Department of Applied Physics, Yale University
Michael Hatridge
Remote entanglement of transmon
qubits
Katrina Sliwa Anirudh Narla Shyam Shankar Zaki Leghtas Mazyar Mirrahimi Evan Zalys-Geller
Chen Wang Luigi Frunzio Steven Girvin Robert Schoelkopf Michel Devoret
35
What is remote entanglement and why is it important?
How do we engineer interactions over arbitrary length scales?
Q1
Q2
Q3
Alice
Bob
Monroe, Hanson, Zeilinger
QN+1
QN+2
QN+3
Direct interaction Remote entanglement via msmt of ancilla
34
Remote entanglement with flying qubits
ALICE
BOB
bert
arnie
|𝐺
|𝑔
|𝑔
|𝐺
1/ 2 |𝐺𝑔 + |𝐸𝑒 ⊗
1/ 2 |𝐺𝑔 + |𝐸𝑒
How can we do this with a superconducting system?
Instead of qubits use coherent states
How do we entangle flying qubits?
How well can we transmit our flying qubit?
How do we build efficient detector?
quantum-limited amplification
𝑅𝑥 𝜋/2 ,
𝑅𝑥 𝜋/2 ,
1/ 2 |𝐺𝐺 + |𝐸𝐸 or
1/ 2 |𝐺𝐺 − |𝐸𝐸 or
1/ 2 |𝐺𝐸 + |𝐸𝐺 or
1/ 2 |𝐺𝐸 − |𝐸𝐺
measure X
(sign)
measure Z
(parity)
33
Part 1: measurement with coherent
states
32
microwave cavity
coherent pulse
phase meter
dispersive cavity/pulse interaction
Dispersive measurement: classical version
|𝑔
transmission line
31
microwave cavity
coherent pulse
phase meter
dispersive cavity/pulse interaction
Dispersive measurement: classical version
|𝑔 |𝑒
transmission line
30
microwave cavity
coherent pulse
dispersive cavity/pulse interaction
Now a wrinkle: finite phase uncertainty
|𝑔 |𝑒
transmission line
phase meter
29
microwave cavity
coherent pulse
dispersive cavity/pulse interaction
|𝑔 |𝑒
phase meter
Measurement with bad meter (still classical)
noise added by amp.
AND signal lost in transmission
• Each msmt tells us only a little • State after msmt not pure! • This example optimistic, best
commercial amp adds 20-30x noise • We fix this with quantum-limited
amplification
28
Ideal phase-preserving amplifier • What is power dissipated in Rload due to noise in Rsource, for Rload = Rsource = R?
BTkRR
TRBkR
R
VP B
Brmsload
22
2
4
2
NBload TTBGkP
𝜌𝑖 → 𝜌𝑓 = 𝑇𝑟𝑈𝜌𝑖𝑈†
𝑆 𝜌𝑓 = 𝑆 𝜌𝑖
MATH
Phase-sensitive amps
180° hybrid (beam splitter)
𝜙 = 0
𝜙 = 𝜋2
|0
|𝛼 𝐺 ≫ 1
𝐺 ≫ 1
Signal in
Idler in
180° hybrid (beam splitter)
These ports are often internal degrees of freedom, in our amp they are accessible. We’ll use this for remote entanglement
Signal out
Idler out
• Adds its inputs, outputs 2 copies of combined inputs • Adds minimum fluctuations to signal output*
* 𝜎𝑜𝑢𝑡2 = 2𝜎𝑖𝑛
2 (Caves’ Thm) Caves, Phys Rev D (1982)
27
microwave cavity
coherent pulse
Quantum-limited amplification: projective msmt
|𝑔 |𝑒
phase meter w/ P. P. pre-amp
• state of qubit pure after each msmt • For unknown initial state
𝑐𝑔|𝑔 + 𝑐𝑒|𝑒 , repeat
many times to estimate 𝑐𝑔2
, 𝑐𝑒2
only quantum fluctuations
coherent superposition
26
microwave cavity
WEAK coherent
pulse
Quantum-limited amplification: ‘partial’ msmt
|𝑔 |𝑒
phase meter w/ P. P. pre-amp
• state of qubit pure after each msmt • counter-intuitive, but is achievable
in the laboratory
only quantum fluctuations
coherent superposition
25
Part 2: Partial measurement with
transmon qubit and JPC
24
200 nm
The Josephson tunnel junction
SUPERCONDUCTING TUNNEL JUNCTION
𝐼 = 𝐼0 sin𝜙
𝜑0
1nm
Al/AlOx/Al tunnel junction
nonlinear inductor shunted by capacitor
02e
LJ
CJ
𝐼
𝜙
23
Superconducting transmon qubit
Potential energy
f
Josephson junction with shunting capacitor anharmonic oscillator
lowest two levels form qubit
fge ~ 5.025 GHz, fef ~ 4.805 GHz
Koch et al., Phys. Rev. A (2007)
|𝑔
|𝑒
|𝑓
22
mixer
Measurement configuration
TR
number of inddent signal modes:
cavity ring-down time 1 2
r
Q
/S RM T
𝑎 𝑔 ⊗ 𝛼𝑔, 0 + 𝑏 𝑒 ⊗ 𝛼𝑒 , 0
isolator circulator
Sig Idl Pump
compact resonator
+ qubit pulses
JPC
HEMT
𝐼𝑚 = 𝐼 𝑡 𝑑𝑡𝑇𝑚
0
readout pulse
Ref
transmon
𝑄𝑚 = 𝑄 𝑡 𝑑𝑡𝑇𝑚
0
on qubit state?
𝐼𝑚 = 𝐼 𝑡 𝑑𝑡𝑇𝑚
0
Readout
phase
tan−1 𝑄𝑚𝐼𝑚
𝜋2
width 𝜅
Readout
amplitude
𝐼𝑚2 + 𝑄𝑚
2 𝑓
dispersive shift 𝜒
Qubit +
resonator + qubit
pulses
JPC
|𝑔 |𝑒
|𝑔 |𝑒 𝜗 = 2 tan−1 𝜒
𝜅
readout
pulse at
𝑓𝑑
𝑓
Ref
𝑄𝑚 = 𝑄 𝑡 𝑑𝑡𝑇𝑚
0
Sig Idl
Pump
𝑓𝑑
HEMT
− 𝜋2
vacuum 50Ω
21
Isolating the transmon from the environment
transmon output coupler
Purcell filter
waveguide-SMA adapter
input coupler
Cavity fc,g = 7.4817 GHz 1/ = 30 ns
10 mm
25 mm
Qubit fQ=5.0252 GHz T1 = 30 s T2R = 8 s
20
The 8-junction Josephson Parametric Converter
G=20 dB
NR= 9 dB
BW= 9 MHz
D.R. ~ 10 photons @ 4.5 MHz
Tunability ~100’s of MHz
~88% of output noise is
quantum noise!
→ quantum fluctuations
on an oscilloscope
Idler
Signal
ADD SOME REF?
Bergeal et al Nature (2010)
See also Roch et al PRL (2012)
20
15
10
5
G (
dB
)
7.507.487.467.447.42
x109
F 7.42 7.46 7.50
Frequency (GHz)
Direct G
(dB
)
20
10
not a defect! quantum jumps of connected qubit
10 m
19
𝐼𝑚/𝜎
𝑄𝑚
/𝜎
|𝑔 10
5
0
10 5 0 -5 -10
102
104
1
|𝑒
𝑓 , … 𝑄𝑚
/𝜎
𝐼𝑚/𝜎
6000
0
10
5
0
10 5 0 -5 -10
|𝑔 |𝑒
MA
𝑓 , …
Rotate to z=0 State preparation Confirm state
8.6 σ
Rotate to z=0 State preparation Confirm state
𝑅𝑥 𝜋/2 𝑅𝑥 𝜋 𝐼𝑑 or
𝑛 11 𝑛 11
640 ns
𝑇𝑚 320 ns
Trep = 20 µs
See also Riste et al PRL (2012) Johnson et al PRL (2012)
Preparation by measurement + post-selection 18
Rotate to z=0 State preparation Confirm state
𝑅𝑥 𝜋/2 𝑅𝑥 𝜋 𝐼𝑑 or
𝑛 11 𝑛 11
640 ns
𝑇𝑚 320 ns
|𝑒
102
104
1
MA MB|MA = |𝑔 Trep = 20 µs
𝐼𝑚/𝜎
𝑄𝑚
/𝜎
10
5
0
10 5 0 -5 -10
|𝑔 |𝑒
𝑓 , …
𝑅𝑥 𝜋 “|𝑒 ”
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
10
5
0
10 5 0 -5 -10
|𝑔 |𝑒
𝑓 , …
𝐼𝑑 “|𝑔 ”
Fidelity=0.994! Error budget:
Msmt ~0.003
T1 ~0.001-0.002
rotation <0.002
Now that we have outcomes MA= |𝑔 either do nothing to retain |𝑔 OR rotate qubit by 𝑅𝑥 𝜋 to create |𝑒
Preparation by measurement + post-selection 17
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
10
5
0
10 5 0 -5 -10
|𝑒
𝐼𝑚/𝜎 𝑄
𝑚/𝜎
10
5
0
10 5 0 -5 -10
102
104
1
|𝑔 |𝑒
𝑓 , …
|𝑔 |𝑒
𝑓 , …
𝑅𝑥 𝜋 “|𝑒 ” 𝐼𝑑 “|𝑔 ”
How ideal is this operation?
Fidelity=0.994!
Strong measurements allow rapid, high-fidelity state preparation and tomography
Say : “practically useful, but doesn’t tell us how ideal we are, and how much room for improvement in signal processing”
16
A picture is worth a thousand math symbols * : Mapping (𝐼𝑚, 𝑄𝑚) to the bloch vector
For weak msmt:
𝜕𝑥𝑓
𝜕𝑄𝑚 𝐼𝑚,𝑄𝑚=0
= η𝐼𝑚𝑔
− 𝐼𝑚𝑒
2
• Now calculate the effect of some added classical noise on the density matrix • Efficiency 𝜂 = 𝜎𝑚
𝜎2
• 𝑥𝑓, 𝑦𝑓 provide important information about efficiency
Rotation (𝑸𝒎) Convergence to poles (𝑰𝒎)
Information loss due to 𝜼 < 𝟏
COWBOY HAT
w/ curvy arrow
𝑄𝑚
𝐼𝑚
𝐼𝑚 gives latitude information
𝑄𝑚 gives longitude information
𝑥
𝑦
𝑧
*Gambetta, et al PRA (2008); Korotkov/Girvin, Les Houches (2011); M. Hatridge et al Science (2013)
The equator is a dangerous place: lost information pulls trajectory
towards the z-axis
15
mixer
Back-action characterization protocol
TR
number of inddent signal modes:
cavity ring-down time 1 2
r
Q
/S RM T
𝑛 11 𝑛 11
Tomography
𝑅𝑥 𝜋/2
𝑅𝑥 𝜋/2 ,
𝑅𝑦 𝜋/2 ,
700ns
qubit
cavity
𝑥𝑓, 𝑦𝑓, 𝑧𝑓
variable 𝑛
𝐼𝑑 or
𝑇𝑚 320 ns
Variable strength
measurement
State preparation
X = 1 𝑥
𝑦
𝑧
or
or Y = 1
Z = 1
(𝐼𝑚, 𝑄𝑚)
14
histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement
Measurement with 𝑰 𝒎 𝝈 = 0.4
Pro
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of
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0
-1
Counts
Max 0
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
6
0
6 0 -6
6
0
6 0 -6
6
0
6 0 -6
𝑋 𝑐 𝑌 𝑐
𝑍 𝑐
6
0
6 0 -6
𝐼 𝑚
𝜎
13
histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement
Measurement with 𝑰 𝒎 𝝈 = 1.0
Pro
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of
gro
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0
-1
Counts
Max 0
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
6
0
6 0 -6
6
0
6 0 -6
6
0
6 0 -6
𝑋 𝑐 𝑌 𝑐
𝑍 𝑐
6
0
6 0 -6
𝐼 𝑚
𝜎
12
histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement
Measurement with 𝑰 𝒎 𝝈 = 2.8
Pro
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of
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-1
Counts
Max 0
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
𝑄𝑚
/𝜎
𝐼𝑚/𝜎
6
0
6 0 -6
6
0
6 0 -6
6
0
6 0 -6
𝑋 𝑐 𝑌 𝑐
𝑍 𝑐
6
0
6 0 -6
𝑓 , … show at ~10-4 contamination
𝐼 𝑚 𝜎
11
𝑛
Measurement strength 𝐼𝑚/𝜎
𝑄𝑚
/𝜎
𝑄𝑚
/𝜎
𝑛 Experiment Theory
x- and y-component along 𝑰𝒎 = 𝟎
0 2 𝐼𝑚/𝜎
0
4
0
4
𝑌 𝑐 along 𝐼𝑚 = 0
𝑋 𝑐 along 𝐼𝑚 = 0
𝑋𝑐,
𝑌𝑐
0 6
0
-1
1
Amplitude determined by one fit parameter: 𝜼 = 0.57 ± 0.02
𝑰 𝒎 𝝈 = 0.82
𝑋 𝑐 = sin𝑄𝑚
𝜎
𝐼 𝑚𝜎
+ 𝜃 exp −𝐼 𝑚𝜎
21 − 𝜂
𝜂
𝑌 𝑐 = cos𝑄𝑚
𝜎
𝐼 𝑚𝜎
+ 𝜃 exp −𝐼 𝑚𝜎
21 − 𝜂
𝜂
-6
𝜼 ≥ 𝟎. 𝟓 → 3 body entanglement (qubit, signal, idler)
𝑄𝑚/𝜎
10
Part 3: remote entanglement
experiment
9
Two qubit readout schematic
𝑇1 = 30 𝜇s 𝑇2𝑅 = 15𝜇s
𝑓𝑄𝑔𝑒
= 4.672 GHz 𝜒
2𝜋 = 2.7 MHz 𝜅 2𝜋 = 6.7 MHz
𝑇1 = 15𝜇s 𝑇2𝑅 = 15 𝜇s
𝑓𝑄𝑔𝑒
= 6.074GHz 𝜒
2𝜋 = 3.5 MHz 𝜅 2𝜋 = 4.1 MHz
𝑓𝑝 = 16.58 GHz
𝑓𝑟𝑑 = 7.464 GHz 𝑓𝑟
𝑑 = 9.116 GHz
50 Ω
8
Simultaneous readout of two qubits
Signal Alone
𝑄𝑚
𝐼𝑚
|𝑒𝑠
|𝑔𝑠
Idler Alone
𝑄𝑚
𝐼𝑚
|𝑒𝑖
|𝑔𝑖
Together
𝑄𝑚
𝐼𝑚
|𝑒𝑒
|𝑔𝑔
|𝑔𝑒
|𝑒𝑔
“Joint Readout”
|𝑒𝑠 |𝑔𝑠
|𝑔𝑖
|𝑒𝑖
𝐼𝑚 encodes Z info
𝑄𝑚 encodes Z info
CHANGE ALL TO FUZZY BALLS
7
|𝑒𝑒
|𝑔𝑔 |𝑔𝑒
|𝑒𝑔
How to perform “entangling readout”
Signal Alone
𝑄𝑚
𝐼𝑚
|𝑒𝑠
|𝑔𝑠
Idler Alone
𝑄𝑚
𝐼𝑚
Together
𝐼𝑚 encodes Z info
𝐼𝑚 encodes Z info
𝑄𝑚 (sign)
𝐼𝑚 (parity)
|𝑒𝑒 |𝑔𝑔
|𝑔𝑒 , |𝑒𝑔
“Entangling Readout”
|𝑒𝑖
|𝑔𝑖
|𝑔𝑖
|𝑒𝑖
• 𝐼𝑚 is now blind to contents of |𝑔𝑒 , |𝑒𝑔
• With/ appropriate initial state, outcome is Bell state w/ 50 % success rate
• 𝑄𝑚 encodes phase of Bell state
6
|𝑒𝑒 |𝑔𝑔
|𝑔𝑒 , |𝑒𝑔
Back action of two qubit msmt creates entanglement
For weak msmt:
𝜕𝑥𝑓
𝜕𝑄𝑚 𝐼𝑚,𝑄𝑚=0
= η𝐼𝑚𝑔
− 𝐼𝑚𝑒
2
• Now calculate the effect of some added classical noise on the density matrix • Efficiency 𝜂 = 𝜎𝑚
𝜎2
• 𝑥𝑓, 𝑦𝑓 provide important information about efficiency
Rotation (𝑸𝒎) Convergence to poles (𝑰𝒎)
Information loss due to 𝜼 < 𝟏
COWBOY HAT
w/ curvy arrow
𝑄𝑚
𝐼𝑚
𝐼𝑚 gives info on even vs. odd
parity (a bit too much, actually)
𝑄𝑚 gives sign info for odd
parity states
Even parity states:
= |𝑔𝑔
= |𝑒𝑒
= |𝑔𝑒 − |𝑒𝑔
= |𝑔𝑒 + 𝑖|𝑒𝑔
= |𝑔𝑒 + |𝑒𝑔
= |𝑔𝑒 − 𝑖|𝑒𝑔
Odd parity states:
Make note about which we keep…which are good ones
The climax… this is my favorite part..
5
Pro
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0
+1
Tomography of strong entangling msmt
Histogram
𝑄𝑚
𝜎
𝑋𝑋 𝑐 𝑍𝐼 𝑐
𝑍𝑍 𝑐
𝐼𝑚 𝜎
10
0
-10
-10 0 10
𝑄𝑚
𝜎
𝐼𝑚 𝜎
10
0
10 0
-10
-10
𝑄𝑚
𝜎
10 0 -10
10
0
-10
10 0 -10
10
0
-10
4
Pro
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of
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+1
Tomography of weak entangling msmt
Histogram
𝑄𝑚
𝜎
𝑋𝑋 𝑐 𝑍𝐼 𝑐
𝑍𝑍 𝑐
𝐼𝑚 𝜎
5
0
-5
-5 0 5
𝑄𝑚
𝜎
𝐼𝑚 𝜎
5
0
5 0
-5
-5
𝑄𝑚
𝜎
5 0 -5
5
0
-5
5 0 -5
5
0
-5
3
XX YY ZZ
Average a strip along 𝑄𝑚 ≃ 0
Blo
ch c
om
p. v
alu
e 5
0
𝑄𝑚
𝜎
𝐼𝑚 𝜎 𝑄𝑚 𝜎
-5 0 5
0
0.5
-0.5
𝑋𝑋 𝑐
5 0
-5
-5
𝑄𝑚
𝜎
Signature of entangling operation
• currently, too much information is lost • correct dependence of qubit correlations • expect to demonstrate entanglement soon (F > 0.5)
2
Evolution of single-qubit readout vs time
Year
1
0
0.5 To
tal e
ffic
ien
cy 𝜂
20
10
20
12
20
14
Conclusions
• Coherent states can be used as flying qubits • Quantum mechanics goes through the amplifier • New tools for building large-scale quantum
entanglement
Compare with optical systems: 𝜂~10−3
due to collector/detector inefficiencies
1