Rob Schoelkopf
Applied Physics, Yale University
PI’s: RS Michel Devoret Luigi Frunzio Steven Girvin Leonid Glazman Liang Jiang Mazyar Mirrahimi
“Putting the Bell on Schrodinger’s Cat”
Postdocs & students wanted!
Overview
• A few remarks on quantum circuits, then and now…
Observing quantum jumps of photon parity: Sun, Petrenko et al., Nature 511, 444 (2014).
• Tracking photon parity of a cat state in real time
CHSH for encoded qubit: Petrenko, Vlastakis, et al., submitted & arXiv (2015).
• “Putting the Bell on a cat”: efficient tomography and single-shot violation for a logical qubit
Before we met…
Some early influences:
• Koch, PhD thesis ~ 1985
• MQT: Martinis, Devoret, Clarke,…
• Control/influence of EM environment!
Forerunner of circuit QED!
In AMO language: strong coupling, bad cavity limit
Devoret, Esteve, Martinis, and Urbina, 1989.
Before we met…
This is sufficient to perform interesting manipulations on the quantum state…
… decoherence…is mainly due to the… electromagnetic environment of the circuit. …we estimate that the life-time of a Q-bit …can be longer than 100 µs.
When I began… First Yale/Chalmers spectroscopy of
Cooper-pair box, ca. 2000 -2002, aka Bouchiat + RF-SET
Linewidth? ~ 1 GHz
T1 = 103 µs
T2E = 145 µs
time (µs)
time (µs)
Rea
dout
Vol
tage
(mV
) R
eado
ut V
olta
ge (m
V)
We’ve come a long way together…
2 1
1 1 12T T Tφ
= +
echo2 1~ 1.5T T
A 3D transmon qubit
Coherence of Photon States in Cavity State preparation by SNAP:
Heeres et al., arXiv:1503.01496,
Thy: Krastanov et al., arXiv: 1502.08015
T1c = 1.2 ms T2c = 0.8 ms
Tφc = 1 ms
Dephasing and relaxation actually limited by qubit… (reverse Purcell and qubit thermal population)
Reagor, Pfaff, et al., in preparation ∆f / f = 100 Hz / 4 GHz ~ 25 ppb !!
“Putting the Bell” on Schrodinger’s Cat
Petrenko, Vlastakis, et al., submitted (2015). Gustave Dore, ca. 1868
“But who will volunteer to place it?”
“The mice in council”
Violation of a CHSH inequality for a macroscopic quantum state
courtesy University of Illinois, Urbana-Champaign
Schrodinger’s Cat = an entangled state between a microscopic object (atom or qubit) and a macroscopic object (easily distinguished by “environment”)
g eor
( )alive d1 ead2
e gΨ = +
“meow?”
“ack!”
??
Non-linear electromagnetic
oscillator
Ene
rgy
g
egeω
efωge efω ω≠
Atom: The Transmon Qubit
2† † †
J0
ˆ 2 ˆ2
cos geH E a aQ a a aaC
π ω λ
Φ = − +…= Φ −
~ 5 10
~ 0.25ge
B
ω
ω
− GHz/k K
Superconductor
Superconductor (Al)
Insulating barrier 1 nm
Josephson junction (dissipation-free?)
Other practitioners (many!): UCSB/Google, Berkeley, Princeton, Delft, Zurich, Chicago…
geH e eω≈ Koch et al., PRA, 2007; Houck et al., PRL, 2008
The Cat: A Cavity Oscillator
E
x
x
ω
2ψzpf / 2x mω=
Glauber (coherent) state 2| |
2
0| |
!
n
ne n
n
β ββ∞−
=
⟩ = ⟩∑max zpfx xβ=
if we can only apply classical controls (e.g. laser, force), can only make displacements
a β β β=
0t =
2Enβ ωω
=
=
What’s a Cat State of an Oscillator?
E
x
x
ω
ψCat state of an oscillator (field)
0t =
/ 2x mω∆ = ( )12
ψ β β= + −
2d xβ= ∆
Size of the superposition, d: 22 4 4d nβ= =
2Enβ ωω
=
=
Bigger cats die faster: rate = 2d κ
What’s a Cat State of an Oscillator?
E
x
x
ω
Cat state of an oscillator (field)
0t =
/ 2x mω∆ =
( )12
ψ β β= + −
This one is EVEN parity!
ψ
2Enβ ωω
=
=
2d xβ= ∆
What’s a Cat State of an Oscillator?
E
x
x
ω
Cat state of an oscillator (field)
0t =
/ 2x mω∆ =
( )12
ψ β β= − −
This one is ODD parity!
ψ
2Enβ ωω
=
=
2d xβ= ∆
What’s a Cat State of an Oscillator?
E
x
x
ω
Cat state of an oscillator (field)
0t =
/ 2x mω∆ =
( )12
ψ β β= − −
This one is ODD parity!
2ψ
2Enβ ωω
=
=
2d xβ= ∆
What’s a Cat State of an Oscillator?
E
x
x
ω
2ψSchrödinger cat state
/ 2x mω∆ =
( )12
ψ β β= + −
2t πω=
What happens now, when packets collide?
~ /x β∆fringes
The sign of fringe = “parity”
2Enβ ωω
=
=
Seeing the Interference: Wigner Function }{( ) ( ) ( )2
m m mW D DTr Pα α ρ απ
= −
( )†
1 ni a aP e π= = −Parity
or xΦ
Thy:
Negative fringes = “whiskers”
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST) Haroche/Raimond , 2008 Rydberg (ENS) Hofheinz et al., 2009 in circuits (UCSB)
orQ p
Using a Cavity as a Logical Qubit?
Qubits 1..7
High-Q (memory)
Ancilla qubit Readout
Our approach : • Cavity is the memory • One error syndrome
Register as memory: 1. More qubits 2. More decay channels
1. 1 qubit! 2. Single readout channel!
Ancilla qubits Readout
earlier ideas: Gottesman, Kitaev & Preskill, PRA 64 , 012310 (2001)
“Hardware-efficient QEC” Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).
Encoding a Quantum Bit in a Cavity State
2-Level System (e.g. transmon)
g e+ g j e+e
gContinuous Variables (e.g. cavity)
cavity 7.22sf GHz=55s sτ µ=
Dispersive cQED Coupling: 2 Cavities + 1 Qubit + Paramp
( )† †/ q qs sH a a e e a aω χ ω= − +
JBA readout 8.17rf GHz=30r nsτ =
5.94qf GHz= 1 10T sµ=2 10T sµ=
qubit
98%RF =
95%PF =
Readout Fidelity
Parity Readout Fidelity
Dispersive Hamiltonian:
Strong* Dispersive Regime
“doubly-QND” interaction
Allows qubit to control many photons at once (and vice-versa)
( )2~ / s qgχ ω ω−,χ γ κ
† †q sa ae e e eaH aω χω= + −
* Schuster et al., 2007; prev. attained only in Rydberg cQED (ENS-Paris)
n=0
n=1
n=0
n=1
n=2
n=2
χ~ 0n
~ 0.5n
~ 1n
0n =1n =2n =
qubi
t abs
orpt
ion
cavity
qubit
Deterministic Cat Creation: QCMAP Gate
Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:
0gψ = ⊗
cavity
qubit
Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:
Deterministic Cat Creation: QCMAP Gate
( ) 0N g eψ = + ⊗
cavity
qubit
Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:
Deterministic Cat Creation: QCMAP Gate
( )N g eψ β= + ⊗
cavity
qubit
Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:
Deterministic Cat Creation: QCMAP Gate †
int e eH a aχ=
( ), , i tN g e e χψ β β −= +
cavity
qubit
Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:
Deterministic Cat Creation: QCMAP Gate
t π χ=after time:
“Here be kittens!”
( ), ,N g eψ β β= + −
†int e eH a aχ=
Efficiently Measuring 𝑋𝑋𝑐𝑐 ,𝑌𝑌𝑐𝑐, 𝑍𝑍𝑐𝑐
−2
−2
0 2
0
2
𝑅𝑅𝑅𝑅(𝛼𝛼)
𝐼𝐼𝐼𝐼(𝛼𝛼
)
P
+1
−1
Cavity state along 𝑋𝑋𝑐𝑐
Measured Wigner function of cavity
Efficiently Measuring 𝑋𝑋𝑐𝑐 ,𝑌𝑌𝑐𝑐, 𝑍𝑍𝑐𝑐
−2
−2
0 2
0
2
𝑅𝑅𝑅𝑅(𝛼𝛼)
𝐼𝐼𝐼𝐼(𝛼𝛼
)
cZ cZcX
cY
P
+1
−1
Cavity state along 𝑋𝑋𝑐𝑐
Efficiently Measuring 𝑋𝑋𝑐𝑐 ,𝑌𝑌𝑐𝑐, 𝑍𝑍𝑐𝑐
−2
−2
0 2
0
2
𝑅𝑅𝑅𝑅(𝛼𝛼)
𝐼𝐼𝐼𝐼(𝛼𝛼
)
cZ cZcX
cY
P
+1
−1 0.07cZ P Pβ β−= − =
0 0.76cX P= =
80.14jcY Pπ
β= =
Cavity state along 𝑋𝑋𝑐𝑐
Efficiently Measuring 𝑋𝑋𝑐𝑐 ,𝑌𝑌𝑐𝑐, 𝑍𝑍𝑐𝑐
−2
−2
0 2
0
2
𝑅𝑅𝑅𝑅(𝛼𝛼)
𝐼𝐼𝐼𝐼(𝛼𝛼
)
cZ cZcX
cY
P
+1
−1
Cavity state along 𝑌𝑌𝑐𝑐
A Bell State
( )12B g e e gψ = +
g e+ g j e+
g
2-Level System (e.g. transmon)
g e+ g j e+
g
2-Level System (e.g. transmon)
The Schrodinger Cat or “Bell-Cat”
( )12B g eψ β β= + −
g e+ g j e+
g
Continuous Variables (e.g. cavity)
2-Level System (e.g. transmon)
0P
8jPπ
β
P Pβ β−−
CX
CZCY
X
ZYBψ
Performing a CHSH Measurement
' ' ' ' ' 'ABA B c c c cO AB AB A B A B= + − +
yRθ
/2yRθ π+
0P
P Pβ β−−
CX
CZ
Vlastakis et al. Science 2013
Violating the CHSH Inequality
g
e
( )12
g e+
XZZX c c c cO XZ XX ZX ZZ= + − +
g
( )12
g e+ 135°
g
( )12
g e+
45°
MAX VIOLATION MAX VIOLATION
Bell
Sign
al 𝑂𝑂
𝛽𝛽 = 1
Violating the CHSH Inequality
2.30 ± 0.04
−2.28 ± 0.04
XZZXO
ZXXZO
WE TAKE ALL MEASUREMENTS: No correction for detector inefficiency
Bell
Sign
al 𝑂𝑂
𝛽𝛽 = 1
Continuously Varying Bell-Cat Size
Increasing 𝛽𝛽: Realizing a Schrodinger’s Cat
Experiment
2.14 ± 0.04
Still violating at 𝑑𝑑2 = 16 photons Finite encoding and
measurement fidelity
dead alivege +
𝑑𝑑 = 2𝛽𝛽 Be
ll Si
gnal
𝑂𝑂
see also Brune et al., 1996
Cavity as a Correctable Memory?
Store a qubit as a superposition of two cats of same parity
0 ( )L α β β+= = + −C N
1 ( )L i i iα β β+= = + −C N
10e eg Lg Lc cc c⇒↑ ++↓
Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).
Z. Leghtas M. Mirrahimi
Cat States for Hardware-Efficient QEC?
even odd even odd odd even
High-Q (memory)
Ancilla qubit Readout
Leghtas, Mirrahimi, et al., PRL 111, 120501(2013). “Cat codes”: much less hardware required
1st tracking of a parity or error syndrome in real-time:
Sun, Petrenko et al., Nature 511, 444 (2014). parity msmt. ala’ Bertet et al.,
Summary
Qubits: T2 ~ 2*T1 ~ 0.0001 sec Cavities: T2 ~ 2*T1 ~ 0.001 sec
• Coherence in circuit QED passing the QEC threshold.
• Tracking the jumps of an error syndrome: photon parity
• Cat-codes: a new shortcut for QEC?
Leghtas, Mirrahimi et al., PRL 2013. Sun, Petrenko et al., Nature 511, 444 (2014). Petrenko, Vlastakis et al., submitted & arXiv (2015)
Next challenge: “breakeven” for error correction!
• Bell violation between qubit and continuous variable system: benchmarking a module
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