z
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BY CAREFUL ENCODING IN A LARGE ENOUGH SPACE,
QUANTUM INFORMATION CAN BE PROTECTED
ELEMENTARY ERRORS E ONLY MOVE OCTAHEDRON ALONG A 3+nth DIMENSION, WITHOUT CHANGING ITS VOLUME
X Y
Z
E
SIMPLE EXAMPLE: IN GENERAL:
cos e sin2 2
i
X AND Y COMPONENTS IN M = 0
MANIFOLD ARE PROTECTEDFROM NOISE IN COMMON MODE
MAGNETIC FIELD B
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3 STRATEGIES FOR PROTECTING QUANTUM INFORMATION
- Encode into a quantum cavity resonator and monitor photon number parity
- Encode into a register of qubits, monitor multi-qubit parities
Examples: Steane & surface codes
Example: cat codes
- Encode using defects in topological quantum matter
Example: Majorana fermions
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1
1
1
1
1
1
1
0
1
1
1
0
0
0
1
1
0
0
0
0
1
0
1
0
0
1
1
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
0
1
0
1
1
1
0
1
0
0
1
0
+ + + +
+++
0
0
0
0
0
0
1
0
0
0
1
1
1
0
0
1
1
0
1
0
1
1
0
0
1
0
1
1
0
0
1
1
1
1
0
1
0
1
0
1
1
1
0
1
0
0
0
1
0
1
1
0
1
+ + + +
+++
0
1
1
0L
1L
REGISTER ERROR CORRECTION: STEANE CODE
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0 1 2e e eX
0 1 2e e eX
0 1 2e e eX
0 1 2e e eX
0 1 2e e eX
0 1 2e e eX
0 1 2e e eX
3 4 5e e eZ
3 4 5e e eZ
3 4 5e e eZ
3 4 5e e eZ
3 4 5e e eZ
3 4 5e e eZ
3 4 5e e eZ
STEANE CODE ERROR CORRECTION PROTOCOL
i = 0Z Z Z
Z Z
Z Z
ZZ
Z
Z
Z
X X X
X X
X X
XX
X
X
Xi = 6
ej
j = 0 j = 5
qi
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in cosA t A t
out cosA t A t f
SUPERCONDUCTING QUBIT PARADIGM:
CONSIDER A SINGLE, DRIVEN ELECTROMAGNETIC MODE...
0
0 DRIVE SIGNAL
REFLECTED SIGNAL
MICROWAVE FREQUENCIES0 Bk T
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Hydrogen atomSuperconducting
LC oscillator
ATOM vs CIRCUIT
L C
metallic latticepositive ions
electron condensate
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Hydrogen atomSuperconducting
LC oscillator
L C
ATOM vs CIRCUIT
metallic latticepositive ions
electron condensate
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Hydrogen atomSuperconducting
LC oscillator
L C
unique electron → whole superconducting condensate velocity of electron → current through inductorforce on electron → voltage across capacitor
ATOM vs CIRCUIT
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Hydrogen atomSuperconducting
LC oscillator
L C
unique electron → whole superconducting condensate velocity of electron → current through inductorforce on electron → voltage across capacitor
ATOM vs CIRCUIT
1mm0.1nm
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+Q
F
-Q
F
E
LC CIRCUIT AS A QUANTUM
HARMONIC OSCILLATOR
r
L C
ˆˆ ,Q i F
F = LI
Q = CV
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F
F
WAVEFUNCTIONS OF LC CIRCUIT
rI
E
YF
Y0
0Y1
In every energy eigenstate,(microwave photon state)current flows in opposite directions simultaneously! F
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F
F
EFFECT OF DAMPING
important: as little
resistance as possible dissipation broadens energy levels
E
11
2 2n r
r
iE n
RC
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F
F
CAN PLACE CIRCUIT IN ITS GROUND STATE
r Bk T10-5 GHz 10mK
rI
E
some cold dissipation isactually needed: provides
reset of circuit!
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UNLIKE ATOMS IN VACUUM,
TWO CIRCUITS EASILY COUPLE
THROUGH THEIR WIRES:
COUPLING CAN BE ARBITRARILY STRONG!
HAMILTONIAN BY DESIGN!
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F
F
E
CAVEAT: THE QUANTUM STATES OF A
PURELY LINEAR CIRCUIT
CANNOT BE FULLY CONTROLLED!
NO STEERING TO AN ARBITRARY STATE
IF SYSTEM PERFECTLY LINEAR
r
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Potential energy
Position coordinate
NEED NON-LINEARITY TO FULLY
REVEAL QUANTUM MECHANICS
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Potential energy
Position coordinate
NEED NON-LINEARITY TO FULLY
REVEAL QUANTUM MECHANICS
Emissionspectrum@ high T
frequency
01122334
0
1
2
3
4
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Potential energy
Position coordinate
NEED NON-LINEARITY TO FULLY
REVEAL QUANTUM MECHANICS
Emissionspectrum
frequency
01122334
absolutenon-linearity
is ratio of peakdistance topeak width
0
1
2
3
4
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JOSEPHSON TUNNEL JUNCTION:
A NON-LINEAR INDUCTOR
WITH NO DISSIPATION
1nmS
I
S
superconductor-insulator-
superconductortunnel junction
CJLJ
100 -1000 nm
MIGHT BE SUPERSEDED SOME DAY BY ANOTHER DEVICE(BASED ON ATOMIC POINT CONTACTS, NANOWIRES, ???)
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JOSEPHSON TUNNEL JUNCTION:
A NON-LINEAR INDUCTOR
WITH NO DISSIPATION
1nmS
I
S
superconductor-insulator-
superconductortunnel junction
f
II f / LJ
0 0sin /I I f f
CJLJ
I
' 't
V t dtf
02e
f
0
0
JLI
f
0I
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LECTURE I: INTRODUCTION AND OVERVIEW
PROTECTING QUANTUM SUPERPOSITIONS
IN JOSEPHSON CIRCUITS
LECTURE II: CAT CODES
LECTURE III: PARITY MONITORING AND CORRECTION
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1nmS
I
S
superconductor-insulator-
superconductortunnel junction
f
0cos /JU E f f
CJLJ
I
' 't
V t dtf
02e
f
2
0J
J
LE
f
bare Josephson potential
2 JE
JOSEPHSON TUNNEL JUNCTION:
A NON-LINEAR INDUCTOR
WITH NO DISSIPATION
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Potential energy
Position coordinate
ELECTROMAGNETIC OSCILLATOR ENDOWED WITH JOSEPHSON
NON-LINEARITY
Emissionspectrum@ high T
frequency
01122334
0
1
2
3
4
Kerr frequency
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QUANTUM STATES OF A HARMONIC OSCILLATOR
REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE
I
Q
vacuum: |n=0>
I
Q
squeezed vacuum
I
Q
coherent state
I
Q
cat state
I
Q
1 photon: |n=1>
2
2
†
; 1a a
W D PD P
I iQ
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I
Q
vacuum: |n=0>
I
Q
squeezed vacuum
I
Q
coherent state
I
Q
cat state
I
Q
1 photon: |n=1>
2
2
QUANTUM STATES OF A HARMONIC OSCILLATOR
REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE †
; 1a a
W D PD P
I iQ
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0L
1L
i i
' '
' 'i i i
CAT ENCODING, AND INFORMATION DECAY
even
even
odd
odd
typical
time
evolution
3
a
1t
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BLOCH SPHERE OF
4-LEG CAT CODE
Leghtas et al. PRL (2013)Vlastakis et al. Science (2013)Mirrahimi et al. NJP (2014)Sun et al. Nature (2014)
+Z
+X
+Y
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TWO TYPES OF CAT ERROR MUST BE ADDRESSED:
- COHERENT STATE SHRINKING, DEPHASING
& DEFORMING
- PHOTON NUMBER PARITY JUMPING
HOW DO WE FULLY CONTROL THE RESONATOR?
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DEVICES
2 mm
0.5 mm
1.2
cm
90 μm3.4 cm
N. Bergeal et al. Nature (2008)
H. Paik et. al. PRL (2011)
G. Kirchmair et. al. Nature (2013)
JosephsonRing Modulator
Al
Si
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EXPERIMENTAL SETUP
instrument rackdilution refrigerator
microwave
generators
atomic clock
arbitrary
waveform
generator
acquisition
card
77 K
100 mK
1 K
4 K
quantum
circuits
20 mK
0.5 m
amplifier
electronics
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sL
2 OSCILLATORS COUPLED
THROUGH A TUNNEL
JOSEPHSON JUNCTION
TE101 modes
E
storagedump
&
readout
junction
,
0
, ,r s r s r sL C 2 2 2 2
0cos /2 2 2 2
s s r rJ r s
s s r s
Q QH E
C L C Lf
F F F F
sC rLrC
SIMPLIFIED MODEL
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A POWERFUL DETERMINISTIC GATE
COMBINING MICROWAVE LIGHT AND A QUBIT
cavity oscillator
superconducting qubit
†
zH a a
1
2c q T
strong dispersive regime: Stark shift per photon > 1000 linewidths
Can combine two cavities, a qubit and a quantum limited Josephson amplifier:
Sstorage cavity
superconducting qubit
readout cavity
S
R
Leghtas et al., Phys. Rev. A87, 042315 (2013)
amplification chain
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s
2 OSCILLATORS COUPLED
THROUGH A TUNNEL
JOSEPHSON JUNCTION
TE101 modes
E
storagedump
&
readout
junction
rq
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2 OSCILLATORS COUPLED
THROUGH A TUNNEL
JOSEPHSON JUNCTION
† † †/ s s s r r r q q qH a a a a a a -cqq
2aq
†( )2
aq( )2
- cqsaq
†aqas
†as - cqraq
†aqar
†ar + ...
TE101 modes
E
storagedump
&
readout
junction
-crs
4as( )
2ar
†( )2
+ h.c.+ ...
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NON-DEMOLITION MEASUREMENT
OF SUPERCONDUCTING ATOM
Readout phase
f
90º
-90º
fc
resonator linewidth ~ c
q ~ p /2
Readout amplitude
f
fc
dispersive shift c
JPC
e
e
g
g
HEMT-amp
I
Q
2 2I Q
1tan /Q I
Circuit QED: Wallraff et al. (2004),
Houck et al., (2007), Gambetta et al. (2008),
Vijay et al., (2012).
ref
readout
frequency
qubit + resonator
Josephson
pre-amp
Cavity QED: Pioneering expts in
Haroche's and Kimble's groups
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0
5
-5
0 50 100 150Time (ms)
I m/σ
7.5σ
~ 90% of fluctuations ofsignal at 300K are quantum noise!
~2000 bits per qubit lifetime
QUANTUM JUMPS OF QUBIT STATE
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TWO TYPES OF CAT ERRORS MUST BE ADDRESSED:
- COHERENT STATE SHRINKING, DEPHASING
& DEFORMING
- PHOTON NUMBER PARITY JUMPING
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REVIEW THE USUAL DRIVEN-DAMPED
HARMONIC OSCILLATOR
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REVIEW THE USUAL DRIVEN-DAMPED
HARMONIC OSCILLATOR
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x
p
REVIEW THE USUAL DRIVEN-DAMPED
HARMONIC OSCILLATOR
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x
p
REVIEW THE USUAL DRIVEN-DAMPED
HARMONIC OSCILLATOR
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H = e1
*a + e1a†
k1a
Hamiltonian :
loss operator :
a = -2ie1 /k1
Re(β)
Im(β)
Wigner function W(β)
REVIEW THE USUAL DRIVEN-DAMPED
HARMONIC OSCILLATOR
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AVERAGE EVOLUTION OF DENSITY MATRIX FOR AN OPEN SYSTEM
† † †
,
1
2
di H c
dt
c c c c c c c
loss operator
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
Force ~ X2
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
Force ~ X2
Damping
~ V2
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
Re(β)
Im(β)
Wigner function W(β)
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
H = e2
*a2 + e(a†)2
k 2 a2
Hamiltonian :
loss operator :
a = ± -2ie2 /k 2
Re(β)
Im(β)
Wigner function W(β)
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LECTURE I: INTRODUCTION AND OVERVIEW
PROTECTING QUANTUM SUPERPOSITIONS
IN JOSEPHSON CIRCUITS
LECTURE II: CAT CODES
LECTURE III: PARITY MONITORING AND CORRECTION
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A SPECIAL DRIVEN-DAMPED OSCILLATOR
H = e2
*a2 + e(a†)2
k 2 a2
Hamiltonian :
loss operator :
a = ± -2ie2 /k 2
Re(β)
Im(β)
Wigner function W(β)
Damping
Force 2X
2X
Mirrahimi et al., New J. Phys. 16, 045014 (2014)
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s
2 OSCILLATORS COUPLED
THROUGH A TUNNEL
JOSEPHSON JUNCTION
TE101 modes
E
storagedump
&
readout
junction
rq
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2 OSCILLATORS COUPLED
THROUGH A TUNNEL
JOSEPHSON JUNCTION
† † †/ s s s r r r q q qH a a a a a a -cqq
2aq
†( )2
aq( )2
- cqsaq
†aqas
†as - cqraq
†aqar
†ar + ...
TE101 modes
E
storagedump
&
readout
junction
-crs
4as( )
2ar
†( )2
+ h.c.+ ...
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p 2 s d
drive pump
H
sr= g
2
*as
2ar
† + h.c.( )
effective loss operator:
k2a
s
2effective drive Hamiltonian:
e
2a
s
†( )2
+ h.c.
TWO-PHOTON EXCHANGE BETWEEN THE 2 CAVITIES
storage
ω
dump
ωωs
ωp
ωs
ωd
ωs
ωs
ωp
d
density
of states
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
Re(β) = I
Im(β) = Q
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
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CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
π/2 W(β)
Re(β)
Im(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
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π/2 W(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
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π/2 W(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)
see also Puri and Blais arXiv:1605.09408
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pumping time
raw data
reconstruction
numerical
simulation
0 μs 2 μs 7 μs 19 μs
W(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
quantum
interference !
CAT SQUEEZES OUT OF VACUUM !
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pumping time
raw data
reconstruction
numerical
simulation
0 μs 2 μs 7 μs 19 μs
W(β)
Re(β)
Im(β
)
-2 0 2
-2
0
2
quantum
interference !
CAT SQUEEZES OUT OF VACUUM !
Data of two years ago…
Leghtas et. al. Science (2015)
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M. Reagor et al., APL (2013), C. Axline et al. APL (2016)
Wigner qubitseparate fromstorage ancilla
0.0
0.3
0.6
-0.3
-0.6
W
Im(𝛽
)
0
-0.5
0.5
-0.5 0.50Re(𝛽)
NOW: 𝜅2
𝜅1= 100 with 1/𝜅1 = 92 μs
PRESENT “CAT-PUMPING” COHERENCE
Touzard, Grimm, et al. arXiv:1705.02401
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M. Reagor et al., APL (2013), C. Axline et al. APL (2016)
Wigner qubitseparate fromstorage ancilla
0.0
0.3
0.6
-0.3
-0.6
W
Im(𝛽
)
0
-0.5
0.5
-0.5 0.50Re(𝛽)
NOW: 𝜅2
𝜅1= 100 with 1/𝜅1 = 92 μs
PRESENT “CAT-PUMPING” COHERENCE
Touzard, Grimm, et al. arXiv:1705.02401
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Im(𝛽
)
-0.5 0.50
0 0.0
0.3
0.6
-0.3
-0.6
Re(𝛽)
-0.5
0.5
W
+ 𝛼| − 𝛼⟩
0 L =𝛼 + −𝛼
2
1 L =𝛼 − −𝛼
2
RABI OSCILLATIONS OF CAT WHILE PUMPING
observation of quantum Zeno dynamics of protected information!
Movie slowed down by 10-6
Misra & Sudarshan (1977); Itano et al., (1990); Fisher, Gutierrez-Medina & Raizen (2001); Facchi & Pascazio (2002); Raimond et al., (2012)
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0.0
0.5
1.0
-0.5
-1.0
W
Re(𝛽)
Im(𝛽
)
-2 20
0
1
-1
MEASUREMENT OF THE 4 PAULI OPERATORS OF THE PROTECTED
QUBIT MANIFOLD
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log
Xπ/2log
+X
+Y
+Z
+XL
+YL
+ZL
+XL
+YL
+ZL
/ : N( 𝛼 ± −𝛼 )
/ : N( 𝛼 ± 𝑖 −𝛼 )
/ : ±𝛼
PROCESS TOMOGRAPHY OF QUANTUM ZENO RABI ROTATION
Encode
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Identity
+X
+Y
+Z
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Zeno 𝜋/2 rotation around X
Xπ/2
+X
+Y
+Z
DURATION OF LOGICALGATES: 1.8 ms
GATE DONEWHILECORRECTING!
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TWO TYPES OF CAT ERRORS MUST BE ADDRESSED:
- COHERENT STATE SHRINKING, DEPHASING
& DEFORMING
- PHOTON NUMBER PARITY JUMPING
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transmon
cavity
transmon cavity
SYNDROMEMEASUREMENT
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transmon
cavitySYNDROMEMEASUREMENT
transmon cavity
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evenn
oddn
transmon
cavitySYNDROMEMEASUREMENT
transmon cavity
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transmon
cavity
evenn
oddn
SYNDROMEMEASUREMENT
transmon cavity
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transmon
cavitySYNDROMEMEASUREMENT
transmon cavity
evenn
oddn
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transmon
cavity
transmon cavity
evenn
SYNDROMEMEASUREMENT
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REAL-TIME QUANTUM CONTROLLER FOR FEEDFORWARD CORRECTION
Analog
Outputs
Analog
Inputs
FPGADACs
ADCs
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Bare Transmon 4Cat Code NO QEC 4Cat Code W/ QEC
Qubit after Qubit after
Fidelity = 0.58 Fidelity = 0.72
4-CAT ENCODING SLOWS DOWN QUANTUM INFORMATION DECAY
Qubit after
Fidelity = 0.26
slide courtesy of R. Schoelkopf and A. Petrenko
QEC = Quantum Error Correction from Parity Monitoring using Field Programmable Gate Array
Ofek & Petrenko et al., Nature (2016)
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MULTI-CAVITY ARCHITECTURES
The “Y-mon”!!other experiments currently
underway in the lab with
5 to 10 qubits and 5 to 10 cavities
M. Reagor et al., APL (2013),
C. Axline et al. (in preparation)
Wang and Gao et al., Science (2016)
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JOINT WIGNER FUNCTION OF THE TWO-MODE CAT
y- = aA
aB
- -aA
-aB
WJ (bA,bB ): a function in a 4D space
(a =1.92)
Predicted Wigner for ideal state:
P. Millman et al. Eur. Phys. J. D 32, 233 (2005)
Quantum
coherence
Photon
distribution
States live in Hilbert space of
dimension ~ 150!
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Ideal:
Measured:
Wang and Gao et al., Science (2016)
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SUMMARY AND PERSPECTIVES
• Quantum information protection can be static and/or dynamic
• Cat code encodes information in dynamical states: hardware
efficiency
• Demonstration of operation while protecting
• Attained break-even point in quantum error correction
• Entanglement of two logical qubits
• Combine protection of cat energy loss and parity monitoring
• Demonstrate hierarchical layers of quantum error correction
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MEET THE TEAM!
R. Schoelkopf L. Frunzio M. Mirrahimi L. Jiang S. GirvinL. Glazman
A. Narla
K. Serniak
Y. LiuE. Zalys-Geller
M. Hatridge
S. Shankar
M. HaysK. Sliwa Z. Leghtas
A. Roy
U. VoolA. Kou
C. SmithS. TouzardZ. Minev
S. Mundhada
N. FrattiniV. Albert
A. GrimmV. Sivak